How to Find a Reference Angle for A given Degree?

TopReference angle is calculated on the basis of Position of the given angle in any of the four quadrants in a rectangular plane. Then how to find a Reference Angle for a given degree? Suppose we have the measure of the main angle as “β” in degree, then the 4 thinkable ways to find the measure of the reference angle are (supposing that the angle β is positive and is less than 360degrees i.e. 2 pi radians):

1. When β is supposed to lies in the 1st quadrant the reference angle will be: β_r= β.

2. When β is supposed to lie in the 2nd quadrant we can write reference angle as: β_r= 180 ⁰ – β and β_r= pi – β when β is in radians.

3. When β is supposed to lie in the 3rd quadrant we can write reference angle as: β_r = β-180 ⁰ and β_r = β – pi if β is present in radians.

4. When β is supposed to lie in the 4th quadrant we can write reference angle as: β_r= 360 ⁰ - β and β_r = 2 pi – β if β is in radians,

We have considered the given degree β in its standard position in all 4 cases. Let us see an example:

Example: Find out the reference angle for a given angle β = 250⁰ in its standard position?

Solution: Here given angle is 250 degree.

β = 250⁰ lies in the III quadrant when drawn in its standard position. So, its corresponding reference angle is:

β_r= β– 180 ⁰,

or β_r = 250⁰ – 180⁰,

or β_r = 70⁰,

Thus, for an angle of 25⁰ degrees, reference angle would be 70⁰.