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Double Angle Formula

Maths: Edexcel A Level Pure Maths, topic 5.4.5

Double Angle Formulae

The double angle formulae are special cases of the angle addition formulae where A=BA = B.

Sine

Sine

  • The double angle formula for sine is:
  • sin⁡2A≡sin⁡Acos⁡A+cos⁡Asin⁡A≡2sin⁡Acos⁡A\sin2A \equiv \sin A\cos A + \cos A\sin A\equiv 2\sin A\cos A
Cosine

Cosine

  • The double angle formula for cosine is:
  • cos⁡2A≡cos⁡Acos⁡A−sin⁡Asin⁡A≡cos⁡2A−sin⁡2A\cos2A \equiv \cos A\cos A - \sin A\sin A\equiv \cos^2A - \sin^2A
Cosine cont.

Cosine cont.

  • We can find equivalent identities by using the fact that cos⁡2A+sin⁡2A=1\cos^2A + \sin^2A = 1:
  • cos⁡2A≡1−2sin⁡2A≡2cos⁡2A−1\cos2A\equiv 1-2\sin^2A\equiv 2\cos^2A-1
Tangent

Tangent

  • The double angle formula for tangent is:
  • tan⁡2A≡tan⁡A+tan⁡A1−tan⁡Atan⁡A≡2tan⁡A1−tan⁡2A\tan2A\equiv \frac{\tan A+\tan A}{1 - \tan A \tan A}\equiv\frac{2\tan A}{1-\tan^2 A}

Test yourself on Double Angle Formula

  • Which of the following expressions is equivalent to tan⁡160\tan160?

  • What is 1−2sin⁡2π121-2\sin^2\frac{\pi}{12} equivalent to?

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