Find Reference Angle
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.
Two or more coterminal angles have the same reference angle.
Assume angle A is positive and less than 360° (2π), we have 4 possible cases (see figure above):
1. If angle A is in quadrant I then the reference angle A_{ r} = A .
2. If angle A is in quadrant II then the reference angle A_{ r} = 180°  A if A is given degrees and A_{ r} = π  A if A is given in radians.
3. If angle A is in quadrant III then the reference angle A_{ r} = A  180° if A is given degrees and A_{ r} = A  π if A is given in radians.
4. If angle A is in quadrant IV then the reference angle A_{ r} = 360°  A if A is given degrees and A_{ r} = 2π  A if A is given in radians.
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 π / 4.
Solution to example 2:
The given angle is not positive and less than 2π. We can use the positive and less than 2π coterminal A_{c} to angle A.
A_{c} =  15 π / 4 + 2 (2 π) = π / 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} = π / 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 π / 6
3. A =  π / 7
Solutions to Above Exercises:
1. A_{ r} = 25°
2. A_{ r} = π / 6
3. A_{ r} = π / 7
More references on angles.

Step by Step Solver to Find Coterminal Angle to a Given Angle.

Step by Step Solver to Find the Reference Angle to a Given Angle.

Angles in trigonometry.

Find The Reference Angle  Trigonometry calculator.

Find the coterminal angle  Trigonometry calculator.

Find the Quadrant of an Angle  Trigonometry calculator.