cos-x-3-sin-x-2-cos-2x-

Question Number 121303 by talminator2856791 last updated on 06/Nov/20

\cos x - \sqrt{3} \sin x = 2 \cos 2 x

Commented by liberty last updated on 06/Nov/20

\Rightarrow \cos x - \sqrt{3} \sin x = k \cos (x - α)

k = \sqrt{1^{2} + {(- \sqrt{3})}^{2}} = 2

\tan α = \frac{- \sqrt{3}}{1} (4^{th} quadrant)

α = 300 °

\Rightarrow 2 \cos (x - 300 °) = 2 \cos 2 x

\Rightarrow \cos 2 x = \cos (x - 300 °)

Answered by Bird last updated on 06/Nov/20

c o s x - \sqrt{3} s i n x = 2 c o s (2 x) \Rightarrow

2 (\frac{1}{2} c o s x - \frac{\sqrt{3}}{2} s i n x) = 2 c o s (2 x)

\Rightarrow c o s x c o s (\frac{π}{3}) - s i n x s i n (\frac{π}{3}) = c o s (2 x)

\Rightarrow c o s (x + \frac{π}{3}) = c o s (2 x) \Rightarrow

x + \frac{π}{3} = 2 x + 2 k π o r x + \frac{π}{3} = - 2 x + 2 k π

\Rightarrow - x = - \frac{π}{3} + 2 k π o r 3 x = - \frac{π}{3} + 2 k π \Rightarrow

x = \frac{π}{3} - 2 k π o r x = - \frac{π}{9} + \frac{2 k π}{3}

k \in Z

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