Cofunction and even/odd identities · trigonometric identity
cos θ = sin(90° − θ) — Cofunction identity for cosine
When to use it
The reverse direction of the sine cofunction: use it to move a cosine into a sine (or back) when only one of them is integrable or listed in a table.
Why it is true
Same complementary pair, read from the other acute angle of the triangle.
The full line-by-line version is on the proof page for cofunction identity for cosine; the “how would I find this myself” version is in the derivation.
Where it comes from
Shift the graph. The cosine curve is the sine curve moved left by π/2 — a fact used constantly in physics and AC circuits.
- Compare the graphs of sin and cosIdentical shape, offset by a quarter period.
- cos θ = sin(θ + π/2)Reading the offset one way.
- cos θ = sin(π/2 − θ)Reading it with the complement, which is the cofunction statement for acute angles.
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° | 1 | 1 | yes |
| 7.5° | 0.991445 | 0.991445 | yes |
| 15° | 0.965926 | 0.965926 | yes |
| 18° | 0.951057 | 0.951057 | yes |
| 22.5° | 0.92388 | 0.92388 | yes |
| 30° | 0.866025 | 0.866025 | yes |
| 37° | 0.798636 | 0.798636 | yes |
| 45° | 0.707107 | 0.707107 | yes |
| 53° | 0.601815 | 0.601815 | yes |
| 60° | 0.5 | 0.5 | yes |
| 67.5° | 0.382683 | 0.382683 | yes |
| 75° | 0.258819 | 0.258819 | yes |
| 90° | 0 | 0 | yes |
| 120° | -0.5 | -0.5 | yes |
| 135° | -0.707107 | -0.707107 | yes |
| 150° | -0.866025 | -0.866025 | yes |
| 180° | -1 | -1 | yes |
| 210° | -0.866025 | -0.866025 | yes |
| 240° | -0.5 | -0.5 | yes |
| 270° | -0 | -0 | yes |
| 300° | 0.5 | 0.5 | yes |
| 330° | 0.866025 | 0.866025 | yes |
| 360° | 1 | 1 | yes |
The mistake students make
Writing cos θ = sin(θ − 90°). The complement is 90° − θ, and note that sin(θ − 90°) = −cos θ, the negative of what you wanted.
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
- sin θ = cos(90° − θ)Cofunction identity for sinedetailsproof
- tan θ = cot(90° − θ)Cofunction identity for tangentdetailsproof
- cos(−θ) = cos θCosine is evendetailsproof
Category hub: Cofunction and even/odd identities · all identities: /identities