Trigonometric Functions and Reference Angles

The right-triangle ratios only handle acute angles. For an angle in standard position of any size, we work through its reference angle $\theta'$ — the acute angle between the terminal side and the x-axis.

Concept The reference angle

The reference angle $\theta'$ of an angle $\theta$ in standard position is the acute angle between the terminal side of $\theta$ and the x-axis.

Here

\theta = 150°

lies in the second quadrant.

Its terminal side makes 30° with the negative x-axis, so

\theta' = 30°

The formula for $\theta'$ depends only on the quadrant the terminal side falls in.

Reference One formula per quadrant

Quadrant	θ lies in	Degrees	Radians
First	0° to 90°	θ′ = θ	θ′ = θ
Second	90° to 180°	θ′ = 180° − θ	θ′ = π − θ
Third	180° to 270°	θ′ = θ − 180°	θ′ = θ − π
Fourth	270° to 360°	θ′ = 360° − θ	θ′ = 2π − θ

Example Finding the reference angle

\theta = 150°

— second quadrant, so

\theta' = 180° - 150°

⟹ θ′ = 30°

\theta = 215°

— third quadrant, so

\theta' = 215° - 180°

⟹ θ′ = 35°

\theta = 300°

— fourth quadrant, so

\theta' = 360° - 300°

⟹ θ′ = 60°

\theta = \dfrac{5\pi}{6}

— second quadrant, so

\theta' = \pi - \dfrac{5\pi}{6}

⟹ θ′ = π/6

Note Two things to watch

A reference angle is always acute: $0° < \theta' \le 90°$ . If your answer is larger, the quadrant was misread.

It is measured to the x-axis — never to the y-axis. Identify the quadrant first, then apply the matching formula.

Summary

The reference angle is the acute angle between the terminal side and the x-axis.
By quadrant: θ, 180° − θ, θ − 180°, 360° − θ.
In radians, replace 180° with π and 360° with 2π.
It is always acute (0° < θ′ ≤ 90°) and always measured to the x-axis.