Cofunction and even/odd identities · derivation ·proof

Where sin(−θ) = −sin θ comes from

Test with numbers, then generalise. If you cannot remember which functions are odd, check 30° on the unit circle: sin(−30°) = −1/2 while sin 30° = 1/2.

Building it step by step

1. sin 30° = 1/2
   Special angle.
2. sin(−30°) = −1/2
   Same point reflected below the axis.
3. sin(−θ) = −sin θ
   The pattern holds for every θ because the reflection argument is not about 30° specifically.

How to recall it under pressure

Re-derive it from the first line rather than searching memory: start from sin(-x) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the sine is odd page.

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7 · Paul's Online Math Notes, Trig Cheat Sheet · revised 2026-09-26 ·how we check ·accuracy policy ·report an error