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Question: The solution set of \((5 + 4\cos\theta)(2\cos\theta + 1) = 0\)in the interval \(\lbrack 0,2\pi\rbrac...

The solution set of (5+4cosθ)(2cosθ+1)=0(5 + 4\cos\theta)(2\cos\theta + 1) = 0in the interval [0,2π]\lbrack 0,2\pi\rbrack is

A

{π3,2π3}\left\{ \frac{\pi}{3},\frac{2\pi}{3} \right\}

B

{π3,π}\left\{ \frac{\pi}{3},\pi \right\}

C

{2π3,4π3}\left\{ \frac{2\pi}{3},\frac{4\pi}{3} \right\}

D

{2π3,5π3}\left\{ \frac{2\pi}{3},\frac{5\pi}{3} \right\}

Answer

{2π3,4π3}\left\{ \frac{2\pi}{3},\frac{4\pi}{3} \right\}

Explanation

Solution

(5+4cosθ)(2cosθ+1)=0(5 + 4\cos\theta)(2\cos\theta + 1) = 0

cosθ=54\cos\theta = \frac{- 5}{4} which is not possible

2cosθ+1=0\therefore 2\cos\theta + 1 = 0 or sin2x=π2.|\sin 2x| = \frac{\pi}{2}. sin2x\mathbf{|}\mathbf{\sin}\mathbf{2}\mathbf{x|} π\mathbf{\pi}

Solution set is sinθ3cosθ\sin\theta - \sqrt{3}\cos\theta