13.2 Angles of Rotation and Radian Measure

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13.2 Angles of Rotation and Radian Measure. ©2002 by R. Villar All Rights Reserved. Angles of Rotation and Radian Measure. An angle of rotation is formed by two rays with a common endpoint (called the vertex ). y. terminal side. One ray is called the initial side .

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### Angles of Rotation and Radian Measure

An angle of rotation is formed by two rays with a common endpoint (called the vertex).

y

terminal side

One ray is called the initial side.

The other ray is called the terminal side.

x

vertex

initial side

In the last section, we looked at angles that were acute. In this section, we will look at angles of rotation whose measure can be any real number.

The measure of the angle is determined by the amount and direction of rotation from the initial side to the terminal side.

The angle measure is positive if the rotation is counterclockwise, and negative if the rotation is clockwise.

A full revolution (counterclockwise) corresponds to 360º.

Example: Draw an angle with the given measure in standard position. Then determine in which quadrant the terminal side lies.

A. 210º b. –45º c. 510º

150º

210º

–45º

510º

Terminal side is in Quadrant III

Terminal side is in Quadrant IV

Terminal side is in Quadrant II

Use the fact that 510º = 360º + 150º.

So the terminal side makes 1 complete revolution and continues another 150º.

510º and 150º are called coterminal (their terminal sides coincide).

An angle coterminal with a given angle can be found by adding or subtracting multiples of 360º.

You can also measure angles in radians.

One radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r.

r

r

Since the circumference of a circle is 2πr, there are 2π radians in a full circle.

Degree measure and radian measure are therefore related by the following:

• To rewrite a degree measure in radians, multiply by π radians 180º

• To rewrite a radian measure in degrees, multiply by 180º π radians

a. 240º b. –90º c. 135º

4

3

240º = 240º • π 180º

–90º = –90º • π 180º

135º = 135º • π 180º

3

4

= –π

2

= 3π

4

= 4π

3

3

4

2

Examples: Rewrite each in degrees

a. 5π b. 16π

8 5

5π = 5π• 180º

8 8 π

16π = 16π• 180º

5 5 π

= 112.5º

= 576º

Two positive angles are complementary if the sum of their measures is π/2 radians

(which is 90º)

Two positive angles are supplementary if the sum of their measures is π radians (which is 180º).

Example: Find the complement of = π8

Thecomplement is π – π28

= 4π – π88

=3π8

Example: Find the supplement of = 3π5

Thesupplement is π – 3π5

= 5π – 3π55

=2π5