Trigonify

Topics · 3 identities

Pythagorean

The three Pythagorean identities relating sine, cosine, tangent, and their reciprocals.

Pythagorean identity: sin²θ + cos²θ = 1

sin⁡2θ+cos⁡2θ=1\sin^{2}\theta + \cos^{2}\theta = 1

The Pythagorean identity says sin²θ + cos²θ = 1 for every angle θ - it is the Pythagorean theorem read off the unit circle, where sine and cosine are the legs and the radius is 1. It is the most-used identity in the subject: rearranged it converts any sine into a cosine (and back), which is how most simplification and integration problems get started.

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Pythagorean identity with tangent: 1 + tan²θ = sec²θ

1+tan⁡2θ=sec⁡2θ1 + \tan^{2}\theta = \sec^{2}\theta

Dividing sin²θ + cos²θ = 1 through by cos²θ gives 1 + tan²θ = sec²θ. Reach for it whenever an expression mixes tangents and secants - it is the substitution behind the trigonometric-substitution integral for √(x² + a²).

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Pythagorean identity with cotangent: 1 + cot²θ = csc²θ

1+cot⁡2θ=csc⁡2θ1 + \cot^{2}\theta = \csc^{2}\theta

Dividing sin²θ + cos²θ = 1 through by sin²θ gives 1 + cot²θ = csc²θ. It is the cotangent-cosecant counterpart of the tangent-secant form, and the one to use when an expression is written in terms of sines rather than cosines.

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