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3.4 Trigonometry (A-level only)

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Question 30

In an investigation of harmonic motion, a physicist requires the double-angle identities for an angle θ\thetaθ.

a.

Starting with the identity for sin⁡(A+B)\sin(A + B)sin(A+B), find an expression for sin⁡2θ \sin 2\theta\,sin2θ in terms of sin⁡θ \sin \theta\,sinθ and cos⁡θ\cos \thetacosθ.

[2]
b.

Starting with the identity for cos⁡(A+B)\cos(A + B)cos(A+B), find an expression for cos⁡2θ \cos 2\theta\,cos2θ in terms of sin⁡θ \sin \theta\,sinθ and cos⁡θ\cos \thetacosθ.

[2]
ci.

Use the result from part (b) to express cos⁡2θ \cos 2\theta\,cos2θ as a function of cos⁡θ \cos \theta\,cosθ only.

[2]
cii.

Use the result from part (b) to express cos⁡2θ \cos 2\theta\,cos2θ as a function of sin⁡θ \sin \theta\,sinθ only.

[2]
d.

Derive the formula for tan⁡2θ \tan 2\theta\,tan2θ in terms of tan⁡θ \tan \theta\,tanθ using the expansion of tan⁡(A+B)\tan(A + B)tan(A+B).

[2]

3.4 Trigonometry (A-level only) Questions

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