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3.5 Trigonometry (A-level only)

3.5 Trigonometry (A-level only)

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Question 28

Points P(cos⁡A,sin⁡A)P(\cos A,\sin A)P(cosA,sinA) and Q(cos⁡B,−sin⁡B)Q(\cos B,-\sin B)Q(cosB,−sinB) lie on the unit circle with centre OOO, where A>0A>0A>0, B>0 B>0\,B>0 and A+B<πA+B<\piA+B<π.

a.

Explain why ∠POQ=A+B\angle POQ=A+B∠POQ=A+B and use the cosine rule in triangle OPQ OPQ\,OPQ to show that

PQ2=2−2cos⁡(A+B)PQ^2=2-2\cos(A+B)PQ2=2−2cos(A+B).

[2]
b.

By finding PQ2 PQ^2\,PQ2 from the coordinates of P P\,P and QQQ, show that

PQ2=2−2(cos⁡Acos⁡B−sin⁡Asin⁡B)PQ^2=2-2(\cos A\cos B-\sin A\sin B)PQ2=2−2(cosAcosB−sinAsinB).

[2]
c.

Hence prove geometrically that

cos⁡(A+B)≡cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B)\equiv\cos A\cos B-\sin A\sin Bcos(A+B)≡cosAcosB−sinAsinB.

[1]
Markscheme

3.5 Trigonometry (A-level only) Questions

  1. A Level
  2. /Maths
  3. /3.5 Trigonometry (A-level only)

225 exam-style questions on WJEC A Level Maths 3.5 Trigonometry (A-level only), covering 3.5.1 Trigonometry (A-level only), 3.5.2 Trigonometry (A-level only), 3.5.3 Trigonometry (A-level only), 3.5.4 Trigonometry (A-level only), 3.5.5 Trigonometry (A-level only), 3.5.6 Trigonometry (A-level only), 3.5.7 Trigonometry (A-level only), and 3.5.8 Trigonometry (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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