Reference Angle - Meaning, Formula, Examples - Cuemath

Reference Angle

In math, a reference angle is generally an acute angle enclosed between the terminal arm and the x-axis. It is always positive and less than or equal to 90 degrees. Let us learn more about the reference angle in this article.

1.	Reference Angle Definition
2.	Rules for Reference Angles in Each Quadrant
3.	How to Find Reference Angles?
4.	FAQs on Reference Angle

Reference Angle Definition

The reference angle is the smallest possible angle made by the terminal side of the given angle with the x-axis. It is always an acute angle (except when it is exactly 90 degrees). A reference angle is always positive irrespective of which side of the axis it is falling.

How to Draw Reference Angle?

To draw the reference angle for an angle, identify its terminal side and see by what angle the terminal side is close to the x-axis. The reference angle of 135° is drawn below:

Here, 45° is the reference angle of 135°.

Rules for Reference Angles in Each Quadrant

Here are the reference angle formulas depending on the quadrant of the given angle.

Quadrant	Angle, θ	Reference Angle Formula in Degrees	Reference Angle Formula in Radians
I	lies between 0° and 90°	θ	θ
II	lies between 90° and 180°	180 - θ	π - θ
III	lies between 180° and 270°	θ - 180	θ - π
IV	lies between 270° and 360°	360 - θ	2π - θ

If the angle is in radians, then we use the same rules as for degrees by replacing 180° with π and 360° with 2π.

Example: Find the reference angle of 120°.

Solution: The given angle is, θ = 120°. We know that 120° lies in quadrant II. Using the above rules, its reference angle is,

180 - θ = 180 - 120 = 60°

Therefore, the reference angle of 120° is 60°.

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.

Related Articles on Reference Angles

Check these interesting articles related to the concept of reference angles.

• Reference Angle Calculator
• Coterminal Angles
• Trigonometry Formulas
• Trigonometric Table