An angle in the interval satisfies the equation

MATHEMATICS · Multi-part · ISC · Mathematics · Sets and Functions · Sets and Functions

Problem

An angle θ\theta in the interval 0<θ<2π0<\theta<2\pi satisfies the equation 3sin⁡θ−4cos⁡θ=13\sin\theta-4\cos\theta=1. Answer the following. (i) Express sin⁡θ\sin\theta in terms of cos⁡θ\cos\theta from the given equation. (ii) Eliminate sin⁡θ\sin\theta to form a quadratic equation in cos⁡θ\cos\theta and hence find all possible values of cos⁡θ\cos\theta. (iii) For each value of cos⁡θ\cos\theta obtained, determine the corresponding value(s) of sin⁡θ\sin\theta. (iv) Using the signs of sin⁡θ\sin\theta and cos⁡θ\cos\theta in the relevant quadrants, obtain all angles θ\theta that satisfy the original equation.

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