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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.
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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