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Trigonometrical Equations
hard
If $0 \le x < 2\pi $ , then the number of real values of $x,$ which satisfy the equation $\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is . . .
A
$7$
B
$9$
C
$3$
D
$5$
(JEE MAIN-2016)
Solution
$\cos x+\cos 4 x+\cos 2 x+\cos 3 x=0$
$\Rightarrow 2 \cos \left(\frac{5 x}{2}\right) \cos \left(\frac{3 x}{2}\right)+2 \cos \left(\frac{5 x}{2}\right) \cos \left(\frac{x}{2}\right)=0$
$\Rightarrow 2 \cos \left(\frac{5 x}{2}\right) 2 \cos x \cos \left(\frac{x}{2}\right)=0$
$\cos x=0 \Rightarrow x=\frac{\pi}{2}, \frac{3 \pi}{2}$
$\cos \frac{x}{2}=0 \Rightarrow x=\pi$
$\frac{5 x}{2}=0 \Rightarrow x=\frac{\pi}{5}, \frac{3 \pi}{5}, \frac{7 \pi}{5}, \frac{9 \pi}{5}$
Standard 11
Mathematics
