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4 marksTrigonometric Functions, Identities and Equations

CAPE Pure Mathematics Unit 1 · May/June 2000 · Paper 2 · Question 1(b)(iii)

A semicircle PQR has PR as diameter, O as centre, and radius r cm. The angle POQ = θ radians, where 0 < θ < π/2. S₁ is the segment bounded by chord PQ, and S₂ is the segment bounded by chord QR. A diagram illustrates this setup.

Given that the area of S₂ is three times the area of S₁, show that 4θ = π + 2 sin θ.

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Other parts of this question

  1. 1(a)Find the coordinates of A and B.[9 marks]
  2. 1(b)(i)Write down, in terms of θ and r, an expression for the area of S₁.[3 marks]
  3. 1(b)(ii)Write down, in terms of θ and r, an expression for the area of S₂.[4 marks]
  4. 1(b)(iv)Show that when r = 6, θ = π/3, the area of triangle PQR = 18√3.[5 marks]

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