The number of solutions of the equation sin x | vedclass

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

Question

(B) Given the equation $\sin x = \cos^{2} x$.Using the identity $\cos^{2} x = 1 - \sin^{2} x$,we get $\sin x = 1 - \sin^{2} x$.Rearranging the terms,w

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(B) Given the equation $\sin x = \cos^{2} x$.Using the identity $\cos^{2} x = 1 - \sin^{2} x$,we get $\sin x = 1 - \sin^{2} x$.Rearranging the terms,we obtain the quadratic equation $\sin^{2} x + \sin x - 1 = 0$.Let $t = \sin x$. Then $t^{2} + t - 1 = 0$.Using the quadratic formula $t = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$,we find $t = \frac{-1 \pm \sqrt{1 - 4(1)(-1)}}{2} = \frac{-1 \pm \sqrt{5}}{2}$.Since $-1 \le \sin x \le 1$,we must have $\sin x = \frac{-1 + \sqrt{5}}{2} \approx 0.618$ (which is positive) or $\sin x = \frac{-1 - \sqrt{5}}{2} \approx -1.618$ (which is impossible).Thus,$\sin x = \frac{\sqrt{5} - 1}{2}$.In the interval $(0, 10)$,the value $10$ radians is approximately $10 / 3.14 \approx 3.18\pi$.Since $\sin x = k$ (where $0 In $(0, 2\pi)$,there are $2$ solutions.In $(2\pi, 3\pi)$,there are $0$ solutions (as $\sin x$ is negative).In $(3\pi, 3.18\pi)$,there are $2$ solutions.Total number of solutions is $2 + 2 = 4$.

The number of solutions of the equation $\sin x = \cos^{2} x$ in the interval $(0, 10)$ is

Similar Questions

If $x \in(-\pi, \pi)$ then the number of solutions of the equation $2 \sin x \sin 3 x \sin 5 x+\sin 5 x \cos 4 x=0$ is

The most general value of $\theta$ which satisfies both the equations $\tan \theta = -1$ and $\cos \theta = \frac{1}{\sqrt{2}}$ is

The number of solutions of the given equation $a \sin x + b \cos x = c$,where $|c| > \sqrt{a^2 + b^2}$,is

The solutions of $\sin x + \sin 5x = \sin 3x$ in the interval $(0, \frac{\pi}{2})$ are

The general solution of the trigonometric equation $(\sqrt{3}-1) \sin \theta + (\sqrt{3}+1) \cos \theta = 2$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module