Find Reference Angle

What is a Reference Angle?

The reference angle is the positive acute angle that can represent an angle of any measure.

The reference angle $$ \text{ must be } < 90^{\circ} $$

In radian measure, the reference angle $$\text{ must be } < \frac{\pi}{2} $$

Basically, any angle on the x-y plane has a reference angle, which is always between 0 and 90 degrees. The reference angle is always the smallest angle that you can make from the terminal side of an angle (ie where the angle ends) with the x-axis. A reference angle always uses the x-axis as its frame of reference.

Rules of Angles and Reference angle

Positive angles go in a counter clockwise direction. Below is a picture of a positive fifty degree angle


Quadrant I

Every positive angle in quadrant I is already acute...so the reference angle is the measure of the angle itself:


Quadrant II

To find the reference angle measuring x ° for angle in Quadrant III, the formula is $$ 180 - x^{\circ} $$.

quadrant II general pic

Quadrant III

To find the reference angle measuring x ° for angle in Quadrant III, the formula is $$ x - 180 ^{\circ} $$.

quadrant II general pic

Quadrant IV

To find the reference angle measuring x ° for angle in Quadrant IV, the formula is $$360 ^{\circ} -x $$.

quadrant II general pic

Practice Problem

Problem 1

What is the reference angle for the angle in the graph below?

Remember that the reference angle always uses the x-axis as a frame of reference.

Problem 2

What is the reference angle for a 210° angle?

Remember that the reference angle always uses the x-axis as a frame of reference.

Problem 3

What is the reference angle for a 300° angle?

Remember that the reference angle always uses the x-axis as a frame of reference.

Word Problems

Problem 4

What is the reference angle for an angle that measures 91°?

For a quadrant 2 angle, the reference angle is always 180° - given angle

In this case, $$ 180 - 91 = \color{Red}{89} $$

91 degree reference angle
Problem 5

What is the reference angle for an angle that measures 250°?

For a quadrant 3 angle, the reference angle is always given angle - 180°

In this case, $$ 250 - 180= \color{Red}{ 70 } $$

91 degree reference angle