Using Pythagorean Identities

Pre-Calculus · Axiom Academy

EXAMPLE Using Pythagorean Identities Find a trig value from another using a Pythagorean identity — and let the quadrant fix the sign. Find using a Pythagorean identity. Given: , and lies in Quadrant I . In Quadrant I every trig function is positive, so the square root resolves to the + root. (Q II: only sin; Q III: only tan; Q IV: only cos.) Nice work — you turned into with a single identity, then used the quadrant to nail the sign. Match the identity to your functions: links sine and cosine; use for tan/sec, and for cot/csc. Square, substitute, solve: plug in the known value, then isolate the squared function you want. The square root gives both signs: means — the algebra alone can't decide. The quadrant breaks the tie: in Quadrant I all functions are positive, so . (In Q II/III/IV cosine could be negative.) Same recipe works for any of the three identities — only the quadrant's sign rule changes the final answer.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.