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Triple-angle sine identity, a cubic equation in sine, and polynomials from sine of a multiple of arcsine

Triple-angle sine identity, a cubic equation in sine, and polynomials from sine of a multiple of arcsine

IB Year 6 | Grade 12
11 marks
  1. Starting from \(\sin3\theta=\sin(2\theta+\theta)\), prove that \(\sin3\theta=3\sin\theta-4\sin^3\theta\).[3]
  2. Find the exact values of \(\sin\theta\) for which \(\sin3\theta=\sin^2\theta\).[2]
  3. Hence or otherwise solve \(\sin3\theta=\sin^2\theta\) for \(0\le\theta\le\pi\), giving any answer that is not a simple multiple of \(\pi\) correct to \(3\) significant figures.[2]
  4. The function \(\mathrm{S}_n\) is defined by \(\mathrm{S}_n(x)=\sin(n\arcsin x)\) for \(-1\le x\le1\), where \(n\) is an odd positive integer.
    1. Write down \(\mathrm{S}_1(x)\).
    2. Show that \(\mathrm{S}_3(x)=3x-4x^3\).
    3. Find the exact values of \(x\) for which \(\mathrm{S}_3(x)=0\).[4]

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Answer:(b) \(\sin\theta=0,\ \dfrac34,\ -1\) (c) \(\theta=0,\ 0.848,\ 2.29,\ \pi\) (d)(i) \(\mathrm{S}_1(x)=x\) (d)(ii) \(\mathrm{S}_3(x)=3x-4x^3\) (d)(iii) \(x=0,\ \pm\dfrac{\sqrt3}{2}\)

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