Trig Identities

Pythagorean Identity

Use sin²θ + cos²θ = 1 to find an unknown trig value given the other, verified symbolically in the Pythagorean polynomial ring.

Given sin θ = p/q in quadrant Q, solve for cos θ: cos²θ = 1 − sin²θ = 1 − (p/q)² = (q²−p²)/q² cos θ = ±√((q²−p²)/q²) [sign from quadrant] Example: sin θ = 3/5 (Q1) → cos²θ = 1 − 9/25 = 16/25 → cos θ = 4/5.

Example

Use sin squared plus cos squared equals one to find the missing trig value.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

sin⁡θ=35Quadrant I\sin\theta= \frac{3}{5} \quad \text{Quadrant I}

Step 2 — Identity

Use the Pythagorean identity.

sin⁡2θ+cos⁡2θ=1\sin^{ \hl{2} }\theta+\cos^{ \hl{2} }\theta= \hl{1}

Step 3 — Substitute sine

Substitute the sine square: 9 over 25.

925+cos⁡2θ=1\hlmath{\frac{9}{25}} +\cos^{ 2 }\theta= 1

Step 4 — Solve for cosine squared

Subtract from 1 to get 16 over 25.

cos⁡2θ=1625\cos^{ 2 }\theta= \hlmath{\frac{16}{25}}

Step 5 — Quadrant sign

Quadrant I makes cosine positive.

positive\hl{positive}

Step 6 — Result

Take the positive square root: 4 over 5.

cos⁡θ=45\cos\theta= \hlmath{\frac{4}{5}}
pythagorean-identity The Pythagorean identity: sin²θ + cos²θ = 1.