trigonometry 101
DESCRIPTION
types of angle, angle measurement, pythagorean theorem, trigonometric function, trigonometric relationship, circle function, co function, reference angle, odd even function,graphing of trigonometric function, special angles and terminology and history of trigonometryTRANSCRIPT
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TRIGONOMETRY MATH 102
2ndSEM/SY2014-2015
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Consultation Time: 2:30 4:30 PM
Main Book:
Algebra and Trigonometry by Loius Liethold
Reference Book:
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VMG
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UM Core Values
Excellence
Honesty and Integrity
Teamwork
Innovation
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Course Description
Trigonometric functions; identities and
equations; solutions of triangles; law of sines;
law of cosines; inverse trigonometric
functions; spherical trigonometry
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Course Objectives
After completing this course, the student must be able to:
1. Define angles and how they are measured;
2. Define and evaluate each of the six trigonometric functions;
3. Prove trigonometric functions;
4. Define and evaluate inverse trigonometric functions;
5. Solve trigonometric equations;
6. Solve problems involving right triangles using trigonometric
function definitions for acute angles; and
7. Solve problems involving oblique triangles by the use of the sine
and cosine laws.
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Course Outline
1. Trigonometric Functions
1.1. Angles and Measurement
1.2. Trigonometric Functions of Angles
1.3. Trigonometric Function Values
1.4. The Sine and Cosine of Real Numbers
1.5. Graphs of the Sine and Cosine and Other Sine Waves
1.6. Solutions of Right Triangle
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Course Outline
2. Analytic Trigonometry
2.1. The Eight Fundamental Identities
2.2. Proving Trigonometric Identities
2.3. Sum and Difference Identities
2.4. Double-Measure and Half-Measure Identities
2.5. Inverse Trigonometric Functions
2.6. Trigonometric Equations
2.7. Identities for the Product, Sum, and Difference of
Sine and Cosine
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Course Outline
3. Application of Trigonometry
3.1. The Law of Sines
3.2. The Law of Cosines
4. Spherical Trigonometry
4.1. Fundamental Formulas
4.2. Spherical Triangles
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TRIGONOMETRY
A branch of Geometry
Developed from a need to compute angles
and distances
Until about the 16th century, trigonometry
was chiefly concerned with computing the
numerical values of the missing parts of a
triangle when the values of other parts were
given.
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Branches of TRIGONOMETRY
Plane
Problems involving angles and distances in one
plane/flat surfaces
Spherical-
Applications to similar problems in more than
one plane of three-dimensional space
curved surfaces
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Plane vs Spherical
The sum of the angles of a spherical triangle is always greater than 180
In the planar triangle the angles always sum to exactly 180.
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Application of Trigonometry
Carpentry
Mechanics
Machine work
Astronomy
Land survey and
measurement
Map making,
Artillery range
finding.
And others
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Greek Word (origin)
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History of TRIGONOMETRY
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History of Trigonometry
Several ancient civilizationsin particular, the Egyptian, Babylonian, Hindu, and Chinesepossessed a considerable knowledge of practical geometry, including some concepts that were a prelude to trigonometry.
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The Rhind papyrus, an Egyptian collection of 84 problems in arithmetic, algebra, and geometry dating from about 1800 BC, contains five problems dealing with the seked.
History of Trigonometry
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For example, problem 56 asks: If a pyramid is 250 cubits high and the side of its base is 360 cubits long, what is its seked? The solution is given as 51/25 palms per cubit; and since one cubit equals 7 palms, this fraction is equivalent to the pure ratio 18/25.
History of Trigonometry
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History of Trigonometry
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Trigonometry began with the
Greeks.
Hipparchus (c. 190120 BC) was
the first to construct a table of
values for a trigonometric function. Astronomer
founder of trigonometry
History of Trigonometry
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He considered every triangleplanar or sphericalas being inscribed in a circle, so that each side becomes a chord (that is, a straight line that connects two points on a curve or surface.
History of Trigonometry
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To compute the various parts of the triangle, one has to find the length of each chord as a function of the central angle that subtends itor, equivalently, the length of a chord as a function of the corresponding arc width.
History of Trigonometry
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The first major ancient work on trigonometry to reach Europe intact after the Dark Ages was the Almagest by Ptolemy (c. AD 100170).
He lived in Alexandria, the intellectual centre of the Hellenistic world, but little else is known about him.
History of Trigonometry
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History of Trigonometry
Chapters 10 and 11 of the first book of the Almagest deal with the construction of a table of chords, in which the length of a chord in a circle is given as a function of the central angle that subtends it, for angles ranging from 0 to 180 at intervals of one-half degree.
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Ptolemy used the Babylonian sexagesimal numerals and numeral systems (base 60), he did his computations with a standard circle of radius r = 60 units.
History of Trigonometry
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The next major contribution to trigonometry came from India.
The first table of sines is found in the ryabhaya.
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History of Trigonometry
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Its author, ryabhaa I (c. 475550), used the word ardha-jya for half-chord, which he sometimes turned around to jya-ardha (chord-half); in due time he shortened it to jya or jiva.
Later, when Muslim scholars translated this work into Arabic, they retained the word jiva without translating its meaning.
History of Trigonometry
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Thus jiva could also be pronounced as jiba or jaib, and this last word in Arabic means fold or bay.
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History of Trigonometry
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History of Trigonometry
When the Arab translation was later translated into Latin, jaib became sinus, the Latin word for bay.
The word sinus first appeared in the writings of Gherardo of Cremona (c. 111487), who translated many of the Greek texts, including the Almagest, into Latin.
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History of Trigonometry
Other writers followed, and soon the word sinus, or sine, was used in the mathematical literature throughout Europe.
The abbreviated symbol sin was first used in 1624 by Edmund Gunter, an English minister and instrument maker.
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The first table of tangents and cotangents was constructed around 860 by abash al-sib (the Calculator), who wrote on astronomy and astronomical instruments.
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History of Trigonometry
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Another Arab astronomer, al-Bttni (c. 858929), gave a rule for finding the elevation of the Sun above the horizon in terms of the length s of the shadow cast by a vertical gnomon of height h.
Al-Bttni's rule, s = h sin (90 )/sin , is equivalent to the formula s = h cot .
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History of Trigonometry
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Based on this rule he constructed a table of shadowsessentially a table of cotangentsfor each degree from 1 to 90.
It was through al-Bttni's work that the Hindu half-chord functionequivalent to the modern sinebecame known in Europe.
History of Trigonometry
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The first definition of a spherical triangle is contained in Book 1 of the Sphaerica, a three-book treatise by Menelaus of Alexandria (c. AD 100) in which Menelaus developed the spherical equivalents of Euclid's propositions for planar triangles.
History of Trigonometry
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Several Arab scholars, notably Nar al-Dn al-s (120174) and al-Bttni, continued to develop spherical trigonometry and brought it to its present form.
s was the first (c. 1250) to write a work on trigonometry independently of astronomy.
History of Trigonometry
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History of Trigonometry
But the first modern book devoted entirely to trigonometry appeared in the Bavarian city of Nrnberg in 1533 under the title On Triangles of Every Kind.
Its author was the astronomer Regiomontanus (143676).
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History of Trigonometry
On Triangles was greatly admired by future generations of scientists; the astronomer Nicolaus Copernicus (14731543) studied it thoroughly, and his annotated copy survives.
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History of Trigonometry
The final major development in classical trigonometry was the invention of logarithms by the Scottish mathematician John Napier in 1614.
His tables of logarithms greatly facilitated the art of numerical computationincluding the compilation of trigonometry tablesand were hailed as one of the greatest contributions to science.
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History of Trigonometry
Leonhard Euler
Established the modern trigonometry
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Direction of Angles
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Angles
The opening between two
straight lines drawn from
a single point
The lines are called Sides
The point where they
meet is called Vertex
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TERMINOLOGY
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Adjacent Angles
Two angles having same
Vertex and one common Side
Notation
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TERMINOLOGY
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Coterminal Angles
Two angles have the same
initial and terminal sides
Coterminal angles = 2 -
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TERMINOLOGY
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When to straight lines
meet with other straight
lines as to make two
adjacent equal angles, the
lines are said to be
Perpendiular and each of
the adjacent angles is
called Right Angle
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TERMINOLOGY
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Types of Angles
Acute Angle
Smaller than right angle
Obtuse angle
Greater than right angle but less than two right
angle
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Types of Angles
Complementary angle
The sum of two angles is equal to right angle
Supplementary angle
The sum of two angles is equal to two right
angles
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Terminology
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O
B
r
r A
Chord,
AOB is the angle
subtended at O
by Central Angle,
- Subtended by a
chord
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Angle Measures and Unit
Angle is dependent on the direction of the
sides
Unit of Measurement
Degree System
Radian System
Gradient System
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Degree System First Developed by the Babylonians
Sexagesimal System
Believe that
The four season of the earth repeated
themselves
The sun completed a circuit around the
heavens among the stars in 360 days
1 circle = 360 days = 360 steps or grade
1 circle = 4 season = 4 quadrant
Fourth Quadrant
First Quadrant Second Quadrant
Third Quadrant
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Degree System
A
1 circle = 6 sextant
1 sextant = 60 degree
1 degree = 60 minute
1 minute = 60 seconds
= =
O
B C
r
r r
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Radian System
Circular system
O A
B
= =
= ,
,
r
r
=
2
2 =C = 360
=180
1 =180/
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Gradient System
Conceptualized by the French
1 circle = 400 part =400 Grades
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Measure of Usual Angles
Right Angle
90 degrees
Straight Angle
180 degrees
First Quadrant Second Quadrant
Fourth Quadrant Third Quadrant
2700
1800
900
00
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Problem Solving! Area of triangle = base*height
2 = 2 + 2
O
A B
r r
C C
D
2= 2 +2
2 - 2 = 2
= 2 22
----height
Area of triangle = 2c* 2 22
Area of triangle = c* 2 22
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O
A B
r r
Arc, S
,
, = , 2
2 = 2
S=
Problem Solving!
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Remember!!
1 circle = 360O = 2 = 400 Grades
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Sample Problem 1
75 degrees
=________radians
=________grades
=coterminal angles:____________
=supplementary angle:
=complementary angle:
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Sample Problem 2
350 grades
=________radians
=________degrees
=coterminal angles:____________
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Sample Problem 3
/3 grades
=________Grades
=________degrees
=coterminal angles:____________
=supplementary angle:
=complementary angle:
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TRIANGLES
Formed by three
intersecting lines at
three points
Three sides
Three angles
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Part of the triangle
Base
The side where the triangle
supposed to stand
Altitude
A line drawn perpendicular to the
base and through the opposite
vertex.
Base
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Part of the triangle with respect to
reference angle
Adjacent side
Side near the reference angle
Opposite side
Side opposite to the reference
angle
Hypotenuse (right triangle only)
The longest length of the three
sides
Adjacent
O
p
p
o
s
i
t
e
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Types of Triangles according to
angles
Right
One of the angles is a
right angle
Oblique
Has no right angle
Obtuse
When one of the angle is obtuse
Acute
If all of the angles are acute
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Isoceles
Has equal two sides
Equilateral
(equiangular)
Three sides are equal
Scalene
No two sides are
equal
Types of Triangles according to
sides
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Important Proof
GH is transversal
CHE and BGF or DHE and
AGF are alternate interior
angles
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Properties of Triangle
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Pythagorean Theorem
Sum of area of square
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Trigonometric function of angles
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Sine
Complementary
Sine
Cosine
Secant Cosecant
Tangent
+ = 90
Remember: Cotangent
Secant: Latin "secant-, secans" from Latin
present participle of "secare" (to cut)
Sine: (jaib) Half-Chord
Tangent: Latin "tangent-, tangens" from
present participle of "tangere" (to touch)
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Trigonometric function of angles
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sine
cosine
secan
tangent
cosecant
cotangent
sine
cosine
secan
tangent
cosecant
cotangent
Sine
Cosine
Secant Cosecant
Tangent
Cotangent
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Trigonometric function of angles
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sin =
cos =
tan =
Cosine
Secant
Tangent
Cotangent
Sine
csc =
cot =
sec =
Cosecant
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Trigonometric function
and relations
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sin =
cos =
tan =
csc =
cot =
sec =
sin = 1
csc
= 1
sec
= 1
cot
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Special Angle 45O
45O
From Pythagorean Theorem:
c2 = a2 + b2
45O
1
1
b =
= a
c2 = 12 + 12
c = 1 + 1
c = 2
Sin 45O = 1
2
Cos 45O =
1
2
Tan 45O = 1
1 = 1
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Special Angle 60O, 30O
60O
From Pythagorean Theorem:
= c
= b
12 = a2 + ()2
a2 = 1-1/4 a =3
4
Sin 30O =
1
2
1 = 1/2
Cos 30O =
3
2
1 =
3
2
Tan 30O = 1/2
3
2
= 1
3
60O
30O 30O
1
1 1
1/2 1/2
a =3
2
Sin 60O =
3
2
1 =
3
2
Cos 60O =
1
2
1 = 1
2
Tan 60O =
3
21
2
= 3
c2 = a2 + b2
a
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Hand Technique
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Sign Convention
A -
+
-
+
Sine = +
Cosine = + Tangent = +
Sine = +
Cosine = - Tangent = -
Sine = -
Cosine = - Tangent = +
Sine = -
Cosine = + Tangent = -
90O
180O 1,0
0O
270O
-1,0
0,1
0,-1
Unit Circle = radius is 1 A S
T C
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COFUNCTION THEOREM
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COFUNCTION IDENTITIES
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Sample
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sin30O = cos(90 - 30O)
sin30O = cos(60)
tan x = cot(90 - x) csc 40 = sec (90 - 40)
csc 40 = sec (50)
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Even and Odd Function
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Reference Angle,
Is the acute angle
formed by the
terminal side of and
the horizontal axis
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Sample
Find the exact value of cos 210O
- Solution: 210 is located at III quadrant
Reference Angle 210 - 180 = 30
cos 30 = 3
2
cos 210 = - 3
2 (210 is located at III quadrant)
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Sample
Find the exact value of tan 495O
- Solution:
Reference Angle: 495 - 360 = 135
180 - 135 = 45
tan 45 = 1
tan 45 = - 1 (135 is located at II quadrant)
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Samples
Find the exact values for
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Circular Functions
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More Sample
Determine the exact values
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More Sample
Determine whether the following statements
are true
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The Graphs of Sinusoidal Functions
The following are examples of things that
repeat in a predictable way
heartbeat
tide levels
time of sunrise
average outdoor temperature for the time of
year
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Periodic Function
A function f is called a periodic function if there
is a positive number p such that f(x + p) = f(x)
for all x in the domain of f
If p is the smallest such number for which this equation
holds, then p is called the fundamental period
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sine, cosine, secant, and cosecant functions
have fundamental period of 2 , but that
tangent and cotangent functions have
fundamental period .
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Sine Graph
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Sine Function
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Cosine Graph
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Cosine Function
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PERIOD OF SINUSOIDAL FUNCTIONS
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Finding the Period of a Sinusoidal Function
y = cos(4x)
y = sin (1/3x)
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STRATEGY FOR SKETCHING GRAPHS
OF SINUSOIDAL FUNCTIONS
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Sample
Graph y=3sin(2x)
Step1 A=3, B = 2, p =2/B -----p=
Step2 p/4------- p = /4
Step3
Make a table starting at x=0 to the period x=
in steps of /4
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Sample
Step3
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Sample
Step4 Step5
p5
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Sample
Step6
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Sample
Graph -2cos(1/3x)
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Good Luck for the FIRST EXAM
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Exercise
101
Evaluate the following
expression exactly
Graph the given function
over the given period