Double Angle Formulae

The double angle formulae are special cases of the compound angle formulae with B = A. They are among the most frequently used identities at A-Level (9MA0) and appear in integration, equation solving, and proof questions.

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The Double Angle Formulae

sin(2A)

sin(2A) = 2sinA cosA

cos(2A) — Three Forms

cos(2A) = cos²A - sin²A

cos(2A) = 2cos²A - 1

cos(2A) = 1 - 2sin²A

All three are equivalent (use sin²A + cos²A = 1 to convert between them).

tan(2A)

tan(2A) = 2tanA / (1 - tan²A)

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Deriving from Compound Angle Formulae

sin(2A) = sin(A + A) = sinA cosA + cosA sinA = 2sinA cosA

cos(2A) = cos(A + A) = cosA cosA - sinA sinA = cos²A - sin²A

Using sin²A = 1 - cos²A: cos(2A) = cos²A - (1 - cos²A) = 2cos²A - 1

Using cos²A = 1 - sin²A: cos(2A) = (1 - sin²A) - sin²A = 1 - 2sin²A

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Useful Rearrangements

From cos(2A) = 2cos²A - 1:

cos²A = (1 + cos(2A))/2

From cos(2A) = 1 - 2sin²A:

sin²A = (1 - cos(2A))/2