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Double-angle identities

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Double-Angle Identities

The double-angle identities give the sine, cosine, or tangent of twice an angle, derived from the sum identities. Sine has one form; cosine has three equivalent forms; tangent has one. Learn each formula and how to use them with a worked example.

Where double-angle identities come from

The double-angle identities are the special case of the sum identities where both angles are the same: sin(θ + θ) becomes sin 2θ. Substituting a = b = θ into sin(a + b) = sin a cos b + cos a sin b gives sin 2θ = 2 sin θ cos θ.

Double-angle identities Sine of 2 theta equals 2 sine theta cosine theta. Cosine of 2 theta has three forms: cosine squared theta minus sine squared theta; 2 cosine squared theta minus 1; and 1 minus 2 sine squared theta. Tangent of 2 theta equals 2 tangent theta over 1 minus tangent squared theta. Double-angle identities sinsin 2θ = 2 sin θ cos θ coscos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ tantan 2θ = 2 tan θ / (1 − tan²θ) Each form comes from the sum identity with a = b = θ.
The double-angle formulas for sine, the three equivalent forms of cosine, and tangent.

The three forms of cos 2θ

Cosine's double-angle identity has three equivalent forms: cos 2θ = cos²θ − sin²θ, or (using the Pythagorean identity to substitute) 2cos²θ − 1, or 1 − 2sin²θ. Each version is useful in a different situation — use whichever one matches the information you already have.

Tangent's double-angle identity

Tangent's version is tan 2θ = 2 tan θ / (1 − tan²θ), obtained the same way from the tangent sum identity.

Worked example

If sin θ = 3/5 and θ is in the first quadrant, find sin 2θ. First find cos θ = 4/5 (from the Pythagorean identity). Then sin 2θ = 2 sin θ cos θ = 2(3/5)(4/5) = 24/25.

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