Range of Trigonometric Expressions

The number of integral values of $k$ for which the equation $\mathrm{}\mathrm{}\mathrm{}7\cos x+\mathrm{}\mathrm{}\mathrm{}5\sin x=\mathrm{}\mathrm{}\mathrm{}2k+1$ has a solution, is

The maximum value of the expression, $\left|\sqrt{{\sin }^{2}x+2a^2}-\sqrt{2a^2-1-{\cos }^{2}x}\right|$, where $a$ and $x$ are real numbers is

The all values of expression $\left(5\cos \theta +3\cos \left(\theta +\frac{\pi }{3}\right)+3\right)$ lies between

The maximum value of ($12\sin \theta -9{\sin }^2\mathrm{\theta )}$ is

Let $x$ and $y$ be real numbers such that $x^2+y^2-2x+2y-2=0$. What is the largest possible value of $x^2+y^2-\sqrt{32} ?$

Find the minimum value of $3\cos x+4\sin x+5.$

If $A={\sin }^2\theta +{\cos }^4\theta ,$ then for all values of $\theta ,$ A lies in the interval

If $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x-\frac{1}{2}\sin 2x$ then the range of $f\left(x\right)$ is

The minimum value of $3\cos x+4\sin x+18$ is :

The least value of $\text{'}a\text{'}$ for which the equation $2\sqrt{a}{\sin }^{2}x+\sqrt{a-3}\sin 2x=5+\sqrt{a}$ has at least one solution is _____ .

Number of integral value(s) of $m$ for which the equation $\sin x-\sqrt{3}\cos x=\frac{4m-6}{4-m}$ has solutions, $x\in [0,2\pi ],$ is ______ .

The value of $a+b$ such that the inequality $a\le 5\cos \theta +3\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta +\frac{\pi }{3}\right)+3\le b$ holds true for all the real values of $\theta$ is (equality holds on both sides atleast once for real values of $\theta$)

If $A={\sin }^2\theta +{\cos }^2\theta \left(1-{\sin }^2\theta \right)$ then for all value of $\theta$, A lies in the interval

The maximum value of $12\sin \theta -9{\sin }^2\mathrm{\theta }$ is

If the equation $\sin \theta (\sin \theta +2\cos \theta )=a$ has a real solution, then the shortest interval containing all the possible values of 'a' is

If ${\cos }^{4}x+{\sin }^{4}x-\sin 2x+\frac{3}{4}{\sin }^{2}2x=y$

If $\sin x+\mathrm{cosec}x+\tan y+\cot y=4,$ where $x\in \left(0,\frac{\pi }{2}\right]$ and $y\in \left(0,\frac{\pi }{2}\right),$ then $\tan \frac{y}{2}$ is a root of the equation

If $7\cos x-24\sin x=d\cos \left(x+\alpha \right), 0<\alpha <\frac{\pi }{2}.$ be true for all $x\in R,$ than

The greatest absolute integral value of $k$ for which the statement $7\cos x+5\sin x=2k+1$ is valid

$\sqrt{3}\sin x+\cos x$ is maximum, when $x$ is