Unit Circle Trig

Trig Values of Special Angles

Evaluate sin, cos, and tan at any special angle using its reference angle and the quadrant sign rules.

Example: 225° is in Q3 with reference angle 45°. sin(225°) = −√2/2, cos(225°) = −√2/2, tan(225°) = 1.

Example

Use reference angles and quadrant signs to get exact trig values.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

225∘Quadrant III225 ^\circ \quad \text{Quadrant III}

Step 2 — Reference angle

Use reference angle 45 degrees for 225 degrees.

45∘\hl{45} ^\circ

Step 3 — Sine value

Apply the quadrant sign to sine using the 45-degree reference angle.

sin⁡(225∘)=−22\sin( 225 ^\circ)= \hlmath{-\frac{\sqrt{2}}{2}}

Step 4 — Cosine value

Apply the quadrant sign to cosine using the 45-degree reference angle.

cos⁡(225∘)=−22\cos( 225 ^\circ)= \hlmath{-\frac{\sqrt{2}}{2}}

Step 5 — Tangent value

Divide sine by cosine; the negatives cancel.

tan⁡(225∘)=1\tan( 225 ^\circ)= \hlmath{1}
trig-of-special-angles Steps to evaluate sin/cos/tan at a non-first-quadrant special angle: 1. Find the reference angle (Q3: θ − 180°, Q2: 180° − θ, Q4: 360° − θ). 2. Look up sin and cos at the reference angle from the special-angle table. 3. Apply the quadrant sign: Q3 makes both sin and cos negative. 4. Compute tan = sin/cos (in Q3, both negative so tan is positive).