Reference Angle: Examples

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.

Example :

Find the reference angle of 8π/3 in radians.Solution: The given angle is greater than 2π.

Step 1:

Finding co-terminal angle:We find its co-terminal angle by subtracting 2π from it.

8π/3 – 2π = 2π/3

This angle does not lie between 0 and π/2.

Hence, it is not the reference angle of the given angle.

Step 2:

Finding reference angle:

Let’s check whether 2π/3 is close to π or 2π

and by how much.

Clearly, 2π/3 is close to π by π – 2π/3 = π/3.

Therefore, the reference angle of 8π/3 is π/3.