How to Find Reference Angles?

In the previous section, we learned that we could find the reference angles using the set of rules mentioned in the table. That table works only when the given angle lies between 0° and 360°. But what if the given angle does not lie in this range? Let’s see how we can find the reference angles when the given angle is greater than 360°.

Steps to Find Reference Angles

The steps to find the reference angle of an angle are explained with an example. Let us find the reference angle of 480°.

Step 1: Find the coterminal angle of the given angle that lies between 0° and 360°.

The coterminal angle can be found either by adding or subtracting 360° from the given angle as many times as required. Let’s find the coterminal angle of 480° that lies between 0° and 360°. We will subtract 360° from 480° to find its coterminal angle.

480° – 360° = 120°

Step 2: If the angle from step 1 lies between 0° and 90°, then that angle itself is the reference angle of the given angle. If not, then we have to check whether it is closest to 180° or 360° and by how much.

Here, 120° does not lie between 0° and 90° and it is closest to 180° by 60°. i.e.,

180° – 120° = 60°

Step 3: The angle from step 2 is the reference angle of the given angle.

Thus, the reference angle of 480° is 60°.

This is how we can find reference angles of any given angle.

► Important Notes:

• The reference angle of an angle is always non-negative i.e., a negative reference angle doesn’t exist.
• The reference angle of any angle always lies between 0 and π/2 (both inclusive).

Tricks to Find Reference Angles:

• We use the reference angle to find the values of trigonometric functions at an angle that is beyond 90°. For example, we can see that the coterminal angle and reference angle of 495° are 135° and 45° respectively.

sin 495° = sin 135° = +sin 45°.

We have included the + sign because 135° is in quadrant II, where sine is positive.

sin 495° = √2/2 [Using unit circle]

• If we use reference angles, we don’t need to remember the complete unit circle, instead we can just remember the first quadrant values of the unit circle.