The maximum value of the function $3\cos x-4\sin x$ 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

The value of $\cos \frac{\pi }{2^2}\cdot \cos \frac{\pi }{2^3}\cdot \dots \cdot \cos \frac{\pi }{2^{10}}\cdot \sin \frac{\pi }{2^{10}}$ is:

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 _______.

HARD

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

If $S={\sin }^{2}p-\frac{1}{2}\sin \left(2\mathrm{p}\right)\left({\sin }^{2}p\right)+\frac{1}{3}\sin \left(3\mathrm{p}\right)\left({\sin }^{3}p\right)-\frac{1}{4}\sin \left(4\mathrm{p}\right)\left(\left({\sin }^{4}p\right)+\cdots \infty \right.$ then $\tan S=$

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

MEDIUM

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

The value of $\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}$ is equal to

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

HARD

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

In a triangle $ABC$, if $A:B:C=1:2:4$, then $\sec A\sec B\sec C$ is equal to

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

HARD

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

The sum of the series: $\mathrm{sinp}+x\sin \left(p+q\right)+\frac{x^2}{2!}\sin (p+2\mathrm{q})+\cdots \infty$ is

HARD

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

The value of $2\sin \frac{\pi }{22}\sin \frac{3\pi }{22}\sin \frac{5\pi }{22}\sin \frac{7\pi }{22}\sin \frac{9\pi }{22}$ is equal to:

HARD

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

$\sum _{k=1}^6\left(\sin \frac{2\pi k}{7}-i\cos \frac{2\pi k}{7}\right)=$

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