Power Reduction Formulas
Reduce powers of sine and cosine to first-power expressions.
Essential for integration in calculus.
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