Unit Circle Trig

Unit Circle Values

Find the exact (cos, sin) coordinates at any special angle by identifying the quadrant and the reference angle.

Example: 5π/6 is in Q2 (150°); reference angle = π/6 (30°). cos(5π/6) = −√3/2, sin(5π/6) = 1/2.

unit-circle-values For any special angle: 1. Identify the quadrant (I–IV) from the angle value. 2. Find the reference angle (acute angle to the nearest x-axis). 3. Look up cos/sin at the reference angle from the special-angle table. 4. Apply quadrant sign: cos is negative in Q2 and Q3; sin is negative in Q3 and Q4.

Example

Write the exact unit-circle coordinate using cosine and sine.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

5π6Quadrant II\frac{5\pi}{6} \quad \text{Quadrant II}

Step 2 — Reference angle

The 150-degree angle has reference angle 30 degrees.

1π6=30∘\hlmath{\frac{1\pi}{6}} = \hl{30} ^\circ

Step 3 — Cosine coordinate

Cosine is the x-coordinate and is negative in Quadrant II.

cos⁡(5π6)=−32\cos( \frac{5\pi}{6} )= \hlmath{-\frac{\sqrt{3}}{2}}

Step 4 — Sine coordinate

Sine is the y-coordinate and is positive in Quadrant II.

sin⁡(5π6)=12\sin( \frac{5\pi}{6} )= \hlmath{\frac{1}{2}}

Step 5 — Unit circle point

Write the unit-circle point as (cos theta, sin theta).

(−32,12)( \hlmath{-\frac{\sqrt{3}}{2}} , \hlmath{\frac{1}{2}} )