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.