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Trigonometry

Higher Angles

Double-angle and triple-angle formulas, and related identities.

Higher-angle identities express functions of 2θ and 3θ in terms of θ. They follow from repeated use of the sum formulas: sin(2θ) = sin(θ+θ) expanded with the sine-sum formula gives 2 sin θ cos θ, and the same trick produces every other identity here. Use them to simplify integrals, solve polynomial-in-sine equations, and reduce high powers to linear combinations of multiple angles. The three forms of cos(2θ) are each the right starting point for different problems.

θ2θ(cos 2θ, sin 2θ)
Double-angle identities express functions of 2θ in terms of θ.

Double-angle for sine, with tangent form.

sin⁡(2θ)=2sin⁡ θ cos⁡ θ=2tan⁡θ(1+tan⁡2θ)\sin\left(2\theta\right)=2\sin\text{ }\theta\text{ }\cos\text{ }\theta=\frac{2\tan\theta}{\left(1+\tan^{2}\theta\right)}

Double-angle for cosine, three equivalent forms.

cos⁡(2θ)=cos⁡2θ−sin⁡2θ=1−2sin⁡2θ=2cos⁡2θ−1\cos\left(2\theta\right)=\cos^{2}\theta-\sin^{2}\theta=1-2\sin^{2}\theta=2\cos^{2}\theta-1

Double-angle for tangent.

tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan\left(2\theta\right)=\frac{2\tan\theta}{1-\tan^{2}\theta}

Triple-angle for sine.

sin⁡(3θ)=3sin⁡θ−4sin3θ\sin\left(3\theta\right)=3\sin\theta-4sin^{3}\theta

Triple-angle for cosine.

cos⁡(3θ)=4cos⁡3θ−3cos⁡θ\cos\left(3\theta\right)=4\cos^{3}\theta-3\cos\theta

Triple-angle for tangent.

tan⁡(3θ)=3tan⁡θ−tan3θ1−3tan⁡2θ\tan\left(3\theta\right)=\frac{3\tan\theta-tan^{3}\theta}{1-3\tan^{2}\theta}

Power-reduction from the cosine double-angle form, used to lower powers of cosine in integrals.

1+cos⁡2θ=2cos⁡2θ1+\cos2\theta=2\cos^{2}\theta

Power-reduction for sine.

1−cos⁡2θ=2sin⁡2θ1-\cos2\theta=2\sin^{2}\theta

Perfect-square form, used to factor a sin-cos combination.

1+sin⁡2θ=[cos⁡θ+sin⁡θ]21+\sin2\theta=\left[\cos\theta+\sin\theta\right]^{2}

Difference form.

1−sin⁡2θ=[cos⁡θ−sin⁡θ]21-\sin2\theta=\left[\cos\theta-\sin\theta\right]^{2}