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Chapter 3. Trigonometric Functions of Angles Section 3.1 Angle Measure.

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An angle is formed when two rays are joined at a common endpoint. The point where they join is called the vertex. The ray where the and begins is called the initial side and the ray where it ends is called the terminal side.

To measure the angle is to associate a number with it in a consistent manner. The two most common ways to do that are using degrees and radians as units of measure.

If the direction from the initial side to the terminal side is counterclockwise the angle measure is positive. If the direction from the initial side to the terminal side is clockwise the angle measure is negative.

If the initial side of the angle is a horizontal ray pointing to the right with the vertex of the angle at the origin of an xy-coordinate plane the angle is said to be in standard position.

Terminal side

Initial side

Positive angle measure

Initial side

Terminal side

Negative angle measure

y

x

The degree measurement of an angle is based on the degree measure of an entire circle being 360 and any fraction of the circle will be proportional (i.e. will form the same fraction). The examples below show the measure of angles in standard position.

y

y

y

y

x

x

x

x

½ circle

½ · 360 = 180

¼ circle

¼ · 360 = 90

1/8 circle

1/8 · 360 = 45

5/8 circle

5/8 · 360 = 225

y

y

y

y

x

x

x

x

1 circle (neg)

1 · -360 = -360

3/8 circle (neg)

3/8 · -360 = -135

1/12 circle (neg)

1/12 · -360 = -30

¾ circle (neg)

¾ · -360 = -270

The radian measurement of an angle is based on the degree measure of an entire circle being 2 and any fraction of the circle will be proportional (i.e. will form the same fraction). The examples below show the measure of angles in standard position.

y

y

y

y

x

x

x

x

½ circle

½ · 2 =

¼ circle

¼ · 2 = /2

1/8 circle

1/8 · 2 = /4

5/8 circle

5/8 · 2 = 5/4

y

y

y

y

x

x

x

x

1 circle (neg)

1 · -2 = -2

3/8 circle (neg)

3/8 · -2 = -3/4

1/12 circle (neg)

1/12 · -2 = -/6

¾ circle (neg)

¾ · -2 = -3/2

The symbol used in radian measure stands for the number 3.1415926…. This number is irrational (i.e. its decimal expansion will never end or repeat). The reason that radian measure is used more often in mathematics, physics, engineering and other disciplines is that the angle measure is the length of arc (part of a circle) of a unit circle ( a circle of radius 1). Radian measure represents a physical distance.

y

The part of the unit circle marked in red is called an arc. The length of this (if you straighten it out) is /4 or the measure of the angle in radians. This can be related to a number by the following calculation:

x

1

1/8 circle

1/8 · 2 = /4

For an arc that is part of a circle of radius r its length often we use the symbol s) can be found by taking s=·r where is the measure of the angle in radians.

r

Both radians and degrees are based on a fraction of a circle that you are considering.

To convert back and forth from degrees to radians we use the proportion below where is the measure in degrees and is the measure in radians.

90 and /2

120 and 2/3

60 and /3

135 and 3/4

45 and /4

150 and 5/6

30 and /6

180 and

0 and 0

or

270 and 3/2

Any two angles with the same terminal side are said to be coterminal angles. We have seen that two angles that have the same terminal side can have different measures depending if you are going clockwise (neg) or counterclockwise (pos) around the circle.

y

y

y

x

x

x

5/8 circle (pos)

5/8 ·360 = 225

5/8 · 2 = 5/4

3/8 circle (neg)

3/8 ·-360 = -135

3/8 · -2 = -3/4

= 13/8 circle (pos)

13/8 ·360 = 585

13/8 · 2 = 13/4

Angles exceeding 360 or 2 radians are thought of a wrapping around the circle a certain number of times before hitting the terminal angle. For example a 780 angle is really coterminal with a 60 angle since:

780 - 360 = 420 and 420 - 360 = 60

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