Equation reducing to a quadratic in cosine, then the same equation with the angle doubled
Equation reducing to a quadratic in cosine, then the same equation with the angle doubled
IB Year 6 | Grade 12
7 marks
The functions \(\mathrm{f}\) and \(\mathrm{g}\) are given by \(\mathrm{f}(x)=2\sin^2x\) and \(\mathrm{g}(x)=3+3\cos x\).
- Solve the equation \(\mathrm{f}(x)=\mathrm{g}(x)\) for \(-2\pi\le x\le\pi\).[5]
- Hence solve the equation \(\mathrm{f}(2x)=\mathrm{g}(2x)\) for \(-\pi\le x\le\dfrac{\pi}{2}\).[2]
Solution:
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Answer:(a) \(x=-\dfrac{4\pi}{3},\ -\pi,\ -\dfrac{2\pi}{3},\ \dfrac{2\pi}{3},\ \pi\) (b) \(x=-\dfrac{2\pi}{3},\ -\dfrac{\pi}{2},\ -\dfrac{\pi}{3},\ \dfrac{\pi}{3},\ \dfrac{\pi}{2}\)
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