hbmt4103

HBMT4203 Teaching Mathematics In Form Three

AZYYATI A.RAHIM 761022017628 1

OPEN UNIVERSITY MALAYSIAFaculty of Education and Language

HBMT4103 Teaching Mathematics in Form Three

Semester May 2011

NamaAzyyati bte A.Rahim

Matriculation Number761022017628002

IC Number761022017628

Phone Number017-7337690

[email protected]

Tutor’s Name Dr. Suhaidah bte Tahir

Learning Centre Institut Perguruan Tun Hussein Onn

mailto:[email protected]


ACKNOWLEDGEMENT

Alhamdulillah , I am so happy with this development. The idea of coming up with these

assignment is to provide more ongoing good teacher to teach our students. The main focus of this

assignment is to develop and enhance teachers’ skills. It also aims to encourage myself to

prepared some technique to solve the problem and interesting activity in my lessons with

cooperative learning.

The main focus of the assignment is to develop teachers’ basic communicative skills and

build confidence in preparing the lesson plan. It is aim to assist teachers to manage classrooms

interaction as well as the teaching and learning processes. It will also enhance teachers’ abilities

in using the accurate terminology and classroom language so the students are able to follow the

lesson taught effectively.

I would like to express my sincere thanks and appreciation especially to my tutor’s Dr.

Suhaidah binti Tahir for she commitment as a guider to finish these assignment. Thanks for all

who is participate to help me to complete this assignment. I would also like to acknowledge their

dedication and perseverance guiding me to make a good task.

Likewise I hope that this initiative will grow in scope and make a greater impact from the

small yet significant beginnings we make today.

Thank you.



CONTENT

PAGES

Acknowledgment

Content

1.0 Introduction 4

2.0 History of Trigonometry 5 - 41

3.0 Trigonometry in use 42 - 46

4.0 Activity to determine the tangent, sine and cosine

of an acute angle 47 - 53

5.0 Three Activities To Demonstrate On How To Find And Angle Using The Scientific Calculator 54 – 59

6.0 Reflection 60 - 65

References 66 - 67



1.0 INTRODUCTIONS

Trigonometry developed from the study of right-angled triangles by applying their

relations of sides and angles to the study of similar triangles. The word trigonometry comes from

the Greek words "trigonon" which means triangle, and "metria" which means measure. The term

trigonometry was first invented by the German mathematician Bartholomaeus Pitiscus, in his

work, Trigonometria sive de dimensione triangulea, and first published 1595. The primary use of

trigonometry is for operation, cartography, astronomy and navigation, but modern

mathematicians has extended the uses of trigonometric functions far beyond a simple study of

triangles to make trigonometry indispensable in many other areas. Especially astronomy was

very tightly connected with trigonometry, and the first presentation of trigonometry as a science

independent of astronomy is credited to the Persian Nasir ad-Din in the 13 century.

Trigonometric functions have a varied history. The old Egyptians looked upon

trigonometric functions as features of similar triangles, which were useful in land surveying and

when building pyramids. The old Babylonian astronomers related trigonometric functions to arcs

of circles and to the lengths of the chords subtending the arcs.

Centuries later, trigonometric functions acquired a geometric interpretation when they

came to be looked upon as the lengths of specific line segments related to the central angle.



2.0 HISTORY OF TRIGONOMETRY

The history of trigonometry dates back to the early ages of Egypt and Babylon. Angles

were then measured in degrees. History of trigonometry was then advanced by the Greek

astronomer Hipparchus who compiled a trigonometry table that measured the length of the chord

subtending the various angles in a circle of a fixed radius r. This was done in increasing degrees

of 71. In the 5th century, Ptolemy took this further by creating the table of chords with increasing

1 degree. This was known as Menelaus's theorem which formed the foundation of trigonometry

studies for the next 3 centuries. Around the same period, Indian mathematicians created the

trigonometry system based on the sine function instead of the chords. Note that this was not seen

to be ratio but rather the opposite of the angle in a right angle of fixed hypotenuse.

The history of trigonometry also included Muslim astronomers who compiled both the

studies of the Greeks and Indians. In the 13th century, the Germans fathered modern

trigonometry by defining trigonometry functions as ratios rather than lengths of lines. After the

discovery of logarithms by the Swedish astronomer, the history of trigonometry took another

bold step with Issac Newton. He founded differential and integral calculus. Euler used complex

numbers to explain trigonometry functions and this is seen in the formation of the Euler's

formula.

The history of trigonometry came about mainly due to the purposes of time keeping and

astronomy. It is quite difficult to describe with certainty the beginning of trigonometry... In

general, one may say that the emphasis was placed first on astronomy, then shifted to spherical

trigonometry, and finally moved on to plane trigonometry. An early Hindu work on astronomy,

the Surya Siddhanta, gives a table of half-chords based on Ptolemy’s table. But the first work to

refer explicitly to the sine as a function of an angle is theAryabhatiya of Aryabhata (ca. 500 AD).

Now begins an interesting etymological evolution that would finally lead to our modern word

"sine". When the Arabs translated the Aryabhatiya into their own language, they retained the

wordjiva without translating it’s meaning. When the Arabic version was translated into

Latin, jiva, or jaib was translated into sinus, which means curve.



We find the word sinus in the writings of Gherardo of Cremona, who translated many of

the old Greek works. Other writers followed and soon the word sinus or sine became common in

mathematical texts throughout Europe.The remaining two main functions have a more recent

history. The cosine function, which we today regard as equal in importance to the sine, first arose

from the need to compute the sine of the complementary angle. The name cosinus originated

with Edmund Gunter in 1620: he wrote co.sinus, which was modified to cosinus by John Newton

(not related to Isaac Newton). The word "tangent" comes from the Latin tangere, to touch. It was

introduced by the Dane Thomas Fincke in 1583.

2. 1 Early Astronomy and the Beginnings of a Mathematical Science

1. Ancient Instruments and Measuring the Stars

The most ancient device found in all early civilisations, is a "shadow stick". The shadow

cast from a shadow stick was used to observe the motion of the Sun and thus to tell time. Today

we call this instrument a Gnomon. The name gnomon comes from the Greek and refers to any L-

shaped instrument, originally used to draw a right angle.

In Euclid Book II, where Euclid deals with the transformation of areas, the gnomon takes

the form of an "L-shaped" area touching two adjacent sides of a parallelogram. Today, a gnomon

is the vertical rod or similar device that makes the shadow on a sundial.

For more about sundials go to Leo's article - Brief History of Time Measurement.


http://nrich.maths.org/public/viewer.php?obj_id=6070&part=


At midday the shadow of a stick is shortest, and the civilisations of Mesopotamia, Egypt,

and China took the North - South direction from this alignment. In contrast, the Hindus used the

East - West direction, the rising and setting of the sun, to orient their "fire-altars" for religious

practices. To do this they constructed the "gnomon circle" whose radius was the square root of

the sum of the square of the height of the gnomon and its shadow [See Note 2 below].

The Merkhet is one of the oldest known astronomical instruments. It was developed

around 600 BCE and uses a plumb line to obtain a true vertical, as in the picture. The other

object is the rib of a palm leaf, split at one end to make a thin slit for a sight. Babylonian and

Egyptian astronomers were able to measure the altitude and lateral displacement of heavenly

objects from a particular direction by using a Merkhet, thus giving the earliest ideas of turning,

or angle.



A pair of merkhets were used to establish a North-South direction by lining them up, one

behind the other, with the Pole Star. Viewing the plumb lines through the sight made sure the

two merkhets and the sight were in the same straight line with the Pole Star. Using a water clock

to determine timings, this arrangement of merkhets allowed people to take measurements of

night-time events, for example times when certain stars crossed the vertical plumb line (a "transit

line").

The Egyptians divided the 360 degrees of the ecliptic into 36 sections of 10 degrees

each.. This division was known before 2300 BCE. Each ten degree section (called a decanfrom

the Greek for ten) contained a constellation of stars lined up along the ecliptic. Since the Earth

makes a full rotation in 24 hours, the stars in a new decan will rise above the horizon about every

40 minutes. The system of decans was used for determining the night hours and the seasons.



The divisions in the top part of the chart represent decans. The chart was read from right

to left and the pictures represent Mars (the boat and the bull), Orion with the three stars including

the Sun and Moon, Sirius, Jupiter, Saturn, Mercury and Venus. The lower section contains

pictures of star gods or demons. They represent some of the most important days of the year. The

chart is largely symbolic and functional but does contain pictures of some significant groups of

stars.

2. Babylonian Astronomers

Observations of celestial bodies by the Babylonians from about 1,800 BCE gave rise to

the eventual division of the circle into 360 degrees, and by about 500 BCE, the division of the

heavens into twelve regions of 30 degrees each, often referred to as the 12 houses of the zodiac.

The Babylonians recorded the events of the lunar month, the daily movement of the sun across

the sky over the year, and the rising and setting of the major planets. So, by 750 BCE

astronomers had a reasonably accurate means of measuring the elevation (latitude) and lateral

direction (longitude) of all objects in the heavens. They built up an extensive collection of data,

and made tables of the positions of objects in the sky at any given time through a year (these

tables are called ephemerides).

Using observational techniques like heliacal rising, which occurs when a planet, star or other

body first becomes visible above the eastern horizon at dawn , it was discovered that:

the constellations of the zodiac completed a full circle through the sky once a year

the Sun's apparent movement daily across the sky formed of a circle

the moon moved through about of a circle each day

the Ecliptic was inclined to the horizon (about degrees)

planets were travelling through the starry background in regular paths that were

sometimes moving back on themselves in a loop (retrograde)

eclipses of the moon and the sun could be predicted

transits of planets (e.g. Venus) moving across the face of the sun, and occultations

(where the moon covered the stars) could be observed.



These observations continued over many centuries, slowly becoming more accurate, so

that ancient people were able to make star maps, and detect the regular events in the heavens.

Many seasonal phenomena like the flooding of the Nile, or special events like religious

ceremonies were linked to astronomical phenomena. The ability to predict some of these major

astronomical events gave rise to astrology, where people believed that there was a link between

heavenly and earthly events, and that the stars had some control over their lives. See this BBC

news item about a prehistoric star map.

The Babylonians and Chinese both believed that the earth and the moon were spherical,

that the earth and the moon rotated on an axis, and that the sun and the planets moved in circles

round the earth. This enabled them to be able to explain the phases of the moon, and predict

eclipses of the moon and the sun by believing that the earth cast a shadow on the moon, and the

moon cast a shadow over the sun.

The Babylonian astronomers recorded astronomical data systematically and by the

Seleucid period (330-125BCE) there were a great many astronomical tablets showing

ephemerides for the moon and the major planets. Many of the tablets contain "procedures" or

instructions for how to calculate intervals between astronomical events using the properties of

simple arithmetic progressions. These procedural processes were the earliest steps of a

mathematical astronomy, and both the procedures and the data were used by those who came

later. The Babylonians wrote down lists of numbers, in what we would call an arithmetic

progression and recognised that numbers repeated themselves over periods of time.


http://news.bbc.co.uk/1/hi/sci/tech/871930.stm


As you can see, Neugebauer published the sexagesimal values for twelve measurements

of the position of the Moon taken from a clay tablet dated 133/132 BCE. In the table above, the

top line shows the end of the year 133 BCE with the last month Aires, so the start of the

Babylonian year was at the vernal equinox, and the bottom line represents the end of year 132

BCE. The height of the lines on the zig-zag graph below approximately represent the sequence of

the numerical values in the table. There are two groups of numbers, one starting with 28,

followed by another starting 29. The results for Gemini and Cancer differ only in the third place

of sexagesimals and the minimum on the graph is interpolated from the results in the table.

Similarly the results for Sagitarius and Capricorn indicate the maximum value for the longitude.

Looking at the first three sets of sexagesimal numbers: 28, 55, 57, 58; 28, 37, 57, 58 and

28, 19, 57, 58 we can notice that the significant differences in the second place between 55, 37

and 19 are all giving a constant 18, which is the difference in height of the vertical lines on the

zig-zag graph (except at the minimum and maximum). The graph was drawn to illustrate the

periodicity of the data. It is important to realise that the Babylonians recognised the events

repeated themselves after some time, but they did not see these results as a 'graph' as we can [see

Note 3 below].

The use of graphs as a way of recording the data comes from Neugebauer's book The

Exact Sciences in Antiquity. The Babylonian astronomers recognised the events were periodic

but they did not have a theory of planetary motion.



3. The Hindu Sulbasutras

The Sulbasutras are the only early sources of Hindu mathematical knowledge and

originally come from the Vedic period (during the second millennium BCE). The earliest written

texts we have from this oral tradition date from about 800 BCE. The Sulbasutras are the

instructions for constructing various geometrical shapes to make 'fire-altars' using the "Peg and

Cord" technique. Each 'fire-altar' was a different shape and associated with unique gifts from the

Gods.

For more information on Peg and Cord geometry see: The Development of Algebra Part 1:

Section 4 "Early Indian Mathematics" an article by Leo already published on NRICH.

The Vedic people knew how to find the cardinal directions (NSEW). The Sulbasutras

gave procedures for the construction of the altars by starting with a line marking the E-W

direction (sun rises in east and sinks in the west), thus the E-W direction had special religious

significance.

At the end of the fourth century BCE the Indian part of Alexander the Great's empire

broke up into small kingdoms run by Indian Greeks. Around this time there was a collection of

mathematical knowledge called jyotsia, a mixture of astronomy, calendar calculations and

astrology. The rulers still maintained trading links between western India and the Hellenistic

culture of the Roman Empire. At this time, Indian horoscope astrology became popular needing

precise calendar and astronomical calculations.

The Panca-siddhantica is a collection of five astronomical works composed in the sixth

century CE by Vrahamihira. These works contain earlier mathematical knowledge and here we

find an approximation for as , because they used

the relationship between the circumference of a circle, and its diameter as

. Sines were calculated at intervals of or ,

giving a series of values for Sines of angles in the first quadrant and, using the same terms in

Sanskrit as the Babylonians for the radius of a circle. Also, the use of similar calculation methods

as the Babylonians suggest that this is the earliest surviving Indian sine table. [See Note 4 below]


http://nrich.maths.org/6485&part=


The method of calculation and the values used by Vrahamihira is very similar to a Greek

Chord table for arcs up to and intervals of the quadrant into sixths, namely arcs of

. This suggests that the Indian invention of the trigonometry of Sines was inspired by

replacing the Greek Chord geometry of right triangles in a semicircle by the simpler Sine

geometry of right triangles in a quadrant [See Note 5 below].

This discovery is much earlier than the account usually given of the sine table derived from

chords by Aryabhata the Elder (476-550 CE) who used the word jiya for sine. Brahmagupta

reproduced the same table in 628 CE and Bhaskara gave a detailed method for constructing a

table of sines for any angle in 1150 CE.

4. Chinese Astronomy

The Chinese were the most accurate observers of celestial phenomena before the Arabs.

"Oracle Bones" with star names engraved on them dating back to the Chinese Bronze Age (about

2,000 BCE) have been found, and very old star maps have been found on pottery, engraved on

stones, and painted on the walls of caves.

Surviving records of astronomical observations made by two astronomers Shi Shen and

Gan De date from the 4th century BCE. Shi Shen wrote a book on astronomy, and made a star

map and a star catalogue. In 364 BCE Gan De made the first recorded observation of sunspots,



and the moons of Jupiter and they both made accurate observations of the five major planets.

Their observations were based on the principle of the stars rotating about the pole (equivalent to

the earth rotating on its axis).

A famous map due to Su Song (1020-1101) and drawn on paper in 1092 represents the whole

sky with the positions of some 1,350 stars.

The equator is represented by the horizontal straight line running through the star chart,

while the ecliptic curves above it. The oldest star map found so far is from Dunhuang. Earlier

thought to date from about 940 CE, it was made with precise mathematical methods by the

astronomer and mathematician Li Chunfeng (602-670) and shows 1339 stars in 257 Chinese star

groups with a precision between 1.5 and 4 degrees of arc. In all there are 12 charts each in 30

degree sections displaying the full sky visible from the Northern hemisphere.

Up to now it is the oldest complete preserved star atlas discovered from any civilisation.

It has been on display this year in the British Library to celebrate 2009 as the International Year

of Astronomy. Some elements of Indian astronomy reached China with the expansion of

Buddhism (25-220 CE). Later, during the period (618-907 CE) a number of Indian astronomers

came to live in China and Islamic astronomers collaborated closely with their Chinese

counterparts particularly during (1271-1368).



Very little of the knowledge of the Indians and the Chinese was known in Europe before the

Portuguese navigators and the Jesuit scientist Matteo Ricci in the fifteenth century. Babylonian

astronomy contributed direct empirical data as a foundation for Greek theory and exactly the

same data which provided the information for the "zig-zag" data results in Babylonian theory

were used to calculate the mean motions of the sun and moon by Hipparchus.

5. The Beginnings of Mathematical Astronomy

As we have seen, the Babylonians were interested in predicting the time at which

a particular celestial event would occur whereas Greek astronomy became much more interested

in predicting where a celestial body would be at any particular time. This contrast is clearly

illustrated by the development of a geometrical model of the motions of the heavens by the

Greek mathematician Eudoxus.

Eudoxus (408-355 BCE)

The original belief of an Earth-centred system in perfect circles came from the Pythagoreans

(6th/5th centuries BCE) and was discussed and elaborated over the years. Eudoxus was taught by

Archytas, one of the leading Pythagorean philosophers of his time, who maintained that the most

perfect shape was a circle, so Eudoxus proposed a system of concentric spheres to describe the

movement of the planets. This idea, where the planets moved about the Earth in circles on the

surfaces of different spheres, remained the widely held theory of the universe until it was

challenged by Nicolaus Copernicus in 1543. The simple idea is shown in the diagram below.

However, there were enough data from the Babylonians and others to show that the system was

much more complicated, and Eudoxus finally produced a system with 27 spheres altogether.



Using data derived from the Babylonian astronomers, Eudoxus was able to calculate that

the variation in latitude of the Moon was about +/−5∘ (in our terms), and the angle between the

equator and the ecliptic [See Note 1 below] was measured as 115 of a circle or 24∘ which is the

angle subtended at the centre of a 15 sided polygon. This construction was included in Euclid's

Elements Book IV proposition 16, because it was so useful for astronomy.

Eudoxus also developed the theory of proportion, now contained in Book V of Euclid's

Elements, which became the major logical tool for empirical investigaton until the seventeenth

century, and his method of exhaustion became the foundation of the technique developed by

Archimedes (287-212 BCE) for finding areas of circles and other shapes.

The theory of the solar system that the Greeks developed was principally an "explanatory

device" more descriptive than predictive, and as observations became more accurate, it needed

constant revision and modification.

Aristarchus (310-230 BCE)

Using Eudoxus' theory of proportion, Aristarchus measured the relative sizes and

distances to the Moon and Sun and found the Sun to be bigger than Earth! So, he reasoned that

the Sun rather than Earth is the centre of the Universe and the Earth is one of the planets.



In his book On the Sizes and Distances of the Sun and the Moon he assumed that the

Earth is at the centre of a sphere on which the Moon moves, and at half Moon the edge of the

shadow is in a direct line with the eye. At this time he claimed that its angular distance from the

Sun is 87∘(actually it is about 89∘). Using the time taken for a complete eclipse he estimated that

the width of the Earth's shadow is two Moon diameters (actually it is nearer three), and the Moon

had an angular diameter of 2∘ (this is about four times too large). Using these hypotheses he

obtained the ratio of the distance from the earth to the sun as:

18≤d1dm≤20

Today, this would simply involve finding cos87∘(=sin3∘) but the concepts

of cos and sinwere not thought of until much later. Applying his logical reasoning to the theory

of proportion, he arrived at an estimate for the distance to the Sun of about 19 times the distance

to the Moon. This estimate was generally accepted for the next 2,000 years.

Aristarchus' observations showed that Eudoxus' model was not able to account for some

of the simplest motions of the planets, and so Greek mathematicians like Apollonius (262-190

BCE) managed to find more complicated geometrical constructions in order to keep the Earth at

the centre of the universe.



Although it might seem like a missed opportunity to us for Greek scholars to abandon

Aristarchus' model of the solar system, even though he had shown that the Sun was larger than

the Earth, there was no real evidence to show that the Sun was the centre of our system. The fact

that the Sun might be larger than the Earth was no reason to suppose that the Sun was at the

centre when the obvious apparent rotation of the Sun, Moon and Planets around the Earth was so

strong.

Hipparchus (190-120 BCE)

The first known table of chords was produced by the Greek mathematician Hipparchus in

about 140 BC. Although nothing has survived, it is claimed that he wrote twelve books of tables

of chords, but Neugebauer shows this to be impossible. Hipparchus developed great

observational skills, improved the design of instruments and compiled a catalogue of about 850

stars. He also tried to improve the geometrical model of the universe. His most important

discovery was the "Precession of the Equinoxes". Since the Earth's axis is tilted with respect to

the stars it slowly describes a circle. This results in the intersection points of the celestial equator

and the ecliptic (the equinoxes) changing slowly. Precession also explains why the "Polar Star"

changes its position in the heavens over the centuries.



Hipparchus is credited with constructing the first known table of chords. The importance of

Hipparchus' achievement in doing this was to change the mathematical tools involved from the

arithmetical 'procedures' of the Babylonian scholars to a geometrical device' , namely the use of

arcs of a circle imagined to be on the surface of the 'heavenly sphere'. Even so, he was still using

the new technique to investigate the location of heavenly bodies, and the process was still clearly

embedded in astronomy. Trigonometry, as a separately identifiable science in its own right, does

not appear before the Arab scholars developed it much further in the eleventh and twelfth

centuries CE.

Menelaos (70-130 BCE)

In about 100 BCE Menelaos compiled a Book of Spherical Proportions Sphaerica, in

which he set up the basis for treating spherical triangles by using arcs of great circles instead of

arcs of parallel circles on the sphere. The plane triangle version of the theorem states:

If a straight line crosses the three sides of a triangle (one of the sides has to be produced) then

the product of three of the non-adjacent segments thus formed is equal to the product of the other

three segments of the triangle.

This proof uses similar triangles

ABC is a triangle with a transversal DE that cuts AC produced in F

First, insert a construction line CX parallel to AD

(CX will be eliminated later). Now triangles ADF and CXF are similar, so

ADCX=FAFC , and in the cyclic quadrilateral, BFCD,BEEC=DBCX

Multiply ADCX=FAFC by equal quantities,

and ADCX×BEEC=FAFC×DBCX



Insert the ratio CXCX(=1) into the Left Hand Side product before cancellation, and

ADDB×BEEC×FCFA=1

Note: The orientation of FC is important.

Menelaus produced a spherical triangle version of this theorem and a modification of it appears

in Ptolemy's Almagest.

Claudius Ptolemy (c85-c165 CE)

Ptolemy was the most influential Greek astronomer of this time. He supported the

geocentric theory of the universe, and his book The Mathematical Compilation dominated

astronomical thought until Copernicus (1473-1543) published his heliocentric theory in 1543.

The aim of the Almagest was to provide numerical data for astronomical phenomena and

observation and to explain the empirical foundations and theoretical reasons for the data.

The Almagest contains a collection of all the then known astronomical knowledge: geometrical

and numerical procedures, the longitude and latitude of heavenly bodies, information about

parallax, the distance and relative sizes of the Sun and the Moon, lunar theory, solar motion, and

the occurrence of eclipses, transits and occultations. This was the reason the Arab scholars called

it "Al Megiste" (the great one) and the name has remained.

Ptolemy was the author of a new table of chords, based on exactly the same principles

inherited from the Babylonians, and using more sophisticated observational techniques, more

accurate data, and the new mathematics of Euclid, Apollonius Archimedes and Menelaos.

He divided the circle into 360 parts and used a diameter of 120 units, and calculated in

the traditional sexagesimal numbers that astronomers had been using for two thousand years.

Ptolemy calculated chords by constructing a series of regular polygons in a circle as shown by

Euclid.



The Chords of regular polygons of 3, 4, 5, 6 and 10 sides respectively subtend angles

of 120∘,90∘,72∘,60∘, and 10∘ at the centre of a circle. By continued bisection and interpolaton

he could find chords for many other angles which he expressed as ratios of chord to

diameterfinally arriving at chords of angles of 12∘ where in our terms, chord α=120 Sinα

In sexagesimal notation, the chords for the regular polygons are:

chord 36∘ = 37; 4, 55,

chord 60∘ = 1, 0; 0

chord 72∘ = 1, 10; 32. 3

chord 90∘ = 1, 24; 51, 10

from Pythagoras' theorem:

chord 2α + chord2(180−α)=1202

he obtained chord 120∘ = 1, 43; 55, 23. and chord 144∘ = 1, 54; 7, 37.

Converting Ptolemy's chord calculations with base ten:



for chord 602 the sexagesimal 1, 0; 0. is 60 in base ten and 120 sin 30∘ = 120 ×12 or 60 units,

which is the radius of the circle.

similarly, chord 90∘ = 1, 24; 51, 10. or in base ten,

chord 90∘ = 60 + 24 + 3060+103600 which is 84.8527 units

and chord 902 = 120 x sin 450 or 120 x or 84.8536 units

Using what is now called "Ptolemy's theorem" (this property of cyclic quadrilaterals was known

much earlier) he set up a system where one of the sides of the quadrilateral was a diameter, that

enabled him to calculate sums and products of chords.

Converting the chords expression to modern notation, and putting x=β2 and y=α2 this is

equivalent to:

sin(x−y)=sinxcosy−cosxsiny

Similarly he obtained

sin(x+y)=sinxcosy+cosxsiny

and also derived

2sin2x=1−cos2x

Using the table of chords with the geometry and formulae above, Ptolemy was able to

solve all the triangles he needed to create his table of chords in steps of half degrees

from 12∘ to 180∘ which is equivalent to a table of sines from 14∘ to 90∘.



6. Ptolemy's use of Spherical Triangles

Menlaos' Sphaerica was the major work that established the science of spherical

geometry, and this was, in a sense, a non-Euclidean geometry where the sides of a triangle were

formed by the intersections of three great circles on a sphere, and the angle sum of a triangle

could be more than1800. At the time, it was not seen as a radically new geometry, but more as an

extension of Euclid, and most of the time the triangles used were constructed with only one right

angle. At this time, spherical geometry was included in the quadrivum (arithmetic, geometry,

music, astronomy), and taught as part of astronomy. The new relationships developed were

possible by the use of proportional reasoning, a direct result of Eudoxus' theory. Ptolemy was

well aware of the new possibilities, because finding the distance between two stars was

equivalent to measuring an arc of a circle, and he adapted the spherical geometry for use with

tables of chords.

7. Passing on the Knowledge

By this time, mathematicians and astronomers had developed a complex mathematically

based science, had a wide range of geometrical techniques whereby they measured the Earth,

estimated the distances of the Moon and the Sun, developed a theory of the movement of the

planets, and precisely catalogued hundreds of stars. A substantial body of geometrically based

mathematics had been developed and scholars made commentaries on the works of Euclid,

Apollonius, Archimedes, and others. In the next centuries, Diophantos wrote his Arithmetica,

which was to inspire Fermat centuries later, Pappus recorded much of the earlier learning for

later generations, and contacts along the trade routes began to be made with people in India and

China. In September 622 Mohammed made his famous escape from persecution in Mecca to

safety in Medina, and within two hundred years, the Arab culture had established an empire from

India through the Middle East and North Africa and into Spain.

8. The Arabs collect knowledge from the known world

The Arab civilisation traditionally marks its beginning from the year 622 CE the date

when Muhammad, threatened with assassination, fled from Mecca to Medina where Muhammad

and his followers found safety and respect. Over a century later, the Arabs had established

themselves as a powerful unified force across large parts of the Middle East and The Caliph Abu



Ja'far Al-Mansour moved from Damascus to establish the city of Baghdad during the years 762

to 766. Al-Mansour sent his emissaries to search for and collect knowledge. From China, they

learnt how to produce paper, and using this new skill they started a programme of translation of

texts on mathematics, astronomy, science and philosophy into Arabic. This work was continued

by his successors, Caliphs Mohammad Al-Mahdi and Haroun Al-Rasheed. The quest for

knowledge became a lasting and significant part of Arab culture.

Al-Mansour had founded a scientific academy that became called 'The House of

Wisdom'. This academy attracted scholars from many different countries and religions to

Baghdad to work together and establish the traditions of Arabic science that were to continue

well into the Middle Ages. Some of this work was later translated into Latin by Mediaeval

scholars and passed on into Europe. The dominance of Baghdad and the influence of the Arab

World was to last for the next 500 years.

The scholars in the House of Wisdom came from many cultures and translated the works

of Egyptian, Babylonian, Greek, Indian and Chinese astronomers and mathematicians. The

Mathematical Treatise of Ptolemy was one of the first to be translated from the Greek into

Arabic by Ishaq ben Hunayn (830-910). It was admired for its extensive content and became

known in Arabic as Al-Megiste (the Great Book). The name 'Almagest' has continued to this day

and it is recognized as both the great synthesis and the culmination of mathematical astronomy

of the ancient Greek world. It was translated into Arabic at least five times and constituted the

basis of the mathematical astronomy carried out in the Islamic world.

9. India: The Sine, Cosine and Versine

Greek astronomy began to be known in India during the period 300-400 CE. However, Indian

astronomers had long been using planetary data and calculation methods from the Babylonians,

and even though it was well after Ptolemy had written the Almagest, 4th century Indian

astronomers did not entirely take over Greek planetary theory. Ancient works like the Panca-

siddhantica (now lost) that had been transmitted through the version by Vrahamihira and

Aryabhata's Aryabhatiya (499 CE) demonstrated that Indian scholars had their own ways of

dealing with astronomical problems and that they had great skill in calculation.



Even in the oldest Indian texts, the Chord [to remind yourself about Chords see the

section on Claudius Ptolemy is not used, and instead there appear some very early versions of

trigonometric tables using Sines. However, the Indian astronomers divided the 90∘arc

into 24 sections, thus obtaining values of Sines for every 3∘45′ of arc.

In this diagram, SB is the arc for the angle θ and AS is the jiya. So the relation between

the jiyaand our sine is:

jiya(θ)=Rsinθ

where R is the radius of the circle.

Many Indian Sine tables use R=3438 which is the result if the circumference of the circle

is 360×60 or 21,600 minutes.

By the 5th century, two other functions had been defined and used. The length EA was

called thekotijya (our cosine), and AB was called the utkrama-jya (our versine). This was

sometimes called the sama meaning an 'arrow', or sagitta in Latin

The versine function for a circle radius R is: vers θ=R−cosθ .

In Aryabhata's work, he uses R=3438 and took this value to calculate his table of Sines.

This became the standard for later works. Comparison with Varhamihira's Sines (in sexagesimal

numbers) and Hipparchs' table (in lengths of chords) suggests a possible transmission of at least

some of the Greek works to the Hindus. However, we have no way of knowing this for certain,

and it is quite possible that the Hindus calculated their values independently.

The 'Great Work' (the Mahabhaskariya) of Bhaskara I was written in about 600 CE. He

produced a remarkable method for approximating values for the Sines, by using the ratio of two

quadratic functions. This was based entirely on comparing the results of his calculations with

earlier values.



However, since these tables only gave values for every 3∘45′, there was cnsiderable room

for improvement. It is curious that since Ptolemy's table of chords enabled him to find values

equivalent to Sines from 14∘ to 90∘ that the Indian scholars did not go further at this stage. Later,

Brahmagupta (598-670) produced an ingenious method based on second order differences to

obtain the Sine of any angle from an initial set of only six values from 0∘ in 15∘ intervals to 90∘.

10. Trigonometry in the Arab Civilisation

The introduction and development of trigonometry into an independent science in the

Arab civilisation took, in all, some 400 years. In the early 770s Indian astronomical works

reached the Caliph Al-Mansur in Baghdad, and were translated as the Zij al-Sindhind, and this

introduced Indian calculation methods into Islam.

Famous for his algebra book, Abu Ja'far Muhammad ibn Musa al-Khwarizmi had also

written a book on Indian methods of calculation (al-hisab al-hindi) and he produced an improved

version of the Zij al-Sindhind. Al-Khwarizmi's version of Zij used Sines and Versines, and

developed procedures for tangents and cotangents to solve astronomical problems. Al-

Khwarizmi's Zij was copied many times and versions of it were used for a long time.

Many works in Greek, Sanskrit, and Syriac were brought by scholars to Al-Mansur's

House of Wisdom and translated. Among these were the works of Euclid, Archimedes

Apollonius and of course, Ptolemy. The Arabs now had two competing versions of astronomy,

and soon the Almagest prevailed.

The Indian use of the sine and its related functions were much easier to apply in

calculations, and the sexagesimal system from the Babylonians continued to be used, so apart

from these two changes, the early Arabic versions of the Almagest remained faithful to Ptolemy.

Abu al-Wafa al-Buzjani (Abul Wafa 940-998) made important contributions to both

geometry and arithmetic and was the first to study trigonometric identities systematically. The

study of identities was important because by establishing relationships between sums and

differences, and fractions and multiples of angles, more efficient astronomical calculations could

be conducted and more accurate tables could be established.

The sine, versine and cosine had been developed in the context of astronomical problems,

whereas the tangent and cotangent were developed from the study of shadows of the gnomon. In

his Almagest, Abul Wafa brought them together and established the relations between the six



fundamental trigonometric functions for the first time. He also used R=1 for the radius of the

basic circle.

From these relations Abul Wafa was able to demonstrate a number of new identities

using these new functions:

sec2θ=1+tan2θ. . . . . . . cosec2θ=1+cot2θ

Abul Wafa also devised methods for calculating trigonometric tables by an improved

differencing technique to obtain values that were accurate to 5 sexagesimal (8 decimal) places.

Greek astronomers had long since introduced a model of the universe with the stars on

the inside of a vast sphere. They had also worked with spherical triangles, but Abul Wafa was

the first Arab astronomer to develop ways of measuring the distance between stars using his new

system of trigonometric functions including the versine.



In the diagram above, the blue triangle with sides a, b, and c represents the distances

between stars on the inside of a sphere. The apex where the three angles α, β, and γ are marked,

is the position of the observer. The blue curves are Great Circles on the sphere, and by measuring

the angles, finding more accurate values for their functions, and assuming a value for R the

radius of the sphere, it became possible to find the great-circle distances between the stars.

By an ingenious application of Menelaos' Theorem using special cases of great circles

with two right angles, Abul Wafa showed how the theorem could be applied in spherical

triangles. This was a considerable advance in Spherical Trigonometry that enabled the

calculation of the correct direction for prayer (the quibla) and was to have important applications

in Navigation and Cartography.



The Abul Wafa crater of the Moon is named in recognition of his work in astronomy.

Abu al-Rayhan Muhammad ibn Ahmad Al-Biruni (973-1050) was an outstanding scholar

reputed to have written over 100 treatises on astronomy, science, mathematics, geography,

history, geodesy and philosophy. Only about twenty of these works now survive, and only about

a dozen of these have been published.

Al-Biruni's treatise entitled Maqalid 'ilm al-hay'a (Keys to the Science of Astronomy) ran

to over one thousand pages and contained extensive developments in on trigonometry. Among

many theorems, he produced a demonstration of the tangent formula, shown below.

From the diagram, O is the centre of the semicircle, and AED a right-angled triangle with

a perpendicular from E to C.



Consequently, triangles AEC and EDC are similar.

Angle EOD is twice angle EAD, and angles EAC and DEC are equal.

If the radius of the circle R=1, then EC=sinθ and OC=cosθ

So tan(θ2)=ECAC=sinθ1+cosθ . . . and . . . tan(θ2)=DCEC=1−cosθsinθ

From which he derived the half angle and multiple angle formulae.

While many new aspects of trigonometry were being discovered, the chord, sine, versine

and cosine were developed in the investigation of astronomical problems, and conceived of as

properties of angles at the centre of the heavenly sphere. In contrast, tangent and cotangent

properties were derived from the measurement of shadows of a gnomon and the problems of

telling the time.

In his Demarcation of the Coordinates of Cities he used spherical triangles for finding the

coordinates of cities and other places to establish local meridian (the quibla) and thereby finding

the correct direction of Mecca, and in his Exhaustive Treatise on Shadows he showed how to use

gnomons for finding the time of day.

Abu Muhammad Jabir ibn Aflah (Jabir ibn Aflah c1100 - c1160) probably worked in

Seville during the first part of the 12th century. His work is seen as significant in passing on

knowledge to Europe. Jabir ibn Aflah was considered a vigorous critic of Ptolemy's astronomy.

His treatise helped to spread trigonometry in Europe in the 13th century, and his theorems were

used by the astronomers who compiled the influential Libro del Cuadrante Sennero (Book of the

Sine Quadrant) under the patronage of King Alfonso X the Wise of Castille (1221-1284).

A result of this project was the creation of much more accurate astronomical tables for

calculating the position of the Sun, Moon and Planets, relative to the fixed stars, called the

Alfonsine Tables made in Toledo somewhere between 1252 and 1270. These were the tables



Columbus used to sail to the New World, and they remained the most accurate tables until the

16th century.

By the end of the 10th century trigonometry occupied an important place in astronomy

texts with chapters on sines and chords, shadows (tangents and cotangents) and the formulae for

spherical calculations. There was also considerable interest in the resolution of plane triangles.

But a completely new type of work by Nasir al-Din al-Tusi (Al-Tusi 1201-1274) entitled Kashf

al-qina 'an asrar shakl al-qatta (Treatise on the Secrets of the Sector Figure), was the first

treatment of trigonometry in its own right, as a complete subject apart from Astronomy. The

work contained a systematic discussion on the application of proportional reasoning to solving

plane and spherical triangles, and a thorough treatment of the formulae for solving triangles and

trigonometric identities. Al-Tusi originally wrote in Persian, but later wrote an Arabic version.

The only surviving Persian version of his work is in the Bodleian Library in Oxford.

This was a collection and major improvement on earlier knowledge. Books I, II and IV

contain parts of the Elements, the Almagest and a number of other Greek sources. Book III deals

with the basic geometry for spherical triangles and the resolution of plane triangles using the sine

theorem:

asinA=bsinB=csinC

Book V contains the principal chapters on trigonometry dealing with right-angled

triangles and the six fundamental relations equivalent to those we use today; sine, cosine,

tangent, cotangent, secant and cosecant. He provided many new proofs and showed how they

could be used to solve many problems more easily.



Al-Tusi invented a new geometrical technique now called the 'Al-Tusi couple' that

generated linear motion from the sum of two circular motions. He used this technique to replace

the equant used by Ptolemy, and this device was later used by Copernicus in his heliocentric

model of the universe. Al-Tusi was one of the greatest scientists of Mediaeval Islam and

responsible for some 150 works ranging from astronomy, mathematics and science to philosophy

and poetry.

11. Arab Science and Technology Reaches Europe

The Arab astronomers had learnt much from India, and there was contact with the

Chinese along the Silk Road and through the sea routes, so that Arab trading posts were

established in India and in China. Through these contacts Indian Buddhism spread into China

and was well established by the 3rd century BCE, probably later carrying with it some of the

calculation techniques of Indian astronomy. However few, if any, technological innovations

seemed to have passed from China to India or Arabia.

By 790 CE, the Arab empire had reached its furthest expansion in Europe, conquering

most of the Iberian peninsula, an area called Al-Andalus by the Arabs.



At this time many religions and races coexisted in Iberia, each contributing to the culture.

The Muslim religion was generally very tolerant towards others, and literacy in Islamic Iberia

was more widespread than any other country in Western Europe. By the 10th century Cordoba

was said to have equally good libraries and educational establishments as Baghdad, and the cities

of Cordoba and Toledo became centres of a flourishing translation business.

Between 1095 and 1291 a series of religiously inspired military Crusades were waged by

the Christians of Europe against the Arab Empire. The principal reason was the restoration of

Christian control over the Holy Land, but there were also many other political and economic

reasons.

In all this turmoil and conflict there were periods of calm and centres of stability, where

scholars of all cultures were able to meet and knowledge was developed, translated and

transmitted into Western Europe. The three principal routes through which Greek and Arab

science became known were Constantinople (now Istanbul) Sicily and Spain. Greek texts

became known to European monks and scholars who travelled with the armies through

Constantinople on their way South to the Holy Land. These people learnt Greek and were able to

translate the classical works into Latin. From Sicily, Arabs traded with Italy, and translation took

place there, but probably the major route by which Arabic science reached Europe was from the

translation houses of Toledo and Cordoba, across the Pyrenees into south-western France.



During the twelfth and thirteenth century hundreds of works from Arabic, Greek and

Hebrew sources were translated into Latin and the new knowledge was gradually disseminated

across Christian Europe.

Geometrical knowledge in early Mediaeval Europe was a very practical subject. It dealt

with areas, heights, volumes and calculations with fractions for measuring fields and the building

of large manors, churches, castles and cathedrals.

Hugh of St. Victor (1078-1141) in his Practica Geometriae divides the material into

Theorica (what is known and practised by a teacher) and Practica (what is done by a builder or

mason). Theoretical geometry in the Euclidean sense was virtually unknown until the first

translations of Euclid appeared in the West.

The astrolabe was commonly used to measure heights by using its 'medicline' (a sighting

instrument fixed at the centre of the circle) and the shadow square engraved in the centre of the

instrument, and then comparing the similar triangles. The horizontal distance from the centre of

the astrolabe to the edge of the square was marked with twelve equal divisions.

This system was in use well into the 16th century as seen in the illustration below:



This is from Thomas Digges' Pantometria of 1571. The same system is still used, but the square

in the quadrant is marked with six divisions.

A popular twelfth century text, the Artis Cuiuslibet Consummatio shows the gradual

insertion of more technical knowledge, where the measuring of heights (altimetry) was much

more related to astronomy, showing how to construct gnomons and shadow squares. Gradually

the translations made on the continent of Europe came to England.

Richard of Wallingford (1292-1336)

After entering Oxford University in about 1308, Richard entered monastic life at St

Alban's in 1316. After his ordination as a priest, his Abbot sent him back to Oxford where he

studied for nine years. In 1327 he became Abbot of St Albans.

Richard's early work was a series of instructions (canons) for the use of astronomical tables that had been drawn up by John Maudith, the Merton College Astronomer. Later he wrote an important work, the Quadripartitum, on the fundamentals of trigonometry needed for the solution of problems of spherical astronomy. The first part of this work is a theory of trigonometrical identities, and was regarded as a basis for the calculation of sines, cosines, chords and versed sines. The next two parts of the Quadripartitum dealt with a systematic and rigorous exposition of Menelaos' theorem. The work ends with an application of these principles to astronomy. The main sources of the work appear to be Ptolemy's Almagest, and Thabit ibn Qurra (826-901 CE).



The Quadripartitum was probably the first comprehensive mediaeval treatise on

trigonometry to have been written in Europe, at least outside Spain and Islam. When Richard

was abbot of St. Albans, he revised the work, taking into account the Flores of Jabir ibn Afla.

In 1326 to 1327 Richard also designed a calculation device, called an equatorium, a

complex geared astrolabe with four faces. He described how this could be used to calculate

lunar, solar and planetary longitudes and thereby predict eclipses in his Tractatus Albionis. It is

possible that this led to his design for an astronomical clock described in his Tractatus Horologii

Asronomici, (Treatise on the Astronomical Clock) of 1327, which was the most complex clock

mechanism known at the time. The mechanism comprised a rotating star map that modeled the

lunar eclipse and planets by gearing, presented as a geocentric model. It appeared at a

transitional period in clock design, just before the advent of the escapement. This makes it one of

the first true clocks, and certainly one of first self powered models of the heavens. Unfortunately

it was destroyed during Henry VIII's reformation at the dissolution of St Albans Monastery in

1539.

Georg von Peuerbach (1423-1461)

Peuerbach's work helped to pave the way for the Copernican conception of the world

system; he created a new theory of the planets, made better calculations for eclipses and

movements of the planets and introduced the use of the sine into his trigonometry.

His early work, Tabulae Eclipsium circulated in manuscript was not published until 1514, contained tables of his eclipse calculations that were based on the Alfonsine Tables. He calculated sines for every minute of arc for a radius of 600,000 units and he introduced the Hindu-Arabic numerals in his tables. [See Note 9 below]


http://www.wallingfordclock.talktalk.net/

http://www.wallingfordclock.talktalk.net/


Peuerbach's Theoricae Novae Planetarum, (New Theories of the Planets) was composed

about 1454 was published in 1473 by Regiomontanus' printing press in Nuremburg. While the

book was involved in attempting a technical resolution of the theories of Eudoxus and Ptolemy,

Peuerbach claimed that the movement of the planets was determined by the Sun, and this has

been seen as a step towards the Copernican theory. This book was read by Copernicus, Galileo

and Kepler and became the standard astronomical text well into the seventeenth century.

In 1460 he began working on a new translation of Ptolemy's Almagest, but he had only

completed six of the projected thirteen books before died in 1461.

Johannes Muller von Konigsberg or Regiomontanus (1436-1476)

Regiomontanus had become a pupil of Peuerbach at the University of Vienna in 1450. Later, he undertook with Peuerbach to correct the errors found in the Alfonsine Tables. He had a printing press where he produced tables of sines and tangents and continued Puerbach's innovation of using Hindu-Arabic numerals.



As promised, he finished Peuerbach's Epitome of the Almagest, which he completed in 1462 and was printed in Venice. The Epitome was not just a translation, it added new observations, revised calculations and made critical comments about Ptolemy's work.

Realising that there was a need for a systematic account of trigonometry, Regiomontanus

began his major work, the De Triangulis Omnimodis (Concerning Triangles of Every Kind)

1464. In his preface to the Reader he says,

"For no one can bypass the science of triangles and reach a satisfying knowledge of the

stars .... You, who wish to study great and wonderful things, who wonder about the

movement of the stars, must read these theorems about triangles. Knowing these ideas will

open the door to all of astronomy and to certain geometric problems. For although certain

figures must be transformed into triangles to be solved, the remaining questions of

astronomy require these books."

The first book gives the basic definitions of quantity, ratio, equality, circles, arcs, chords

and the sine function. Next come a list of axioms he will assume, and then 33 theorems for right,

isosceles and scalene triangles. The formula for the area of a triangle is given followed by the

sine rule giving examples of its application. Books III to V cover the all-important theory of

spherical trigonometry. The whole book is organised in the style of Euclid with propositions and

theorems set out in a logical hierarchical manner. This work, published in 1533 was of great

value to Copernicus.

Regiomontanus also built the first astronomical observatory in Germany at Nuremburg

with a workshop where he built astronomical instruments. He also took observations on a comet

in 1472 that were accurate enough to allow it to be identified as Halley's Comet that reappeared

210 years later. Regiomontanus died during an outbreak of plague in Rome in 1476.



12. The Final Chapter: Trigonometry Changes the World System

Nicolaus Copernicus (1473 - 1543)

Copernicus wrote a brief outline of his proposed system called the Commentariolus that

he circulated to friends somewhere between 1510 and 1514. By this time he had used

observations of the planet Mercury and the Alfonsine Tables to convince himself that he could

explain the motion of the Earth as one of the planets. The manuscript of Copernicus' work has

survived and it is thought that by the 1530s most of his work had been completed, but he delayed

publishing the book.

His student, Rheticus read the manuscript and made a summary of Copernicus' theory

and published it as the Narratio Prima (the First Account) in 1540. Since it seemed that

the Narratiohad been well accepted by colleagues, Copernicus was persuaded to publish more of

his main work, and in 1542 he published a section on his spherical trigonometry as De lateribus

et angulis traingulorum (On the sides and angles of triangles). Further persuaded by Rheticus

and others, he finally agreed to publish the whole work, De Revolutionibus Orbium

Coelestium (The Revolutions of the Heavenly Spheres) and dedicated it to Pope Paul III. It

appeared just before Copernicus' death in 1543.



Georg Joachim von Lauchen called Rheticus (1514-1574)

Rheticus had facilitated the publication of Copernicus' work, and had clearly understood the basic principles of the new planetary theory.

In 1551, with the help of six assistants, Rheticus recalculated and produced the Opus Palatinum de Triangulis (Canon of the Science of Triangles) which became the first publication of tables of all six trigonometric functions. This was intended to be an introduction to his greatest work, The Science of Triangles.

When he died his work was still unfinished, but like Copernicus, Rheticus acquired a student, Valentinus Otho who supervised the calculation (by hand) of some one hundred thousand ratios to at least ten decimal places filling some 1,500 pages. This was finally completed in 1596. These tables were accurate enough to be used as the basis for astronomical calculations up to the early 20th century.

Bartholomaeus Pitiscus (1561 - 1613)

The term trigonometry is due to Pitiscus and as first appeared in his Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus, published in 1595. A revised version in1600 was the Canon triangularum sive tabulae sinuum, tangentium et secantium ad partes radii 100000 (A Canon of triangles, or tables of sines tangents and secants with a radius of 100,000 parts.) The book shows how to construct sine and other tables, and presents a number of theorems on plane and spherical trigonometry with their proofs.



However, soon after Rheticus' Opus Palatinum was published, serious inaccuracies were

found in the tangent and secant tables at the ends near 1∘ and 90∘. Pitiscus was commissioned to

correct these errors and obtained a manuscript copy of Rheticus' work. Many of the results were

recalculated and new pages were printed incorporating the corrections. Eventually, Pitiscus

published a new work in 1613 incorporating that of Rheticus with a table of sines calculated to

fifteen decimal places entitled the Thesaurus Mathematicus.

By the beginning of the seventeenth century, the science of trigonometry had become a

sophisticated technique used in calculating more and more accurate tables for use in astronomy

and navigation, and had been instrumental in fundamentally changing man's concept of his

world.



3.0 TRIGONOMETRY IN USE

The Babylonians, sometime before 300 B.C.E. were using degree measurement for

angles. The Babylonian numerals were based on the number 60, so it may be conjectured that

they took the unit measure to be what we call 60°, then divided that into 60 degrees. Perhaps 60°

was taken as the unit because the chord of 60° equals the radius of the circle, see below about

chords. Degree measurement was later adopted by

Hipparchus.

The Babylonians were the first to give coordinates for

stars. They used the ecliptic as their base circle in the

celestial sphere, that is, the crystal sphere of stars. The sun

travels the ecliptic, the planets travel near the ecliptic, the

constellations of the zodiac are arranged around the ecliptic, and

the north star, Polaris, is 90° from the ecliptic. The celestial sphere rotates around the axis

through the north and south poles. The Babylonians measured the longitude in degrees

counterclockwise from the vernal point as seen from the north pole, and they measured the

latitude in degrees north or south from the ecliptic.

Hipparchus of Nicaea (ca. 180 - ca. 125 B.C.E.)

Hipparchus was primarily an astronomer, but the beginnings of trigonometry apparently

began with him. Certainly the Babylonians, Egyptians, and earlier Greeks knew much astronomy

before Hipparchus, and they also determined the positions of many stars on the celestial sphere

before him, but it is Hipparchus to whom the first table of chords is attributed. It has been

hypothesized that Apollonius and even Archimedes constructed tables of chords before him, but

there is no reference to any such earlier table.

Some of Hipparchus' advances in astronomy include the

calculation of the mean lunar month, estimates of the sized and

distances of the sun and moon, variants on the epicyclic and eccentric

models of planetary motion, a catalog of 850 stars (longitude and

latitude relative to the ecliptic), and the discovery of the precession of

the equinoxes and a measurement of that precession.



According to Theon, Hipparchus wrote a 12-book work on chords in a circle, since lost.

That would be the first known work of trigonometry. Since the work no longer exists, most

everything about it is speculation. But a few things are known from various mentions of it in

other sources including another of his own. It included some lengths of chords corresponding to

various arcs of circles, perhaps a table of chords. Besides these few scraps of information, others

can be inferred from knowledge that was taken as well-known by his successors.

Aristarchus of Samos (310 – 250 BC), a mathematician and astronomer was the first man

to propound a heliocentric theory of the universe – eighteen centuries before Copernicus. He

made an attempt to compare the distance from Earth to the Sun and to the Moon. His reasoning

was perfectly sound but instrument he used to determine the angle of sight between the sun and

the half moon failed him by fault calibration. He found the distance to the sun to be about

eighteen - twenty times that to the moon, instead of the correct figure of approximately 390

times.

The first known attempt to calculate circumference of the Earth was made by

Eratosthenes of Alexandria. Having heard that, at the summer solstice, the sun was at zenith at

Aswan, he decided to determine the height of the sun also at Alexandria. From his measurement

he deduced, correctly, that the distance between Alexandria and Aswan must equal 1/50 th of the

Earth’s circumstances, but all his other data were inaccurate or pure guesswork. But remember

that this was approximately 1700 years before Columbus tried to find India, and found America.

Hippoarchus of Nicea (? – 127 BC) is also famous and most determination of the length

of the lunar month ( the time the moon use in one revolution around the Earth), to within one

second of today’s accepted value. As a mathematician, Hippoarchus introduced to Greek the

Babylonian method of dividing the circle into 360 degrees.

The most important work in the history of trigonometry and astronomy is Almagest,

which means the greatest of the great. It was written by the great mathematician and astronomer

Ptolemy of Alexandria in the second century AD. It is a thirteen-book mathematical collection,

and it contains over 1000 pages in modern edition. In the first book, there are tables of chords for

all arcs 0 – 180 , at 0,5 intervals to at least 5 places of decimals. The Almagest also



contains theorems corresponding to the present day law of sines. That so much of early Greek

work on astronomy has been lost, could be a result of the completeness and elegance of

presentation of the Almagest. The Almagest had become the basic textbook in astronomy for

more than a thousand years. Building further on the Almagest the Persian Abu al-Wafa

systematised theorems and proves of trigonometry and prepared extensive trigonometric tables

of sines and tangens.

The earliest work on spherical trigonometry was

Menelaus' Spherica(ca. 100 C.E.) . It included what is now called

Menelaus' theorem which relates arcs of great circles on spheres. Of

course, Menelaus stated his result in terms of chords, but in terms of

modern sines, his theorem reads

sin CE

sin EA =

sin CF

sin FD

sin BD

sin BA

andsin CA

sin EA =

sin CD

sin FD sin BF

sin BE



He proved this result by first proving the plane version, then "projecting" back to the sphere. The

plane version says

CE

EA

=

CF

FD

BD

BA

andCA

EA =

CD

FD BF

BE

Claudius Ptolemy's(ca. 100 - 178 C.E.) famous mathematical work was the Mathematike

Syntaxis (Mathematical Collection) usually known as the Almagest. It is primarily a work on

astronomy which included mathematical theory relevant to astronomy. It included trigonometric

table, a table of chords for angles from 1/2° to 180° in increments of 1/2°, the chords were

rounded to two sexagesimal places, about five digits of accuracy. He also included the geometry

necessary to construct the table. He computed the chord of 72°, an central angle of a pentagon, a

constructable angle. Along with the chord of 60° (the radius which Ptolemy took to be 60), that

gives crd 12°, then crd 6°, crd 3°, crd 1 1/2°, and crd 3/4°. He used interpolation to find crd 1°

and crd 1/2°.



Ptolemy proved the theorem that gives the sum and difference formulas for chords.

Theorem. For a cyclic quadrilateral (that is, a quadrilateral inscribed in a circle), the product of

the diagonals equals the sum of the products of the opposite sides.

AC BD = AB CD + AD BC

When AD is a diameter of the circle, then the theorem says crd AOC crd BOD =

crd AOB crd COD + d crd BOC. Where O is the center of the circle and d the diameter. If we

take a to be angle AOB and b to be angle AOC, then we have crd b crd (180° - a) = crd a crd

(180° - b) + d crd (b - a) which gives the difference formula

crd (b - a) =

crd b crd (180° - a) - crd a crd (180° - b)

d

With a different interpretation of a and b, the sum formula results:

crd (b + a) =

crd b crd (180° - a) + crd a crd (180° - b)

d

These, of course, correspond to the sum and difference formulas for sines.

In a modern presentation of trigonometry, the sine and cosine of an angle a are the y-

and x-coordinates of a point on the unit circle, the point being the intersection of the unit circle

and one side of the angle a; the other side of the angle is the positive x-axis. For this description

of trigonometry, we'll leave the radius unspecified as r and it's double, the diameter, we'll

denote d. The chord of an angle AOB where O is the center of a circle and A and B are two points

on the circle, is just the straight line AB. Chords are related to the modern sine and cosine by the

formulas

crd a = d sin (a/2)

sin a = (1/d) crd 2a

crd (180° - a) = d cos (a/2)

cos a = (1/d) crd (180° - 2a)

where a is an angle, d the diameter, and crd an abbreviation for chord.



4.0 Activity to determine the tangent, sine and cosine of an acute angle

Before defining the trigonometric functions, we must see how to relate the angles and

sides of a right triangle. A right triangle is

composed of a right angle, the angle at C, and

two acute angles, which are angles less than a

right angle. It is conventional to label the acute

angles with Greek letters. We will label the angle

at Bwith the letter θ ("THAY-ta"). And we will

label the angle atA with the letter φ ("fie").

As for the sides, the side AB, opposite the right angle, is called thehypotenuse ("hy-

POT'n-yoos"). Each acute angle is formed by the hypotenuse and the side adjacent to the

angle. Thus, angle θ is formed by the hypotenuse and side BC. Angle φ is formed by the

hypotenuse and side AC. With respect to angle θ, though, side AC is its opposite side. While

side BC is the side opposite φ.

Any two sides of the triangle will have a ratio a relationship to one another. It is

possible to form six such ratios: the ratio of the opposite side to the hypotenuse; the adjacent

side to the hypotenuse; and so on. Those six ratios have historical names and abbreviations,

with which the student will have to make peace. They are the following.

sine of θ = sin θ

=

opposit e hypotenuse

cosecant of θ

= csc θ

=

hypotenuse opposite

cosine of θ

= cos θ

=

adjace nt hypotenuse

secant of θ

= sec θ

=

hypotenuse adjacent


http://www.themathpage.com/aTrig/ratio-and-proportion.htm

http://www.themathpage.com/aTrig/theorems-of-geometry.htm#right


tangent of θ

= tan θ

= opposite adjacent

cotangent of θ

= cot θ

= adjacen topposite

Notice that each ratio in the right-hand column is the inverse, or thereciprocal, of the

ratio in the left-hand column.

The reciprocal of sin θ is csc θ ; and vice-versa.

The reciprocal of cos θ is sec θ.

And the reciprocal of tan θ is cot θ.

Each ratio moreover is a function of the acute angle. That is, one quantity is a

"function" of another if its value depends on the value of the other. The circumference of a

circle is a function of the radius, because the size of the circumference depends on the size of

the radius, and when the radius changes, the diameter also will change. As we will see in the

next topic, the value of each ratio depends only on the value of the acute angle. That is why

we say that those ratios are functions of the acute angle. We call them the trigonometric

functions of an acute angle. All of trigonometry is based on the definitions of those functions.

Problem 1.

Complete the following with either "opposite," "adjacent to," or "hypotenuse."

To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload").

a) In a right triangle, the side opposite the right angle is called the

a) hypotenuse.

b) CA is called the side opposite angle θ.

c) BC is called the side adjacent to angle θ.


http://www.themathpage.com/aTrig/definitions-trigonometric.htm#defs

http://www.themathpage.com/aTrig/trigonometry-of-right-triangles.htm


d) AC is called the side adjacent to angle φ.

e) BC is called the side opposite angle φ.

Problem 2.

The sides of a right triangle are in the ratio 3 : 4 : 5, as

shown. Name and evaluate the six trigonometric functions

of angle θ.

sin θ

=

4

5

csc θ

=

5

4

cos θ

=

3

5

sec θ

=

5

3

tan θ

=

4

3

cot θ

=

3

4

Problem 3.

The sides of a right triangle are in the ratio 8 : 15 : 17, as

shown. Name and evaluate the six trigonometric functions



of angle φ.

sin φ

=

1517

csc φ

=

1715

cos φ

=

8

17

sec φ

=

17 8

tan φ

=

15 8

cot φ

=

8

15

Notice that the sides of this triangle satisfy, as they must, the Pythagorean theorem:

8² + 15² = 17²

64 + 225 = 289

PLANE TRIGONOMETRY is based on the fact of similar figures.

Figures are similar if they are equiangularand the sides that make the equal angles are proportional.



For triangles to be similar, however, it is sufficient that they be equiangular. (Theorem

15 of "Some Theorems of Plane Geometry.") From that it follows:

Right triangles will be similar if an acute angle of one is equal to an acute angle of the other.

In the right triangles ABC, DEF, if the acute angle at B is equal to the acute angle at E,

then those triangles will be similar. Therefore the sides that make the equal angles will be

proportional. If CA is half of AB, for example, then FD will also be half of DE.

A trigonometric Table is a table of ratios of sides. In the Table, each value of

sin θ represents the ratio of the opposite side to the hypotenuse in every right triangle with

that acute angle. If angle θ is 28°, say, then in every right triangle with a 28° angle, its

sides will be in the same ratio. We read in the Table, sin 28° = .469

This means that in a right triangle having an acute angle of 28°, its opposite side is 469

thousandths of the hypotenuse, which is to say, a little less than half. It is in this sense that in a

right triangle, the trigonometric ratios the sine, the cosine, and so on are "functions" of the

acute angle. They depend only on the acute angle.

Example. Indirect measurement. Trigonometry is used typically to measure things that

we cannot measure directly.

For example, to measure the height h of a flagpole, we could measure a distance of,

say, 100 feet from its base. From that point P, we could then measure the angle required to

sight the top. If that angle (called the angle of elevation) turned out to be 37°, then

so that h = tan 37°


http://www.themathpage.com/aTrig/trigonometric-table.htm


http://www.themathpage.com/aTrig/theorems-of-geometry.htm#theorem15

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100

so that

so that h = 100 × tan 37°.

From the Table, we findtan 37° = .754

Therefore, on multiplying by 100,

h = 75.4 feet.

(Skill in Arithmetic: Multiplying and dividing by powers of 10.)

If we know the value of any one trigonometric function, then with the aid of the Pythagorean

theorem we can find the rest.

In a right triangle, sin θ = 5 13

Sketch the triangle, place the ratio numbers, and evaluate the remaining functions of θ.

To find the unknown side x, we have

x² + 5² = 13²

x² = 169 − 25 = 144.Therefore,

x = = 12.

We can now evaluate all six functions of θ:


http://www.themathpage.com/Alg/pythagorean-distance.htm#origin

http://www.themathpage.com/Alg/pythagorean-distance.htm#origin

http://www.themathpage.com/arith/multiply-by-powers-of-10.htm



sin θ = 5 13

csc θ = 13 5

cos θ = 1213

sec θ = 1312

tan θ = 5 12

cot θ = 12 5

Sine, Cosine, and Tangent

Sine, Cosine and Tangent are all based on a Right-Angled Triangle

Before getting stuck into the functions, it helps to give a name to each side of a right triangle:



"Opposite" is opposite to the angle θ

"Adjacent" is adjacent (next to) to the angle θ

"Hypotenuse" is the long one

Adjacent is always next to the angle

And Opposite is opposite the angle

The three main functions in trigonometry are Sine, Cosine and Tangent.

They are often shortened to sin, cos and tan.

To calculate them:

Divide the length of one side by another side ... but you must know which sides!

For a triangle with an angle θ, the functions are calculated this way:

Sine Function: sin(θ) = Opposite / Hypotenuse

Cosine Function: cos(θ) = Adjacent / Hypotenuse

Tangent Function: tan(θ) = Opposite / Adjacent

5.0 THREE ACTIVITIES TO DEMONSTRATE ON HOW TO FIND AND

ANGLE USING THE SCIENTIFIC CALCULATOR

ACTIVITY 1



Introduction: In this lesson, three trigonometric ratios (sine, cosine, and tangent) will be defined

and applied. These involve ratios of the lengths of the sides in a right triangle.

In a right triangle, one angle is 90º and the side across from this angle is called the hypotenuse.

The two sides which form the 90º angle are called the legs of the right triangle. We show a right

triangle below. The legs are defined as either “opposite” or “adjacent” (next to) the angle A.

We shall call the opposite side “opp,” the adjacent side “adj” and the hypotenuse “hyp.”

Definitions: In the following definitions, sine is called “sin,” cosine is called “cos” and tangent

is called “tan.” The origin of these terms relates to arcs and tangents to a circle.

i. sin(A) =

ii. cos(A) =

iii. tan(A) =

For example, in the triangle below, in relation to angle θ, opp = 5, adj = 6, and by the

Pythagorean Theorem, hyp = .

By the three definitions we have:



sin(θ) =

cos(θ) =

tan(θ) =

Often, fractions involving radicals are rewritten (Radical Simplifying) so that there is not

a radical expression in the denominator. Then we have sin(θ) = and cos(θ) = .

We can also find the measure of the angle θ when we know any of these three trigonometric

ratios. In this case, tan(θ) = . Using a calculator, we can determine what angle has this

tangent. Using a TI-83 calculator, we press and get

Let's Practice:

i. In the triangle below, what is the sin(θ) and what is the length of the adjacent side? What

is the measure of angle θ?

The adjacent side can be found using the Pythagorean Theorem and is . The sin(θ)

= Using we get . Notice that the adjacent side is . If we

had used we would still get .

ii. In the triangle below, what is the length of side x?


http://www.algebralab.org/lessons/lesson.aspx?file=algebra_radical_simplify.xml


We know that . Therefore, We solve

this equation by multiplying by 6 and get x = 2.9088.

iii. In the triangle below, what is the length of the hypotenuse x?

41º

7

Since 7 is the length of the adjacent side, we have cos(41º) = . Therefore,

cos(41º) . We solve this equation by multiplying by x and dividing by 0.7547. We

get .

ACTIVITY 2

Are you having trouble remembering angles in trigonometry? With all of the different angles in trigonometry it can be hard to remember what the exact value of each angle is. Here is an easy way to find the exact value of the basic angles in trigonometry without remembering the whole table.

Difficulty:

Moderate



InstructionsThings You'll Need

Calculator Pencil Paper

1.

o 1First you need to know the three basic Trigonometric functions which are sine cosine and tangent. Also make sure your calculator is set in degrees and not radians.

o 2

Remember that there are two known triangles in Trigonometry. The known triangles are the 30-60-90 triangle which has a 1,2,√3 ratio, and the 45-45-90 triangle which has a 1,1,√2 ratio.

o 3

For example: You are asked to find the exact value cos (√3/2). Type in your calculator ((√3)/2) and you'll get 0.866025404. Now to find the exact value of 0.866025404 you need to plug in your known angles 30, 45, 60, and 90. After trying the known angles you will find that cos(30) = 0.866025404. That means sin (√3/2)= sin(30) = ╥/6.

o 4

Now if you were asked to find sin (√2/2) or sin (1/√2). Type in your calculator √(2)/2 and press enter you'll get 0.707106781. Now to find the exact value of 0.707106781 you plug in your known angles



again 30,45,60,90. type in your calculator sin(30) which is 0.5 so we can eliminate 30 degrees. Now try sin(60) which equals 0.866025404 so we can eliminate 60 degrees. Now try sin(45) which equals 0.707106781. That means sin(√(2)/2) (or sin (1/√2)) = sin (45) = ╥/4.

o 5After using sine and cosine to find exact values of angles now we will use tangent. Lets say you were asked to find tan(√3/3) or tan(1/√3). Again type in your calculator √(3)/3 and you will get 0.577350269. Now try plugging in your known angles to find the exact value. Try plugging in 30, 45, 60, and 90. you'll notice that tan(30) = 0.577350269 which means tan(√3/3)(or tan(1/√3)) = ╥/6.

ACTIVITY 3

Trigonometry uses triangles and the angles between their sides to compute a variety of useful information. The mathematics of triangles is used in architecture to calculate where the sun will shine within a structure as the seasons change; it is also applied to figure the slope of a roof.

The Pythagorean theorem says that for a right triangle -- one with a 90-degree angle between two of its sides -- the length of the sloped side squared is equal to the sum of the squared lengths of the other two sides.

Difficulty:



Moderately Easy

InstructionsThings You'll Need

Calculator with trigonometric functions Protractor

1. Draw Right Triangle

o 1Use the flat side of a protractor to draw a right triangle that measures 3 inches high by 4 inches long by 5 inches on the sloped side. Label the vertical portion "a" the bottom "b" and the remaining sloped side "c." Note that "c" is called the hypotenuse of the triangle.

o 2Use the calculator to prove to yourself that a^2 plus b^2 equals c^2: 9 + 16 = 25.

o 3Try drawing several more right triangles. Measure the sides on each one and crunch the numbers. Use the square root function on a calculator to solve for triangles that don't come out to round numbers. Verify that all adhere to the formula. If they don't, you measured incorrectly.

2. Using Sines

o 1Look at the first triangle you drew with sides measuring 3, 4 and 5. Label the angle between the hypotenuse c and the "bottom" b of the triangle as A. Label the opposite angle between sides c and a as B. Divide the length of the side directly opposite angle A (side a) over the hypotenuse of the triangle. Write down the answer: You will get 0.6.

o 2Understand the following relationship: Sine equals the length of the opposite side (from the angle in question) divided by the hypotenuse. Reverse this concept using the inverse sine function on your calculator to solve for the angle: Enter the number "0.6" in your calculator and press the "arc-sin" or "sin -1" key. Examine the display to verify that it reads "36.86." This is the angle A in degrees -- the angle between sides c and b.

o 3Determine the remaining angle B inside the triangle. Add the right angle, which is 90 degrees to the angle A, which is 36.86 degrees. Subtract this number from 180 to solve for angle B. Verify that your answer reads 53.14 degrees. Write down this formula: sin = opposite/hypotenuse.

3. Using Cosines



o 1Use the protractor to draw a right triangle with sides 10, 24 and 26. Label the vertical side "a," the bottom "b" and the hypotenuse "c."

o 2Use the formula cos B = sin A to understand the relationship between sines and cosines. Because the three angles together must equal 180 degrees, and the right angle takes up 90 degrees, the angles A and B adjacent to the hypotenuse will always add up to 90 degrees. In a similar fashion, the sine and cosine of angle A or B are related through the equation sin A = cos B.Note that just as the sine of an angle is given by the length of the opposite side (from the angle) divided by the hypotenuse, the cosine is given by the adjacent side divided by the hypotenuse. Write down this formula: cos = adjacent/hypotenuse.

o 3Explore this concept further. Use your calculator to compute the cosine of the angle A for the triangle where a = 10, b = 24 and C = 26. Check your answer: It should read 0.923.

o 4Take the inverse cosine of this angle to get 22.62. Use the 180 degrees formula: 180 - 90 - 22.62 = 67.38. This is angle B.

Use the relationship cos A=sin B to find another way to arrive at this result. Note that the sine of angle B is also 0.923, the cosine of angle A.

6.0 REFLECTION

Educational technology is recognized as an essential component of the instructional

process. In particular, scientific calculator has emerged as a useful tool for teaching and learning

of Mathematics. This was intended to avoid wastage of time in solving mathematical problems

and improve students’ performance as before. This has raised questions as to the potential

contribution of scientific calculators to the teaching and learning of Mathematics as performance


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did not improve with their use. It was also not clear what challenges were faced by users of this

gadget. The purpose of this study therefore was to establish challenges and benefits that may

result from the use of scientific calculators in the teaching and learning of Mathematics.

Mathematics is a compulsory subject in the current curriculum. It involves the process of

manipulating mathematical procedures and algorithms concerning specified instructions and

axioms relating to mathematical concepts under investigation. This calls on the learner to be able

to think logically, present information in logical steps and proofs of theorems systematically.

Several studies have been carried out to find out the real cause of the poor performance in the

subject with no tangible results. It is widely believed that performance in the subject depends

largely on the effective teaching and learning process.

Ouko (2004) notes that for the last ten years, performance in national examinations

indicates that, national examinations mean grade for Mathematics has been grade E which is a

score below 20%. The same was noted by Aduda (2005), and Kituku (2004). A report from the

KNEC newsletter found out that, the greatest challenge the learners face in the process of

examination was time management. The report further noted that during examinations,

candidates were not able to complete both Mathematics examination papers on time, which

greatly contributes to the poor performance in the subject. In line with this, factors like

teaching methods, attitude of both the teacher and the learner and teaching and learning

resources in Mathematics have been addressed through studies, yet the root cause of this

performance has not been discovered. Of much focus was the use of teaching and learning aids

in the learning process.

The study established that, there are various challenges linked to scientific calculator use

in Mathematics in Mathematics class. These included provision of scientific calculators to the

learners. On provision, it was established that parents and guardians were to provide learners

with scientific calculators. This was a problem because the study established that, most learners

did not have scientific calculators because of poverty. In addition, since most learners did not

access scientific calculators frequently, they were unable to use scientific calculators effectively

in Mathematics, especially when provided with one during examination time. In addition, there

were learners with sight problems who were not able to use the calculators effectively. These



learners were grouped as those with special needs, under an integrated system of learning. Time

management for this group of learners was a real problem. These challenges should be addressed

to make the teaching and learning of Mathematics effective.

The research was conceptualized on the benefits and difficulties in the use of scientific

calculators in the teaching and learning of Mathematics. To address this problem, there was

need to introduce technology in Mathematics that could aid learners in time management in their

examinations. Much of technology commonly in use in the teaching and learning process

included visual aids (charts, hand-outs, pictures, photographs, cut-outs, models, flash cards,

muted videos and drawings), audio aids (radio and cassette recorded sounds) and audio-visual

aids (video and film clips) have been used in Mathematics, mainly to facilitate concept

understanding of facts, which would otherwise be thought abstract to the learner. The afore-

mentioned media in Mathematics are specifically designed to make learners understand concepts

faster, than merely observing the teacher manipulate mathematical procedures and algorithms.

This means that their integration in Mathematics makes teaching and learning process effective

but not to aid in time management. The difficulties include inability to use scientific calculators

properly in Mathematics under topical requirements, de-linking the mind of the learner from

basic computation capabilities, accessibility to scientific calculators during Mathematics learning

process and sight problem in some of the learners.

Although, a study by Ambuko (2008) points out that, availability of various media

resources and their advantages in classroom instruction has necessitated the integration of such

media in the teaching and learning process, these media may present difficulties to the teachers

and learners in the learning process, such as accessibility, negative attitudes and lack of training.

Therefore, challenges may arise from the methods used in the teaching and learning of

Mathematics using a scientific calculator. Difficulties may also arise from a situation where a

learner does not access a scientific calculator in the teaching and learning process. This may

affect attitude of both the teacher and the learner towards the use of scientific calculators in

Mathematics.

To effectively use calculator technology in a Mathematics learning process, each learner

in the class should have access to a calculator during learning sessions. The calculators can be



provided to the learners through a system that ensures all learners have the gadget. Whereas the

study by Burrill was conducted to compare the availability of scientific calculators between the

rural and urban schools, this study sought to establish the extent to which learners accessed

scientific calculators during the teaching and learning of Mathematics in secondary schools in

Emuhaya District.

The aspect of time management especially when it comes to computations was addressed

by the introduction of Slide rules and Mathematical tables. They were integrated in the teaching

and learning process in secondary school Mathematics both as topical requirement and purely as

tools to aid in computations. Learners used to spend a lot of time in computations at the expense

of mathematical processes when confined to both slide rules and mathematical tables. The

integration of the calculators was to impact positively on the teaching and learning process by

reducing drudgery of applying arithmetic and algebraic procedures and improving

manipulative skills.

According to the National Council of Teachers of Mathematics (NCTM, 1989 (a)), the

use of calculators along with traditional paper-and-pencil instruction enhances the learning of

basic skills. This is concurred by Roberts (1991) who noted that the integration of the calculator

into the school Mathematics program should be at all grade levels in class work, homework, and

evaluation" (Roberts, 1991, p.51). In addition, Roberts observes that the use of calculators should

not eliminate the teaching of the basic algorithmic skills and processes of Mathematics. It should

be properly integrated to reinforce the basic concepts, that are being taught and to aid in the

application of these Mathematics processes in the real-world situations. The implication is that

teacher supervised activities relating to the mathematical concepts being learned are reduced.

Learning therefore becomes student-centered.

The study concludes that, calculator use can benefit the learner and the teacher. This is

observed when the learners are actively involved in the learning process with minimum teacher

supervision and direction. The benefits include making Mathematics concepts well understood,

increasing the mastery of computing skills and amount of calculations, displaying accurate

answers on the screen and using it to confirm answers, motivate learners to want to work more,

and are convenient for confidential working for those who know how to use the calculator. These

benefits should be exploited to increase the number of learners who are proficient in the use of



the calculator to make teaching and learning more effective and learner centred. This will go

towards improving performance in the subject.

The benefits included making Mathematics concepts well understood, increasing the

mastery of computing skills and amount of calculations, as well as using it to confirm answers,

displaying accurate answers on the screen, motivating learners to want to work more and the

calculator being convenient for confidential working.

To solve these problem or to encourage pupils The Ministry of Education should fund the

purchase of scientific calculators for all learners at secondary school level, to ensure every

learner accesses a scientific calculator during the learning sessions and during examination

times. This should be carried out under similar scheme of Free Secondary Education (FSE)

currently underway. District policy on calculator technology acquisition use and integration in

Mathematics should be formulated. Provide a calculator that caters for learners with sight

problems and schools should set up Mathematics laboratory (Technology in Education Centre)

that pays special attention to practical sessions in Mathematics, to promote participatory and

collaborative learning in the use of scientific calculators in Mathematics. This will reduce the

role of the teacher to a facilitator rather than a provider of knowledge.

Refresher courses on the use of calculator technology to be regular at the district level, to

upraise teachers’ knowledge and skills on calculator use while at the same time developing

positive attitude, toward calculator technology integration in Mathematics. The government

should set out guidelines to suppliers on the quality of calculator expected to be supplied to the

learners. The Ministry of Education should source funds from development partners and donors

in the education sector, to be able to purchase calculators for every learner, as well as

formulating a policy on provision of scientific calculators to the learners, to eliminate problems

associated with accessibility and use of scientific calculators in Mathematics in secondary

schools, to ensure equitable accessibility to calculators by the learners.

The Ministry of Education should organize seminars that concern calculator use in

Mathematics and other related disciplines across the secondary school curriculum for teachers.

Do away with topical requirement of scientific calculator, since this tends to make teachers and

learners spend a lot of time mastering calculator manipulative skills, instead of using calculators

as tools to aid computations. The manufacturers through the Ministry of Education to provide a

calculator that caters for learners with sight problems; a calculator made with ear-phones could



be ideal where sounds will be produced upon pressing the relevant keys (talking calculators).

More materials on calculator technology to be supplied to schools as teaching and learning aids,

to cater for individual differences amongst learners. A special panel set up to design and produce

materials related to calculator technology use, such as texts and charts to be accessed by the

learners in specially designed places like Mathematics labs, classroom walls displays.

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Van Brummelen, G. (2009) The Mathematics of the Heavens and the Earth: The Early History of

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Evans, J. (1998) The History and Practice of Ancient Astronomy. Oxford.O.U.P.

Leo Rogers. (2010). History of Trigonometry . Speacilists in Rich Mathematics.

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Katz, V. (1998) A History of Mathematics. New York. Addison Wesley.

Katz, V. (Ed.) (2007) The Mathematics of Egypt, Mesopotamia, China, India, and Islam.

Princeton. Princeton University Press.


http://nrich.maths.org/6853%E2%88%82=

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