Consider the trigonometric equation

MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

Consider the trigonometric equation sin⁡x cos⁡x=0\sin x\,\cos x = 0. Answer the following questions. (i) State the zero–product principle for real numbers. (ii) Explain why the equation sin⁡x cos⁡x=0\sin x\,\cos x = 0 can be treated as two separate trigonometric equations. (iii) Obtain the general solution of sin⁡x=0\sin x = 0 and of cos⁡x=0\cos x = 0, and hence write the complete general solution of sin⁡x cos⁡x=0\sin x\,\cos x = 0. (iv) List all principal solutions of sin⁡x cos⁡x=0\sin x\,\cos x = 0 lying in the interval [0,2π)[0,2\pi).

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