# 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. 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.