Answer:
714° ----> 6°
23π/5 ----> 2π/5
120° ----> 60°
31π/6 ----> π/6
Step-by-step explanation:
The reference angle depends on the quadrant of the coterminal angle, so
If the coterminal angle is between 0 and 90°, or between 0 and π/2, the reference angle is the same coterminal angle.
If the coterminal angle is between 90° and 180° or between π/2 and π, the reference angle is (180 - angle) or (π - angle).
If the coterminal angle is between 180° and 270° or between π and 3π/2, the reference angle is (angle - 180) or (angle - π).
If the coterminal angle is between 270° and 360° or between 3π/2 and 2π, the reference angle is (360 - angle) or (2π - angle).
Now, for each angle measurement, we get:
714°
A coterminal angle can be calculated subtracting 360° from 714°, so
714° - 360° = 354°
Since 354° is an angle between 270° and 360°, the reference angle is
Reference angle = 360 - angle
Reference angle = 360 - 354
Reference angle = 6°
23π/5
To find the coterminal angle, let's subtract 2π as follows
23π/5 - 2π = 13π/5
13π/5 - 2π = 3π/5
Since 3π/5 is an angle between π/2 and π, the reference angle is
Reference angle = π - 3π/5 = 2π/5
120°
This is an angle between 90° and 180°, so the reference angle is
Reference angle = 180 - 120 = 60°
31π/6
Subtracting 2π, we get that the coterminal angle is
31π/6 - 2π = 19π/6
19π/6 - 2π = 7π/6
Since 7π/6 is an angle between π and 3π/2, the reference angle is
Reference angle = 7π/6 - π = π/6
Therefore, the answers are
714° ----> 6°
23π/5 ----> 2π/5
120° ----> 60°
31π/6 ----> π/6