Find the number of solutions of the following equations.

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Find the number of solutions of the following equations. (i) ∣cot⁡x∣=cot⁡x+1sin⁡x|\cot x|=\cot x+\frac{1}{\sin x}, where x∈[0,2π]x\in[0,2\pi] (ii) sin⁡3xcos⁡x+sin⁡2xcos⁡2x+sin⁡xcos⁡3x=1\sin^3x\cos x+\sin^2x\cos^2x+\sin x\cos^3x=1, where x∈[0,2π]x\in[0,2\pi] (iii) 2cos⁡x=∣sin⁡x∣2^{\cos x}=|\sin x|, where x∈[−2π,2π]x\in[-2\pi,2\pi] (iv) ∣cos⁡x∣=[x]|\cos x|=[x], where [ ⋅ ][\,\cdot\,] denotes the greatest integer function (v) x+2tan⁡x=π2x+2\tan x=\frac{\pi}{2}, where x∈[0,2π]x\in[0,2\pi]

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