How the Power-Reducing Formulas Work
Power-Reducing Formulas:
sin²θ = (1 − cos 2θ) / 2
cos²θ = (1 + cos 2θ) / 2
sin³θ = (3 sin θ − sin 3θ) / 4
cos³θ = (3 cos θ + cos 3θ) / 4
sin⁴θ = (3 − 4 cos 2θ + cos 4θ) / 8
cos⁴θ = (3 + 4 cos 2θ + cos 4θ) / 8
Rewrites a power of sin or cos as a sum of first-power sines/cosines of a multiple angle
Power-reducing formulas (also called power-reduction identities) rewrite a trigonometric function raised to an integer power as a sum involving only first powers of sine or cosine, evaluated at a multiple of the original angle (2θ, 3θ, or 4θ). The squared versions follow directly from the cosine double-angle identity cos 2θ = 1 − 2 sin²θ = 2 cos²θ − 1, solved for sin²θ and cos²θ. The cubic and quartic versions are built by repeating angle-sum identities (or expanding Euler's formula) one step further. This calculator evaluates both the reduced expression and the direct power trig(θ)ⁿ so you can confirm the two agree.
The six formulas used here
- sin²θ = (1 − cos 2θ) / 2 — from cos 2θ = 1 − 2 sin²θ
- cos²θ = (1 + cos 2θ) / 2 — from cos 2θ = 2 cos²θ − 1
- sin³θ = (3 sin θ − sin 3θ) / 4 — from the triple-angle identity for sin 3θ
- cos³θ = (3 cos θ + cos 3θ) / 4 — from the triple-angle identity for cos 3θ
- sin⁴θ = (3 − 4 cos 2θ + cos 4θ) / 8 — square the sin²θ formula and reduce cos²2θ again
- cos⁴θ = (3 + 4 cos 2θ + cos 4θ) / 8 — square the cos²θ formula and reduce cos²2θ again
Common sources of error
- Degrees vs. radians: set the angle unit correctly — the doubled, tripled, or quadrupled angle inside the cosine terms compounds any unit mistake.
- Sign errors: the sine versions use a minus sign (1 − cos 2θ, 3 sin θ − sin 3θ) while the cosine versions use a plus sign; swapping them produces a completely different curve.
- Dropping a multiple-angle term: the quartic formulas need both the 2θ and 4θ terms — omitting either one gives a value that is only approximately right, not exact.
Why the reduced and direct values should match
Because each identity is an algebraic rearrangement of the double- and triple-angle formulas — which themselves follow from the angle-sum identities — sinⁿθ and its power-reduced form are equal for every real θ, not just for special angles. The "Verification difference" result should sit at or near zero, typically on the order of 10⁻¹⁵, reflecting ordinary floating-point rounding in JavaScript's Math library rather than any flaw in the identity itself.
Applications
Power-reducing formulas are the standard technique for integrating even powers of sine and cosine in calculus — for example, ∫sin²x dx becomes straightforward once sin²x is replaced with (1 − cos 2x) / 2. They also appear in signal processing and physics whenever a power of a periodic signal needs to be decomposed into its harmonic (multiple-frequency) components.