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Trigonometric Ratios of Multiple of an Angle

The number of...

Question

The number of solution of the equation $s i n x + s i n 2 x + s i n 3 x = c o s x + c o s 2 x + c o s 3 x, 0 \leq x \leq 2 π$ is

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Solution

The correct option is D
6

$(s i n x + s i n 3 x) + s i n 2 x = (c o s x + c o s 3 x) + c o s 2 x \Rightarrow (s i n 2 x) (2 c o s x + 1) = c o s 2 x (2 c o s x + 1) \Rightarrow c o s x = - \frac{1}{2}, t a n 2 x = 1 \Rightarrow x = 2 n π \pm \frac{2 π}{3}, 2 x = n π + \frac{π}{4} 0 \leq x \leq 2 π x = \frac{π}{8}, \frac{5 π}{8}, \frac{2 π}{3}, \frac{9 π}{8}, \frac{4 π}{3}, \frac{13 π}{8}$

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