Find Reference Angle

Find reference angle Ar 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 Ar is the acute angle formed by the x axis and the terminal side of angle A. See figure below.

reference angle

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 = 180o - 120o = 60o

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 Ac to angle A.

Ac = - 15 Pi / 4 + 2 (2 Pi) = Pi / 4

Angle A and Ac are coterminal and have the same reference angle. Ac 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 = 1620o

2. A = - 29 Pi / 6

3. A = - Pi / 7

Solutions to Above Exercises:

1. A r = 25o

2. A r = Pi / 6

3. A r = Pi / 7



More references on angles.


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Updated: 2 April 2013

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