﻿ Reference Angle: How to find the reference angle as a positive acute angle

# Find 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.

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

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

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

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

### Practice Problem

##### Problem 1

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

##### Problem 2

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

##### Problem 3

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

### Word Problems

##### Problem 4

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

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

##### Problem 5

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

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