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6 marksTrigonometric Functions, Identities and Equations

CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 2 · Question 4(b)(i)

A diagram shows a painting of height h metres on a vertical wall, d metres above an observer O, and x metres away. Angles of inclination to the top and base of the painting are α and β respectively, in radians. The viewing angle is (α – β).

Show that tan (α – β) = hx / (x² + d (d+h)).

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Other parts of this question

  1. 4(a)(i)Solve the equation cos 3A = 0.5 for 0 ≤ A ≤ π.[4 marks]
  2. 4(a)(ii)Show that cos 3A = 4 cos³ A - 3 cos A.[6 marks]
  3. 4(a)(iii)The THREE roots of the equation 4p³ – 3p – 0.5 = 0 all lie between -1 and 1. Use the results in (a) (i) and (ii) to find these roots.[4 marks]
  4. 4(b)(ii)The viewing angle of the painting, (α – β), is at a maximum when x = √h (d+h). Calculate the maximum viewing angle, in radians, when d = 3h.[5 marks]

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