The number of solution(s) of the equation sin | vedclass

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

Question

(B) Given equation: $\sin 2	heta + \cos 2	heta = -\frac{1}{2}$Divide both sides by $\sqrt{2}$:$\frac{1}{\sqrt{2}} \sin 2	heta + \frac{1}{\sqrt{2}}

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(B) Given equation: $\sin 2	heta + \cos 2	heta = -\frac{1}{2}$Divide both sides by $\sqrt{2}$:$\frac{1}{\sqrt{2}} \sin 2	heta + \frac{1}{\sqrt{2}} \cos 2	heta = -\frac{1}{2\sqrt{2}}$$\sin(2	heta + \frac{\pi}{4}) = -\frac{1}{2\sqrt{2}}$Since $	heta \in (0, \frac{\pi}{2})$,we have $2	heta \in (0, \pi)$,so $2	heta + \frac{\pi}{4} \in (\frac{\pi}{4}, \frac{5\pi}{4})$.Let $\alpha = 2	heta + \frac{\pi}{4}$. We need to find the number of solutions for $\sin \alpha = -\frac{1}{2\sqrt{2}}$ in the interval $(\frac{\pi}{4}, \frac{5\pi}{4})$.Since $\sin \alpha$ is negative in the third and fourth quadrants,and our interval $(\frac{\pi}{4}, \frac{5\pi}{4})$ covers the third quadrant,there is exactly one value of $\alpha$ in $(\pi, \frac{5\pi}{4})$ such that $\sin \alpha = -\frac{1}{2\sqrt{2}}$.Thus,there is exactly $1$ solution.

The number of solution$(s)$ of the equation $\sin 2\theta + \cos 2\theta = -\frac{1}{2}$ for $\theta \in \left( 0, \frac{\pi}{2} \right)$ is:

Similar Questions

The number of solutions of $|\cos x| = \sin x$ in the interval $-4 \pi \leq x \leq 4 \pi$ is:

What is the number of roots of the quadratic equation $8\sec^2\theta - 6\sec\theta + 1 = 0$?

The number of solutions of the equation $\text{sgn}(\sin x) = \sin^2 x + 2\sin x + \text{sgn}(\sin^2 x)$ in the interval $\left[ -\frac{5\pi}{2}, \frac{7\pi}{2} \right]$ is (where $\text{sgn}(\cdot)$ denotes the signum function):

The number of values of $x$ in the interval $[0, 5\pi]$ satisfying the equation $3 \sin^2 x - 7 \sin x + 2 = 0$ is

The general solution of $\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module