# Reference Angles

How to find the reference angle of a given angle: formulas, examples, and their solutions.

## Formula

Reference angle is used for

finding the values of trigonometric ratios.

If 0 < *θ* < *π*/2 (quadrant I),

the reference angle is itself: *θ*.

If *π*/2 < *θ* < *π* (quadrant II),

the reference angle is *π* - *θ*.

If *π* < *θ* < 3*π*/2 (quadrant III),

the reference angle is *θ* - *π*.

If 3*π*/2 < *θ* < 2*π* (quadrant IV),

the reference angle is 2*π* - *θ*.

## Commonly Used Angles

These angles are the commonly used angles

when solving trigonometry problems.

Radian measure

## Example 1

*π*/2 < 3*π*/4 < *π*

So the reference angle is the blue angle:*π* - 3*π*/4 = *π*/4.

*π*/4 = 45º

So draw a 45-45-90 triangle

on the coordinate plane

with the signs of the sides:

-1, 1, √2.

Radian measure

Find sin 3*π*/4

from the right triangle above.

Sine: SOH.

So sin 3*π*/4 = sin (blue)

= 1/√2.

## Example 2

*π* < 4*π*/3 < 3*π*/2

So the reference angle is the blue angle:

4*π*/3 - *π* = *π*/3.

*π*/3 = 60º

So draw a 30-60-90 triangle

on the coordinate plane

with the signs of the sides:

-1, -√3, 2.

Radian measure

Find cos 4*π*/3

from the right triangle above.

Cosine: CAH.

So cos 4*π*/3 = cos (blue)

= -1/2.

## Example 3

3*π*/2 < 11*π*/6 < 2*π*

So the reference angle is the blue angle:

2*π* - 11*π*/6 = *π*/6.

*π*/6 = 30º

So draw a 30-60-90 triangle

on the coordinate plane

with the signs of the sides:

1, -√3, 2.

Radian measure

Find tan 11*π*/6

from the right triangle above.

Tangent: TOA.

So tan 11*π*/6 = tan (blue)

= -1/√3.

## Example 4

-5*π*/3 is not between 0 and 2*π*.

So find its coterminal angle

that is between 0 and 2*π*: *π*/3.

0 < *π*/3 < *π*/2

So the reference angle is itself: *π*/3.*π*/3 = 60º

So draw a 30-60-90 triangle

on the coordinate plane

with the signs of the sides:

1, √3, 2.

Radian measure

Find sin (-5*π*/3)

from the right triangle above.

Sine: SOH.

So sin (-5*π*/3) = sin (blue)

= √3/2.