IB Math HL Trigonometry Test 2 2013 P2 Q6

IB Math HL Trigonometry Test 2 2013 P2 Q6

IB Year 6 | Grade 12
11 marks
  1. Using \(\cos(A+B)\) and suitable double-angle identities, prove \(\cos3\theta=4\cos^3\theta-3\cos\theta\).[3]
  2. Find the exact values of \(\cos\theta\) satisfying \(\cos3\theta=11\cos^2\theta\).[2]
  3. Hence or otherwise solve \(\cos3\theta=11\cos^2\theta\) for \(0\le\theta\le\pi\).[2]
  4. Let \(T_n(x)=\cos(n\arccos x)\), where \(x\in[-1,1]\) and \(n\) is a positive integer.
    1. Find \(T_1(x)\).
    2. Show that \(T_2(x)=2x^2-1\).
    3. Express \(T_3(x)\) as a polynomial of degree \(3\).[4]

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Answer:(b) \(\cos\theta=0\) or \(-\frac14\). (c) \(\theta=\frac{\pi}{2}\) or \(1.82\text{ rad}\). (d) \(T_1=x\), \(T_2=2x^2-1\), \(T_3=4x^3-3x\).

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