The number of solutions of the equation sec x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the equation $\sec x \cos 5x + 1 = 0$,we can rewrite it as $\frac{\cos 5x}{\cos x} = -1$,provided $\cos x 
eq 0$.This implies $\cos 5x = -\

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(B) Given the equation $\sec x \cos 5x + 1 = 0$,we can rewrite it as $\frac{\cos 5x}{\cos x} = -1$,provided $\cos x 
eq 0$.This implies $\cos 5x = -\cos x$,which is equivalent to $\cos 5x = \cos(\pi - x)$.The general solution for $\cos 	heta = \cos \alpha$ is $	heta = 2n\pi \pm \alpha$.Case $1$: $5x = 2n\pi + (\pi - x) \implies 6x = (2n+1)\pi \implies x = \frac{(2n+1)\pi}{6}$.For $x \in [0, 2\pi]$,$n = 0, 1, 2, 3, 4, 5$ gives $x = \frac{\pi}{6}, \frac{3\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{9\pi}{6}, \frac{11\pi}{6}$.Checking $\cos x 
eq 0$: $\frac{3\pi}{6} = \frac{\pi}{2}$ and $\frac{9\pi}{6} = \frac{3\pi}{2}$ make $\cos x = 0$,so these are excluded.Remaining values: $\frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$ ($4$ solutions).Case $2$: $5x = 2n\pi - (\pi - x) \implies 4x = (2n-1)\pi \implies x = \frac{(2n-1)\pi}{4}$.For $x \in [0, 2\pi]$,$n = 1, 2, 3, 4$ gives $x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$.None of these make $\cos x = 0$ ($4$ solutions).Total solutions = $4 + 4 = 8$.

The number of solutions of the equation $\sec x \cos 5x + 1 = 0$ in the interval $[0, 2\pi]$ is

Similar Questions

The general value of $\theta$ in the equation $2\sqrt{3} \cos \theta = \tan \theta$ is

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 . . .

Find the general solution of the equation $\cos 4x = \cos 2x$.

If $\sin^2 \theta - 2\cos \theta + \frac{1}{4} = 0$,then the general value of $\theta$ is

The solution set of the trigonometric equation $\tan \theta + 5 \cot \theta = \sec \theta$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module