Formulas & tables/Mathematics/Trigonometry/Pythagorean trigonometric identity

Pythagorean trigonometric identity

sin⁡2α+cos⁡2α=1\sin^2\alpha+\cos^2\alpha=1

Result unit: — (= 1)

What it describes

The fundamental trigonometric identity (the Pythagorean identity).

Symbols

α\alpha
angle

Where it comes from

For any angle, the sum of the squares of the sine and the cosine equals 1 — it follows from the Pythagorean theorem on the unit circle. It lets you compute one function from the other.

When to use it

Finding cos⁡α\cos\alpha from sin⁡α\sin\alpha (and vice versa), simplifying expressions.

Example

sin⁡α=0.6\sin\alpha=0.6. Find cos⁡α\cos\alpha (acute angle).
Substitutioncos⁡2α=1−0.36\cos^2\alpha=1-0.36.
Resultcos⁡α=0.64=0.8\cos\alpha=\sqrt{0.64}=0.8.

Related formulas

Trigonometric functions in a right triangle Tangent as a ratio

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