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Equations with equal cosines of multiple angles, using the complementary-angle symmetry of sine and cosine

Equations with equal cosines of multiple angles, using the complementary-angle symmetry of sine and cosine

IB Year 6 | Grade 12
9 marks
  1. Solve the equation \(\cos5x=\cos x\) for \(0\le x\le\pi\).[4]
    1. Explain why \(\cos\left(\frac{\pi}{2}-x\right)=\sin x\) for all values of \(x\).
    2. Hence solve the equation \(\cos3x=\sin x\) for \(0\le x\le\pi\).[5]

Solution:

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Answer:(a) \(x=0,\ \frac{\pi}{3},\ \frac{\pi}{2},\ \frac{2\pi}{3},\ \pi\) (b)(ii) \(x=\frac{\pi}{8},\ \frac{5\pi}{8},\ \frac{3\pi}{4}\)

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