Solve the equation and analyse the role of domain restrictions.

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

Problem

Solve the equation sin⁡x=tan⁡x\sin x = \tan x and analyse the role of domain restrictions. (i) State precisely the set of real numbers for which the equation sin⁡x=tan⁡x\sin x = \tan x is defined. (ii) Find all solutions of sin⁡x=tan⁡x\sin x = \tan x in the interval (−π,π)(-\pi,\pi), expressing each solution in simplest exact form. (iii) Derive the general solution of sin⁡x=tan⁡x\sin x = \tan x for all permissible real values of xx, clearly indicating any restrictions that arise from the domain of tan⁡x\tan x.

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