Power-reducing identities · trigonometric identity
sin³x = (3 sin x − sin 3x)/4 — Reduction formula for sin³
When to use it
Cubic powers: an integral of sin³x, or a Fourier-style rewrite. Odd powers reduce to a sum of sines of x, 3x, 5x … with binomial-looking coefficients.
Why it is true
From sin 3x = 3 sin x − 4 sin³x — which itself comes from the sine sum formula applied to x + 2x — and then solving for sin³x.
The full line-by-line version is on the proof page for reduction formula for sin³; the “how would I find this myself” version is in the derivation.
Where it comes from
Odd powers keep the same shape. Every odd power of sine reduces to a sum of sines of odd multiples of x. The pattern is worth knowing even if you re-derive the coefficients.
- sin¹: sin xBase case.
- sin³: (3 sin x − sin 3x)/4This identity.
- sin⁵: (10 sin x − 5 sin 3x + sin 5x)/16Next step of the same pattern — re-derive rather than memorise.
Worked examples
- ∫ sin³x dx−cos x + cos³x/3 + C
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° | 0.002224 | 0.002224 | yes |
| 15° | 0.017338 | 0.017338 | yes |
| 18° | 0.029508 | 0.029508 | yes |
| 22.5° | 0.056043 | 0.056043 | yes |
| 30° | 0.125 | 0.125 | yes |
| 37° | 0.217966 | 0.217966 | yes |
| 45° | 0.353553 | 0.353553 | yes |
| 53° | 0.509385 | 0.509385 | yes |
| 60° | 0.649519 | 0.649519 | yes |
| 67.5° | 0.788581 | 0.788581 | yes |
| 75° | 0.901221 | 0.901221 | yes |
| 90° | 1 | 1 | yes |
| 120° | 0.649519 | 0.649519 | yes |
| 135° | 0.353553 | 0.353553 | yes |
| 150° | 0.125 | 0.125 | yes |
| 180° | 0 | 0 | yes |
| 210° | -0.125 | -0.125 | yes |
| 240° | -0.649519 | -0.649519 | yes |
| 270° | -1 | -1 | yes |
| 300° | -0.649519 | -0.649519 | yes |
| 330° | -0.125 | -0.125 | yes |
| 360° | -0 | 0 | yes |
The mistake students make
Guessing the coefficient pattern. For sin³ it is (3 sin x − sin 3x)/4; for cos³ it is (3 cos x + cos 3x)/4 — the signs differ between the two.
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 power-reducing identities.
Related identities
- cos³x = (3 cos x + cos 3x)/4Reduction formula for cos³detailsproof
- sin²x = (1 − cos 2x)/2Power-reducing formula for sin²detailsproof
- sin 2x = 2 sin x cos xDouble-angle identity for sinedetailsproof
- sin(A + B) = sin A cos B + cos A sin BSine of a sumdetailsproof
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