Power reducing formulas lower the exponent of a trigonometric expression. They are essential for integrating sin²x and cos²x, for computing averages of oscillating quantities (the average of sin² over a period is 1/2 – which is why RMS voltage is V/√2) and for Fourier analysis.
How to use the Power Reducing Calculator
- Angle x
Results update as you type, and everything is calculated in your browser.
How it works
Enter the angle x. The calculator computes cos 2x and applies each formula; the plain values sin x, cos x and tan x are shown for comparison.
What each result means
- sin²x = (1 − cos 2x)/2 = (1 – cos(2×rad([Angle x])))÷2
- cos 2x = cos(2×rad([Angle x]))
- sin x = sin(rad([Angle x]))
- cos x = cos(rad([Angle x]))
Example values
How the main result changes when Angle x changes and everything else stays at the example values above:
| Angle x (°) | sin²x = (1 − cos 2x)/2 |
|---|---|
| 15 | 0.067 |
| 22.5 | 0.1464 |
| 30 | 0.25 |
| 37.5 | 0.3706 |
| 45 | 0.5 |
| 60 | 0.75 |
Formula
sin²x = (1 − cos 2x)/2
cos²x = (1 + cos 2x)/2
tan²x = (1 − cos 2x)/(1 + cos 2x)
Worked example
Inputs — Angle x: 30 °.
Results — sin²x = (1 − cos 2x)/2 = 0.25; cos²x = (1 + cos 2x)/2 = 0.75; tan²x = (1 − cos 2x)/(1 + cos 2x) = 0.3333333333; cos 2x = 0.5; sin x = 0.5; cos x = 0.8660254038; tan x = 0.5773502692.
Key terms
- Power reduction
- Rewriting powers of trig functions as first powers of multiple angles.
Frequently asked questions
What is a power reducing formula?
An identity that rewrites sin²x, cos²x or tan²x without squares, using cos 2x.
How do I integrate sin²x?
∫sin²x dx = ∫(1 − cos 2x)/2 dx = x/2 − sin 2x/4 + C.
Are there formulas for higher powers?
Yes – apply the formulas repeatedly, e.g. sin⁴x = (3 − 4cos 2x + cos 4x)/8.
Where do they come from?
From the double angle formula cos 2x = 1 − 2sin²x = 2cos²x − 1.