Cofunction and even/odd identities · trigonometric identity

sin(−θ) = −sin θ — Sine is odd

When to use it

A negative angle inside sine: pull the minus sign out front. This is the standard first move when a problem says sin(−x) and the answers are written with positive x.

Why it is true

On the unit circle, angle −θ is the reflection of θ across the x-axis. Reflection across the x-axis keeps the x-coordinate (cosine) and flips the y-coordinate (sine).

The full line-by-line version is on the proof page for sine is odd; the “how would I find this myself” version is in the derivation.

Where it 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.

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.

Worked examples

• Simplify sin(−x)cos(−x) + tan(−x)
  −sin x cos x − tan x

Checked at these angles

Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.

The mistake students make

Try it

• Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
• Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
• Cheat sheet entry for cofunction and even/odd identities.

Related identities

• cos(−θ) = cos θCosine is evendetailsproof
• tan(−θ) = −tan θTangent is odddetailsproof
• sin θ = cos(90° − θ)Cofunction identity for sinedetailsproof

Category hub: Cofunction and even/odd identities · all identities: /identities

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