1 what you will learn a short history of trigonometry how to convert decimal degree measures to...

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What you will learn

A short history of Trigonometry How to convert decimal degree

measures to degrees, minutes and seconds, and vice versa

How to find the number of degrees in a given rotation

How to identify angles that are coterminal with another angle

How to find a reference angle

Objective: 5-1 Angles and Degree Measure

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A Brief History of Trigonometry

Trigonometry was originally created by the Greeks to aid in the study of astronomy. Hipparchus of Bithynia (190-120 B.C.) tabulated trigonometric ratios, to enable the calculation of a planet's position. Ptolemy continued some of this early work.

The Chinese, in the medieval times, studied astronomy, and hence, trigonometry. They introduced the tangent function. However, most of their work was in the field of astronomy, and many of their trigonometric advancements were not continued.


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History (continued)

The Indians were the next to advance the study of trigonometry. They developed their own sine tables, using the Greek half-angle formula. Later, the cosine table was also constructed. Techniques of approximation to a relatively high accuracy were also introduced.

The Indian works were translated and read by the Islamic mathematicians, who also worked on trigonometry. Similar to the Greeks and Indians, they related trigonometry and astronomy. The Indian sine was used, as well as the chord. The cosine was also formally introduced, by Abu Abdallah Muhammad ibn Jabir al-Battani.


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History (continued)

Trigonometry reached Europe in the medieval times. Richard of Wallingford wrote a text on trigonometry, Quadripartium. In the 16th century, trigonometry was incorporated into geography and navigation. Knowledge of trigonometry was used to construct maps, determining the position of a land mass in relation to the longitudes and latitudes.

Johannes Muller wrote a text On Triangles. He studied plane trigonometry, including results for solving triangles.


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Some Fun Definitions

Vertex

Terminal Side

Initial Side


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More Angle Information

An angle measure provides information about the direction of rotation of a terminal side and the amount of rotation.

- If the rotation is in a counterclockwise direction, the angle formed is a positive angle.

- If the rotation is clockwise, the angle is negative.


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Angles in the Coordinate Plane

x

y

All of these angles are in standard position. The initial side is on the x-axis.

+120o

-120o x

y

x

y

+90o


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Angle Measures We use degrees to measure angles.

Degrees can then be divided into minutes and seconds.

Each minute is 1/60 of a degree.

Each second is 1/60 of a minute or 1/3600 of a degree.


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Converting Decimals to Degrees, Minutes, Seconds

Example: Change north latitude 15.735o to

degrees, minutes and seconds.


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You Try Convert 329.125o to degrees, minutes,

and seconds.


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Converting Degrees, Minutes, Seconds to Decimal

Write north latitude 39o5’34” to a decimal rounded to the nearest thousandth.


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You Try If the bearing from the plane to tower B

is 35o12’7”, write the bearing as a decimal rounded to the nearest thousandth.


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Quadrantal Angles

If the terminal side of an angle that is in standard position coincides with one of the axes, the angle is called a quadrantal angle.

x

y

+90o


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Rotations

A full rotation is 360 degrees.

A measure of more than 360 degrees represents more than one rotation.


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Example Give the angle measure represented by

each rotation.1. 5.5 rotations clockwise:

2. 3.3 rotations counterclockwise:


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You Try Give the angle measure represented by

each rotation.1. 9.5 rotations clockwise.

2. 6.75 rotations counterclockwise.


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Coterminal Angles

Two angles in standard position are called coterminal angles if they have the same terminal side. Since angles differing in degree measure by multiples of 360 degrees are equivalent.

x

y


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Math Definition If is the degree measure of an angle, then

all angles measuring + 360k, where k is an integer; are coterminal with .

Example: Any angle coterminal with 75o can be written 75o + 360ko, where k is the number of rotations around the circle.


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Example Identify all angles that are coterminal

with each angle. Then find one positive angle and one negative angle coterminal with the angle.

1. 45o

2. 225o

x

y


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You Try Identify all angles that are coterminal

with each angle. Then find one positive angle and one negative angle that are coterminal with the angle.

1. 86o

2. 294o


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More Fun If each angle is in standard position,

determine a coterminal angle that is between 0o and 360o. State the quadrant in which the terminal side lies.

1. 775o

2. -1297o


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You Try If each angle is in standard position,

determine a coterminal angle that is between 0o and 360o. State the quadrant in which the terminal side lies.

1. 595o

2. -777o


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VERY IMPORTANT – Reference Angles If is a nonquadrantal angle in standard

position, its reference angle is defined as the acute angle formed by the terminal side of the given angle and the x-axis.


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The Math Definition For an angle , 0o< <360o its reference

angle is defined by:

1. when the terminal side is in Quadrant I

2. 180o - when the terminal side is in Quadrant II

3. - 180o when the terminal side is in Quadrant III

4. 360o - when the terminal side is in Quadrant IV.


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Example Find the measure of the reference angle

for each angle:1. 120o

2. -135o


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You Try Find the measure of the reference angle

for each angle.1. 312o

2. -195o


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Homework Page 281, 18, 20, 24, 26, 30-58 even

1 what you will learn a short history of trigonometry how to convert decimal degree measures to...

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