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Triple-angle identity for cosine used to solve trigonometric equations and evaluate an expression

Triple-angle identity for cosine used to solve trigonometric equations and evaluate an expression

IB Year 6 | Grade 12
15 marks
  1. Show that \(\cos3x=4\cos^3x-3\cos x\).[3]
  2. Hence, solve the equation \(\cos3x=-2\cos x\) for \(0\le x\le\pi\).[5]
  3. Using your result from part (a), solve \(8\cos^3x-6\cos x+\sqrt2=0\) for \(0\le x\le\pi\).[5]
  4. Find the exact value of \(\cos^340^\circ-\frac34\cos40^\circ\).[2]

Solution:

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Answer:(b) \(x=\frac\pi3,\ \frac\pi2,\ \frac{2\pi}3\) (c) \(x=\frac\pi4,\ \frac{5\pi}{12},\ \frac{11\pi}{12}\) (d) \(-\frac18\)

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