Find reference angle A_{r} to a given angle A.
What is the reference angle to an angle in standard position?

If A is an angle in standard positon, 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 postive and less than 360^{ o} (2Pi), we have 4 possible cases:

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^{ o} - A if A is given degrees

and

**A**_{ r} = Pi - A if A is given in radians.

3. If angle A is in quadrant III then the reference angle

**A**_{ r} = A - 180^{ o} if A is given degrees

and

**A**_{ r} = A - Pi if A is given in radians.

4. If angle A is in quadrant IV then the reference angle

**A**_{ r} = 360^{ o} - A if A is given degrees

and

**A**_{ r} = 2Pi - A if A is given in radians.

__Example 1:__ Find the reference angle to angle A = 120 ^{o}.

__Solution to example 1:__

Angle A is in quadrant II and the reference angle is given by

A_{ r} = 180^{o} - 120^{o} = 60^{o}

__Example 2:__ Find the reference to angle A = - 15 Pi / 4.

__Solution to example 2:__

The given angle is not positive and less than 2Pi. We can use the positive and less than 2Pi coterminal A_{c} to angle A.

A_{c} = - 15 Pi / 4 + 2 (2 Pi) = Pi / 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} = Pi / 4

__Example 3:__ Find the reference angle to angle A = - 30^{ o}

__Solution to example 3:__

Angle A is negative, in quadrant IV and its absolute value is less than 90^{ o}. Hence

A_{ r} = | -30^{ o} | = 30^{ o}

__Exercises:__Find the reference angle to angles

1. A = 1620^{o}

2. A = - 29 Pi / 6

3. A = - Pi / 7

__Solutions to Above Exercises:__

1. A_{ r} = 25^{o}

2. A_{ r} = Pi / 6

3. A_{ r} = Pi / 7

More references on angles.