What is the reference angle to an angle in standard position?
If A is an angle in standard position, its reference angle A_{r} is the acute angle formed by the x axis and the terminal side of angle A. See figure below.
Example 1: Find the reference angle to angle A = 120 °.
Solution to example 1:
Angle A is in quadrant II and the reference angle is given by
A_{ r} = 180° - 120° = 60°
Example 2: Find the reference to angle A = - 15 ? / 4.
Solution to example 2:
The given angle is not positive and less than 2?. We can use the positive and less than 2? coterminal A_{c} to angle A.
A_{c} = - 15 ? / 4 + 2 (2 ?) = ? / 4
Angle A and A_{c} are coterminal and have the same reference angle. A_{c} is in quadrant I, therefore
A_{ r} = A_{ c} = ? / 4
Example 3: Find the reference angle to angle A = - 30°
Solution to example 3:
Angle A is negative, in quadrant IV and its absolute value is less than 90°. Hence
A_{ r} = | -30° | = 30°
Exercises:Find the reference angle to angles
1. A = 1620°
2. A = - 29 ? / 6
3. A = - ? / 7
Solutions to Above Exercises:
1. A_{ r} = 25°
2. A_{ r} = ? / 6
3. A_{ r} = ? / 7