Trigonometric equations

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Question

Number of integral values of k $7 \cos x + 5 \sin x = 2 k + 1$ for which the equation has solution

Moderate

Solution

$W e h a v e - \sqrt{a^{2} + b^{2}} \leq a \cos θ + b \sin θ \leq \sqrt{a^{2} + b^{2}}$
$\Rightarrow - \sqrt{74} \leq 7 \cos x + 5 \sin x \leq \sqrt{74}$
$\Rightarrow - \sqrt{74} \leq 2 k + 1 \leq \sqrt{74}$
$\Rightarrow k = - 4, - 3, - 2, - 1, 0, 1, 2, 3$

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