Question

the set of resolutions of sin 5x + sin 3x = square root 3 cos x for 0 degrees smaller is equal to x smaller equal to 360 degrees is .....
Plz write how to get the answer to the question that has multiple choice.
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Answer 1

Answer:

$\sin(5x) + \sin(3x) = \sqrt{3} \cos(x) \\ since \\ \red{ \sin(a) + \sin(b) = 2 \sin( \frac{a + b}{2} \cos( \frac{a - b}{2} ) ) } \\ \\ 2 \sin(4x) \cos(x) = \sqrt{3} \cos(x) \\ \\ \cos(x) \{ 2\sin(4x) - \sqrt{3} \} = 0 \\ \\ \cos(x) = 0 \: \: or \: \: \sin(4x) = \frac{ \sqrt{3} }{2} \\ \\ x = \frac{\pi}{2} or \: \frac{3\pi}{2} \: and \: 4x = 2n\pi \: \pm \frac{\pi}{3} \\ \\ x = \frac{n\pi}{2} \pm \frac{\pi}{12} = 90n + 15 \\ \\ put \: n = 1 \: , \: 2 \: ...\\ x = 15 \:, 105 \: ,195 \:, 285$

So final solution 15° ,90°, 105° ,195° ,270°, 285°