1. Coterminal Angles

2. Basic Terms An angle is formed by rotating a ray around its endpoint.
The ray in its starting position is called the initial side of the angle.
The ray’s location after the rotation is the terminal side of the angle.
terminal side
angle
initial side
The initial side of an angle in standard position is always located on the positive x-axis.

3. Basic Terms
Positive angle: The rotation of the terminal side of an angle counterclockwise.
Negative angle: The rotation of the terminal side clockwise.
When sketching angles, always use an arrow to show direction.

4. Measuring Angles
The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.
There are two common ways to measure angles, in degrees and in radians.
We’ll start with degrees, denoted by the symbol °.
One degree (1°) is equivalent to a rotation of of one revolution.

5. Measuring Angles

6. Draw each angle in standard position
100° ° ° °

7. A complete rotation of a ray results in an angle measuring 360.
We don’t have to stop there!
137 more
137 is coterminal with 497. They have the same terminal side! We can keep adding or subtracting 360 to get more coterminal angles.
360
497 altogether!
Angles that have the same initial and terminal sides are coterminal.

8. Example 2: For the angles below, find the smallest positive coterminal angle.
(Add or subtract 360 as may times as needed to obtain an angle with measure greater than 0 but less than 360.) a) 1115 b) 187 a) 1115° - 360° - 360° - 360° = 35° b) 187 + 360 = 173

9. Ex 3. Find one positive and one negative angle that is coterminal with the angle  = 520° in standard position.
Ex 4. Find one positive and one negative angle that is coterminal with the angle  = -90 in standard position.

10. Reference Angle A reference angle is the acute version of any angle.
It is the smallest angle between the x-axis and the terminal side.

11. Complement and Supplement
Two angles are complementary when they add up to 90 degrees.
Two angles are supplementary when they add up to 180 degrees.

12. Radian Measure A second way to measure angles is in radians.
Definition of Radian:
One radian is the measure of a central angle  that intercepts arc s equal in length to the radius r of the circle.
In general,

13. Conversions Between Degrees and Radians
To convert degrees to radians, multiply degrees by
To convert radians to degrees, multiply radians by