Trig Identities Summary

Pre-Calculus · Axiom Academy

SUMMARY Trigonometric Identities A complete recap of the identities you can now reach for — and when to use each one. Every identity ultimately rests on the Pythagorean relation and the definitions , , , . The sum and difference formulas are the engine: double-angle and half-angle formulas are just special cases of them. To simplify an expression, rewrite everything in terms of and , then combine and cancel. To verify an identity, work one side at a time toward the other — never move terms across the equals sign as if it were an equation. Cosine is even and sine is odd: while . Core Concept Fundamental Identities The reciprocal and quotient identities are definitions — they let you rewrite any of the six functions in terms of and . When to use: the first move on almost any simplification. Watch out for: a hidden zero denominator ( for / ). Core Concept Even–Odd Identities Cosine (and its reciprocal ) is even; sine, tangent, and their reciprocals are odd. These let you strip a negative sign out of an argument. When to use: the angle inside a function is negative. Watch out for: keeps the sign — only the odd functions flip. Core Concept Pythagorean Identities All three come from the unit circle. The second and third follow from the first by dividing through by and . When to use: to trade for (or vice versa). Break an unfamiliar angle into a sum or difference of familiar ones (e.g. ) to find an exact value. When to use: exact values, or combining two angles.

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