Solving for the reference angle in degrees is much easier than trying to determine a trig function for the original angle. To compute the measure (in degrees) of the reference angle for any given angle theta, use the rules in the following table.
| Quadrant | Measure of Angle Theta | Measure of Reference Angle |
|---|---|---|
| I | 0° to 90° | theta |
| II | 90° to 180° | 180° – theta |
| III | 180° to 270° | theta – 180° |
| IV | 270° to 360° | 360° – theta |
Find the reference angle for 200 degrees:
Determine the quadrant in which the terminal side lies.
A 200-degree angle is between 180 and 270 degrees, so the terminal side is in QIII.
Do the operation indicated for that quadrant.
Subtract 180 degrees from the angle, which is 200 degrees. You find that 200 – 180 = 20, so the reference angle is 20 degrees.
Now find the reference angle for 350 degrees:
Determine the quadrant in which the terminal side lies.
A 350-degree angle is between 270 and 360 degrees, so the terminal side is in QIV.
Do the operation indicated for that quadrant.
Subtract 350 degrees from the angle, which is 360 degrees. You find that 360 – 350 = 10, so the reference angle is 10 degrees.
Sometimes angle measures don’t fit neatly in the ranges shown in the table. For example, you may need to find the reference angle for a negative angle or a multiple of an angle.
To find the reference angle for –340 degrees:
Determine the quadrant in which the terminal side lies.
A –340-degree angle is equivalent to a 20-degree angle. (You get the positive angle measure by adding 360, or one full revolution around the origin, to the negative measure.) A 20-degree angle has its terminal side in QI.
Do the operation indicated for that quadrant.
Angles in the first quadrant are their own reference angle, so the reference angle is 20 degrees.
On the other end of the spectrum, to find the reference angle for 960 degrees:
Determine the quadrant in which the terminal side lies.
A 960-degree angle is equivalent to a 240-degree angle. (You get this measure by subtracting 360 from 960 twice.) A 240-degree angle is between 180 and 270 degrees, so its terminal side is in QIII.
Do the operation indicated for that quadrant.
Subtract 180 from 240. You find that 240 – 180 = 60, so the reference angle is 60 degrees.