Trigonometric Inequalities

Find the solution set of $\left|\sqrt{3}\cos x-\sin x\right|\ge 2$ for $x\in (0,4\mathrm{\pi }]$.

Solve the inequality $\sin 2x>\sqrt{2}{\sin }^{2}x+\left(2-\sqrt{2}\right){\cos }^{2}x$.

$2\sin \left(\frac{A}{2}\right)=\sqrt{1+\sin A}-\sqrt{1-\sin A};$ then

Solve, $\sin x>0 \& \cos x<0$, $x\in (0,2\pi )$

Solve, $\sin x+\tan x>0$, $x\in \left(0, \frac{\pi }{2}\right)$

Solve $\left|\cos x\right|\le \frac{1}{2}$

Solve ${\tan }^{3}x-{\tan }^{2}x-3\tan x+3>0$

Show ${\sin }^2\left(\frac{A}{2}\right)+{\sin }^2\left(\frac{B}{2}\right)+{\sin }^2\left(\frac{C}{2}\right)\ge \frac{3}{4}$, where, $A,B,C$ are the angles of a triangle?

The solution set of the inequality $\sqrt{5-2\sin x}\ge 6\sin x-1$ is $\left[2n\pi ,2n\pi +{\theta }_1\right]\cup [2n\pi +{\theta }_2,2n\pi +2\pi )(n\in Z)$ then find the value of $\frac{{\theta }_1+{\theta }_2}{\pi }$, given that ${\theta }_1$ is an acute angle and ${\theta }_2$ is an obtuse angle.

Let $\mathrm{\theta }\in [0, 2\mathrm{\pi }],$ then find the number of integral values of $\mathrm{\theta }$ satisfying $\cos \left(\mathrm{sin\theta }\right)>\sin \left(\mathrm{cos\theta }\right)$.

The complete interval of values of $x$ in $\left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$ satisfying the inequality $\cos x·\cos 2x>-\frac{1}{4}$ and $3\cos x>1,$ is

Embibe Multiple Choice

The number of values of $x\in [0,4\pi ]$ satisfying $|\sqrt{3}\cos x-\sin x|\ge 2$ is

The set of values of $\theta$ satisfying the inequation $2 {\sin }^2\theta -5 \sin \theta +2>0,$ where $0<\theta <2\pi ,$ is

The solution set of the system of inequations $2{\sin }^{2}x-3\sin x+1\le 0$ and $x^2+x-12\le 0$ has

The expression $\cos 3\theta +\sin 3\theta +(2\sin 2\theta -3)(\sin \theta -\cos \theta )$ is positive for all $\theta$ in

If $\sin x+\sin y\ge \cos \alpha \cos x \forall x\in R$, then $\sin y+\cos \alpha$ is equal to ________ .

If $\sin x+\sin y\ge cos\alpha \cos x \forall x\in R$, then ($\sin y+\cos \mathrm{\alpha )}$ is equal to

If $x\in \left(0,\pi \right)$ and $\sin x·{\cos }^{3}x>\cos x·{\sin }^3\mathrm{x }$ then complete set of values of $x$ is

The solution set of $\left|\sin x\right|\le \left|\cos 2x\right|$ contains