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.

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°.