Double Angle Formulae

This lesson covers the double angle formulae for sin, cos, and tan — the expressions for sin 2A, cos 2A (in three forms), and tan 2A. These are derived directly from the addition formulae and are used extensively in solving equations, proving identities, and integration.

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Derivation from Addition Formulae

The double angle formulae are simply the addition formulae with B = A.

sin 2A

sin(A + A) = sin A cos A + cos A sin A = 2sin A cos A

sin 2A ≡ 2sin A cos A

cos 2A

cos(A + A) = cos A cos A − sin A sin A = cos²A − sin²A

cos 2A ≡ cos²A − sin²A — (Form 1)

Using sin²A + cos²A ≡ 1:

Replace sin²A with 1 − cos²A:

cos 2A = cos²A − (1 − cos²A) = 2cos²A − 1

cos 2A ≡ 2cos²A − 1 — (Form 2)

Replace cos²A with 1 − sin²A:

cos 2A = (1 − sin²A) − sin²A = 1 − 2sin²A

cos 2A ≡ 1 − 2sin²A — (Form 3)