Cofunction and even/odd identities · trigonometric identity
sin θ = cos(90° − θ) — Cofunction identity for sine
When to use it
When the two angles in a problem add to 90° (π/2), the sine of one is the cosine of the other. Typical case: sin 32° asked as a cosine, or a calculus substitution that needs cos written as a shifted sine.
Why it is true
In a right triangle the two acute angles are complementary. The side opposite one is the side adjacent to the other, so 'opposite over hypotenuse' for one angle is 'adjacent over hypotenuse' for the other.
The full line-by-line version is on the proof page for cofunction identity for sine; the “how would I find this myself” version is in the derivation.
Where it comes from
Measure from the other corner. The 'co-' in cosine means complement: cos θ started as 'the sine of the complement of θ'.
- sin of an angle = opp/hypDefinition.
- Same ratio from the other acute angleOpposite and adjacent swap roles.
- sin θ = cos(90° − θ)Name the swapped ratio.
Worked examples
- If sin 32° = 0.53, what is cos 58°?cos 58° = 0.53 — the two angles are complementary.
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.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 0 | 0 | yes |
| 7.5° | 0.130526 | 0.130526 | yes |
| 15° | 0.258819 | 0.258819 | yes |
| 18° | 0.309017 | 0.309017 | yes |
| 22.5° | 0.382683 | 0.382683 | yes |
| 30° | 0.5 | 0.5 | yes |
| 37° | 0.601815 | 0.601815 | yes |
| 45° | 0.707107 | 0.707107 | yes |
| 53° | 0.798636 | 0.798636 | yes |
| 60° | 0.866025 | 0.866025 | yes |
| 67.5° | 0.92388 | 0.92388 | yes |
| 75° | 0.965926 | 0.965926 | yes |
| 90° | 1 | 1 | yes |
| 120° | 0.866025 | 0.866025 | yes |
| 135° | 0.707107 | 0.707107 | yes |
| 150° | 0.5 | 0.5 | yes |
| 180° | 0 | 0 | yes |
| 210° | -0.5 | -0.5 | yes |
| 240° | -0.866025 | -0.866025 | yes |
| 270° | -1 | -1 | yes |
| 300° | -0.866025 | -0.866025 | yes |
| 330° | -0.5 | -0.5 | yes |
| 360° | -0 | -0 | yes |
The mistake students make
Pairing the wrong functions. Sine pairs with cosine, tangent with cotangent, secant with cosecant — always the complementary partner, never itself.
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 = sin(90° − θ)Cofunction identity for cosinedetailsproof
- tan θ = cot(90° − θ)Cofunction identity for tangentdetailsproof
- sin(−θ) = −sin θSine is odddetailsproof
Category hub: Cofunction and even/odd identities · all identities: /identities