Reference Angles

Reference Angles

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

Formula

Quadrant I: theta. Quadrant II: pi - theta. Quadrant III: theta - pi. Quadrant IV: 2pi - theta

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 < 3π/2 (quadrant III),
the reference angle is θ - π.

If 3π/2 < θ < 2π (quadrant IV),
the reference angle is 2π - θ.

Commonly Used Angles

Reference Angles: Commonly Used Angles

These angles are the commonly used angles
when solving trigonometry problems.

Radian measure

Example 1

Find the value of the given expression. sin(3pi/4)

π/2 < 3π/4 < π

So the reference angle is the blue angle:
π - 3π/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 3π/4
from the right triangle above.

Sine: SOH.
So sin 3π/4 = sin (blue)
= 1/√2.

Rationalizing a denominator

Example 2

Find the value of the given expression. cos(4pi/3)

π < 4π/3 < 3π/2

So the reference angle is the blue angle:
4π/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 4π/3
from the right triangle above.

Cosine: CAH.
So cos 4π/3 = cos (blue)
= -1/2.

Example 3

Find the value of the given expression. tan(11pi/6)

3π/2 < 11π/6 < 2π

So the reference angle is the blue angle:
2π - 11π/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 11π/6
from the right triangle above.

Tangent: TOA.
So tan 11π/6 = tan (blue)
= -1/√3.

Rationalizing a denominator

Example 4

Find the value of the given expression. sin(-5pi/3)

-5π/3 is not between 0 and 2π.

So find its coterminal angle
that is between 0 and 2π: 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 (-5π/3)
from the right triangle above.

Sine: SOH.
So sin (-5π/3) = sin (blue)
= √3/2.