Unit Circle Trig

Reference Angles

Find the reference angle (acute angle to the nearest x-axis) for any angle and determine the sign of its trig values by quadrant.

Sign rules ("All Students Take Calculus"): Q1: sin > 0, cos > 0 Q2: sin > 0, cos < 0 Q3: sin < 0, cos < 0 Q4: sin < 0, cos > 0 Example: 210° is in Q3; reference angle = 210° − 180° = 30°. Both sin(210°) and cos(210°) are negative.

Example

Find the acute reference angle and quadrant sign information.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

210∘210 ^\circ

Step 2 — Quadrant

The angle 210 degrees is in Quadrant III.

Quadrant III\hl{Quadrant III}

Step 3 — Reference angle

Subtract 180 degrees: 210 - 180 = 30 degrees.

210∘−180∘=30∘210 ^\circ- 180 ^\circ= \hl{30} ^\circ

Step 4 — Sign check

In Quadrant III, both sine and cosine are negative.

sin⁡(210∘)<0cos⁡(210∘)<0\sin( 210 ^\circ)< \hl{0} \quad \cos( 210 ^\circ)< \hl{0}
reference-angles The reference angle is the acute angle between the terminal side and the nearest x-axis: Q1: ref = θ Q2: ref = 180° − θ Q3: ref = θ − 180° Q4: ref = 360° − θ