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1.5 Trigonometry

1.5 Trigonometry

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Question 6

A student is asked to derive several double-angle identities for a classroom presentation using the compound angle formulae.

a.

By substituting B=xB = xB=x into the addition formula for sin⁡(x+B)\sin(x + B)sin(x+B), show that sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos xsin2x=2sinxcosx.

[2]
b.

Using the identity for cos⁡(A+B)\cos(A + B)cos(A+B), derive an expression for cos⁡2x \cos 2x\,cos2x in terms of sin⁡x \sin x\,sinx and cos⁡x\cos xcosx.

[2]
ci.

Hence, show that cos⁡2x=2cos⁡2x−1\cos 2x = 2 \cos^2 x - 1cos2x=2cos2x−1.

[2]
cii.

Hence, show that cos⁡2x=1−2sin⁡2x\cos 2x = 1 - 2 \sin^2 xcos2x=1−2sin2x.

[2]
d.

Use the formula for tan⁡(A+B)\tan(A + B)tan(A+B) to derive the identity for tan⁡2x \tan 2x\,tan2x in terms of tan⁡x\tan xtanx.

[2]
Markscheme

1.5 Trigonometry Questions

  1. A Level
  2. /Maths
  3. /1.5 Trigonometry

317 exam-style questions on OCR A Level Maths 1.5 Trigonometry, covering 1.5.1 Definitions for all arguments, 1.5.2 Sine and cosine rules, 1.5.3 Area of a triangle, 1.5.4 Radian measure (A-level only), 1.5.5 Small angle approximations (A-level only), 1.5.6 Graphs of basic trigonometric functions, 1.5.7 Exact values in radians (A-level only), 1.5.8 Reciprocal and inverse trigonometric ratios (A-level only), 1.5.9 Graphs of reciprocal and inverse functions (A-level only), 1.5.10 Trigonometric identities, 1.5.11 Further trigonometric identities (A-level only), 1.5.12 Double angle and compound angle formulae (A-level only), 1.5.13 Geometrical proofs of formulae (A-level only), 1.5.14 Harmonic form Rcos / Rsin (A-level only), 1.5.15 Trigonometric equations, 1.5.16 Proof involving trigonometric functions (A-level only), and 1.5.17 Trigonometric functions in context. Each one has a worked solution and a mark scheme showing where the marks go.

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