1. Standard Position, Coterminal and Reference Angles

2. Measure of an Angle 1
The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. -1 1
Initial Side
Terminal Side -1

3. Angles that share the same initial and terminal sides.
Coterminal Angles 1
Angles that share the same initial and terminal sides.
Example:
30° and 390° -1 1 -1

4. Angles  and  are coterminal.
Coterminal Angles
Angles that have the same initial and terminal sides are coterminal.
Angles  and  are coterminal.

5. Example of Finding Coterminal Angles
You can find an angle that is coterminal to a given angle  by adding or subtracting multiples of 360º.
Ex 1: Find one positive and one negative angle that are coterminal to 112º.
For a positive coterminal angle, add 360º : 112º + 360º = 472º
For a negative coterminal angle, subtract 360º: 112º - 360º = -248º

6. Ex 2. Find one positive and one negative angle that is coterminal with the angle  = 30° in standard position.
Ex 3. Find one positive and one negative angle that is coterminal with the angle  = 272 in standard position.

7. Ex 4. Find one positive and one negative angle that is coterminal with the angle  = in standard position.
Ex 5. Find one positive and one negative angle that is coterminal with the angle  = in standard position.

8. Reference Angles
The values of the trigonometric functions of angles greater than 90 (or less than 0) can be determined from their values at corresponding acute angles called reference angles. 𝜃’ 𝜃

9. Reference Angles
The reference angles for  in Quadrants II, III, and IV are shown below.
 ′ =  –  (radians)  ′ = 180 –  (degrees)
 ′ =  –  (radians)  ′ =  – 180 (degrees)
 ′ = 2 –  (radians)  ′ = 360 –  (degrees)

10. Special Angles – Reference Angles

11. Example – Finding Reference Angles
Find the reference angle  ′. a.  = 300 b.  = 2.3 c.  = –135

12. Example (a) – Solution
Because 300 lies in Quadrant IV, the angle it makes with the x-axis is
 ′ = 360 – 300
= 60.
The figure shows the angle  = 300
and its reference angle  ′ = 60.
Degrees

13. Example (b) – Solution
cont’d
Because 2.3 lies between  /2  and   , it follows that it is in Quadrant II and its reference angle is  ′ =  – 2.3  The figure shows the angle  = 2.3 and its reference angle  ′ =  – 2.3.
Radians

14. Example (c) – Solution
cont’d
First, determine that –135 is coterminal with 225, which lies in Quadrant III. So, the reference angle is  ′ = 225 – 180 = 45. The figure shows the angle  = –135 and its reference angle  ′ = 45.
Degrees

15. Reference Angles
When your angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0° and 360° or 0 and 2𝜋.

16. Your Turn: Find the reference angle for each of the following. 213°
1.7
−144°
-144 ̊ is coterminal to 216 ̊
216 ̊ ̊ = 36 ̊