Angles of Rotation and Radian Measure

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.

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

initial side

terminal side

vertexOne ray is called the initial side.

The other ray is called the terminal side.

x

y

Angles of Rotation and Radian Measure

initial side

terminal side

vertexx

y

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.

Angles of Rotation and Radian Measure

x

y This is a positive (counter-clockwise) angle

y

x

This is a negative (clockwise) angle

Angles of Rotation

x

y That would be a 90º Angle

x

y

That would be a 180º Angle

Angles of Rotation

x

y That would be a 270º Angle

x

y

That would be a 360º Angle

Angles of Rotation

x

y An Angle of 120º in standard position

y

x

An Angle of -120º in standard position

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

Use the fact that 510º = 360º + 150º.So the terminal side makes 1 complete

revolution and continues another 150º.

Terminal side is in Quadrant III

Terminal side is in Quadrant IV

Terminal side is in Quadrant II

210º

–45º510º

150º

510º

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.

So if you are asked to find coterminal angles you can simply add 360 to the angle or subtract 360 from the angle

Find two angles that are coterminal with 130º (one positive and one negative

130º + 360º = 490º

130º - 360º = -290º

Complimentary and Supplementary Angles

2 angles that are complimentary add up to equal 90 degrees

2 angles that are supplementary add up to equal 180 degrees

Find the supplement to an angle of 24º

180 – 24 = 156Find the compliment to an angle of 24º90 – 24 = 66

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.

Conversion Between Degrees and Radians• To rewrite a degree measure in radians, multiply by π radians

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

π radians

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:

360º = 2π radians

r r

one radian

Examples: Rewrite each in radiansa. 240º b. –90º c. 135º

240º = 240º • π

180º

240º = 4π radians 3

= 4π 3

3

4–90º = –90º • π

180º= –π 2

135º = 135º • π

180º= 3π 4

4

3

–90º = –π radians 2

135º = 3π radians 4

Examples: Rewrite each in degreesa. 5π b. 16π 8 5

5π = 5π • 180º 8 8 π

= 112.5º

16π = 16π • 180º 5 5 π

= 576ºTwo 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 = π 8

The complement is π – π 2 8

= 4π – π 8 8

= 3π 8

Example: Find the supplement of = 3π 5 The supplement is π – 3π

5 = 5π – 3π

5 5 = 2π

5