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WASpecialist MathematicsSyllabus dot point

How do trigonometric identities convert products and powers of sine and cosine into integrable forms?

Integrate trigonometric functions by first applying double-angle, Pythagorean and product identities

WACE Specialist Unit 4 trigonometric integration: using double-angle identities to integrate sine and cosine squared, the Pythagorean identity for odd powers, and product-to-sum identities, with a worked example.

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  1. What this dot point is asking
  2. Even powers: the double-angle identities
  3. Odd powers: split and substitute
  4. Products of different angles

What this dot point is asking

SCSA wants you to choose the right identity to transform a trigonometric integrand into a sum of standard integrals.

Even powers: the double-angle identities

The integrals of sin⁡2x\sin^2 x and cos⁡2x\cos^2 x rely on lowering the power. From the double-angle formula for cosine,

Each right-hand side integrates termwise to a sum of a linear term and a sin⁡2x\sin 2x term. Higher even powers are reduced by applying these identities repeatedly.

Odd powers: split and substitute

For an odd power such as sin⁡3x\sin^3 x, peel off one factor of sin⁡x\sin x and convert the remaining even power using sin⁡2x=1−cos⁡2x\sin^2 x = 1 - \cos^2 x. Then substitute u=cos⁡xu = \cos x, since du=−sin⁡x dxdu = -\sin x\,dx supplies the leftover factor. The same idea handles odd powers of cosine with u=sin⁡xu = \sin x.

Products of different angles

For products like sin⁡mxcos⁡nx\sin mx \cos nx, use the product-to-sum identities to rewrite as a sum of single trigonometric terms, each of which integrates directly.