The graphs of and intersect at points where .

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

Problem

The graphs of y=sin⁡xy=\sin x and y=tan⁡xy=\tan x intersect at points where sin⁡x=tan⁡x\sin x=\tan x. (i) Transform the equation sin⁡x=tan⁡x\sin x=\tan x into an equivalent equation that is easier to solve. (ii) Solve the transformed equation for all solutions in the interval −π<x<π-\pi<x<\pi. (iii) Using the periodicity of the trigonometric functions involved, write the general solution of sin⁡x=tan⁡x\sin x=\tan x for all real xx. (iv) Explain, with reference to the periodicity and key features of the two graphs, why the intersection points found in part (iii) repeat at regular intervals along the xx-axis.

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