The number of solutions of $2{cos}^2\left(\frac{x}{2}\right){sin}^{2}x=x^2+\frac{1}{x^2}\text{, 0}\le x\le \frac{\pi }{2}$ is

Important Questions on Trigonometric Ratios and Identities

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

Let $y=4{\sin }^2\theta -\cos 2\theta .$ If $l$ and $m$ are the minimum and maximum values of $y$ respectively, then

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

The equation $\sin x+\sin y+\sin z=-3$ for $0\le x\le 2\pi ,0\le y\le 2\pi ,0\le z\le 2\pi$ has

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

Let $\alpha =\underset{x\in R}{\max }\left\{8^{2\sin 3x}·4^{4\cos 3x}\right\}$ and $\beta =\underset{x\in R}{\min }\left\{8^{2\sin 3x}·4^{4\cos 3x}\right\}.$ If $8x^2+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5},$ then the value of $c-b$ is equal to :

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

If $\cos \left({\theta }_1\right)+\cos \left({\theta }_2\right)+\cos \left({\theta }_3\right)+\cos \left({\theta }_4\right)=-4,$ then the value of $\cot \left(\frac{{\theta }_1}{2}\right)+\cot \left(\frac{{\theta }_2}{2}\right)+\cot \left(\frac{{\theta }_3}{2}\right)+\cot \left(\frac{{\theta }_4}{2}\right)=$
 

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

Maximum value of $\left(2{\cos }^{2}18°-\sin 18°\right)\left(\cos \theta +3\sqrt{2}\cos \left(\theta +\frac{\pi }{4}\right)+3\right)$ is

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

The number of solutions of $2\sin x+\cos x=3$ is

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

The number of integral values of $k$ for which the equation $3\sin x+4\cos x=k+1$ has a solution, $k\in R$ is _______.

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

The maximum and minimum value of function $f\left(x\right)=\sin 3x+4$ are respectively : 

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

The maximum value of $3\cos \theta +5\sin \left(\theta -\frac{\pi }{6}\right)$ for any real value of $\theta$ is :

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

The maximum and minimum values of the function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by
$f\left(x\right)=5\cos x+3\cos \left(x+\frac{\pi }{3}\right)+8$ for all $x\in \mathrm{ℝ},$ are respectively

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

If $A=\left\{x\in [0,2\pi ]/\tan x-{\tan }^{2}x>0\right\}$ and $B=\left\{x\in [0,2\pi ]/\left|\sin x\right|<\frac{1}{2}\right\},$ then $\mathrm{A}\cap \mathrm{B}=$

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

If the equation ${\cos }^4\theta +{\sin }^4\theta +\lambda =0$ has real solutions for $\theta$, then $\lambda$ lies in interval

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

The graphs of the polynomial $x^2-1$ and $\cos x$ intersect

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

In a $∆ABC,$ if $\angle C=90°, \frac{a^2+b^2}{a^2-b^2}\sin (A-B)=1$ and $0<B<45°,$ then

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

If  $f\left(x\right)=\sqrt{\left[\right(\sin 2x+2\left)\right(\cos 2x+2\left)\right]}$ for $x$ is a real number, then which of the following values lies in the domain of $f\left(x\right)$ but NOT in the range of $f\left(x\right) ?$

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

If $x>0$, then values of $f\left(x\right)=-3\sin \sqrt{3+x+x^2}$ lie in the interval

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

Identify the following function.

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

If $e^{\sin x}-e^{-\sin x}-4=0,$ then the number of real values of $x$ is

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

Equation $\sin x+\cos \left(t+x\right)+\cos \left(t-x\right)=2$ has real solution, then $\sin t$ can be

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Range of Trigonometric Expressions

The maximum value of $5$$\cos \theta +3\cos \left(\theta +\frac{\pi }{3}\right)+3$ is