Power Reduction Formulas
Reference for power reduction formulas that rewrite sin2(x) and cos2(x) as first-power trig expressions.
Essential for integration and Fourier analysis.
The Formulas
sin²(θ) = (1 - cos(2θ)) / 2
cos²(θ) = (1 + cos(2θ)) / 2
tan²(θ) = (1 - cos(2θ)) / (1 + cos(2θ))
cos²(θ) = (1 + cos(2θ)) / 2
tan²(θ) = (1 - cos(2θ)) / (1 + cos(2θ))
Power reduction formulas convert squared trig functions into expressions with no exponents. They are derived from the double angle formulas and are extremely useful in calculus for integration.
Variables
| Symbol | Meaning |
|---|---|
| θ | The angle (in radians or degrees) |
| sin²(θ) | Sine of θ, squared |
| cos(2θ) | Cosine of twice the angle |
Example 1
Rewrite sin²(30°) using the power reduction formula
sin²(30°) = (1 - cos(60°)) / 2
= (1 - 0.5) / 2 = 0.5 / 2
= 0.25 (which matches sin(30°) = 0.5, and 0.5² = 0.25)
Example 2
Simplify cos⁴(θ)
cos⁴(θ) = (cos²(θ))² = ((1 + cos(2θ))/2)²
= (1 + 2cos(2θ) + cos²(2θ)) / 4
Apply power reduction again to cos²(2θ): (1 + cos(4θ))/2
= (3 + 4cos(2θ) + cos(4θ)) / 8
When to Use Them
Use power reduction formulas when:
- Integrating sin²(x), cos²(x), or higher powers in calculus
- Simplifying trigonometric expressions for easier computation
- Converting squared trig functions in physics (e.g., energy equations)
- Working with Fourier analysis or signal processing
Key Notes
- The formulas are derived from the double angle identity cos(2θ) = 1 − 2sin²θ = 2cos²θ − 1; solving for sin²θ and cos²θ gives the power reduction forms directly
- Primary use in calculus: ∫sin²x dx is unsolvable without substitution, but applying power reduction gives ∫(1 − cos 2x)/2 dx = x/2 − sin(2x)/4 + C immediately
- Applying the formula repeatedly reduces sinⁿ(x) or cosⁿ(x) for any even power n to a sum of first-power cosine terms — the foundation of Fourier series analysis
- These are exact identities, not approximations — they hold for every value of θ without restriction
Key Notes
- Core formulas: sin²θ = (1 − cos2θ)/2; cos²θ = (1 + cos2θ)/2; sin²θ cos²θ = (1 − cos4θ)/8. These reduce even powers of trig functions to first-power cosines of double (or quadruple) angles — eliminating powers entirely.
- Derived from double-angle formulas: cos2θ = 1 − 2sin²θ → sin²θ = (1 − cos2θ)/2. Similarly cos2θ = 2cos²θ − 1 → cos²θ = (1 + cos2θ)/2. Both derivations are immediate rearrangements of the same double-angle identity.
- Integration of even powers: ∫sin²θ dθ = ∫(1−cos2θ)/2 dθ = θ/2 − sin2θ/4 + C. Without power reduction, integrating sin²θ directly is impossible using elementary methods. Higher powers (sin⁴θ, cos⁴θ) require applying the formula twice.
- Connection to Fourier series: Power reduction formulas appear when computing Fourier coefficients of periodic signals — products and powers of sinusoids must be reduced to sums of pure sinusoids before integrating over a period.
- Applications: Power reduction formulas are essential for integrating even powers of trig functions (physics: average power in AC circuits, average kinetic energy in oscillating systems), simplifying trig expressions in signal processing, and proving further trigonometric identities by systematic degree reduction.