JEE Main 2022 — Trigonometry

The number of solutions of $|\mathrm{cos}x|=\mathrm{sin}x$, such that $-4\pi \leq x\leq 4\pi$ is

If tan θ = 3/4 and θ is in the third quadrant then sin θ is:

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation ${\mathrm{cos}}^{-1}(x)-2{\mathrm{sin}}^{-1}(x)={\mathrm{cos}}^{-1}(2x)$ is equal to

If ${\mathrm{sin}}^{2}(10^{\circ})\mathrm{sin}(20^{\circ})\mathrm{sin}(40^{\circ})\mathrm{sin}(50^{\circ})\mathrm{sin}(70^{\circ})=\alpha -$ $\frac{1}{16}\mathrm{sin}(10^{\circ})$, then $16+{\alpha }^{-1}$ is equal to _____.

$\alpha =\mathrm{sin}36^{\circ}$ is a root of which of the following equation

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

Let $S={\theta \in (0,\frac{\pi }{2}):\sum _{m=1}^{9}\mathrm{sec}(\theta +(m-1)\frac{\pi }{6})\mathrm{sec}(\theta +\frac{m\pi }{6})=-\frac{8}{\sqrt{3}}}$. Then

$16\mathrm{sin}(20^{\circ})\mathrm{sin}(40^{\circ})\mathrm{sin}(80^{\circ})$ is equal to

If the sum of solutions of the system of equations $2{\mathrm{sin}}^{2}\theta -\mathrm{cos}2\theta =0$ and $2{\mathrm{cos}}^{2}\theta +3\mathrm{sin}\theta =0$ in the interval $[0,2\pi ]$ is $k\pi$, then $k$ is equal to _______.

The number of solutions of the equation $\mathrm{sin}x={\mathrm{cos}}^{2}x$ in the interval $(0,10)$ is ______.

If $\mathrm{cot}\alpha =1$ and $\mathrm{sec}\beta =-\frac{5}{3}$, where $\pi <\alpha <\frac{3\pi }{2}$ and $\frac{\pi }{2}<\beta <\pi$, then the value of $\mathrm{tan}(\alpha +\beta )$ and the quadrant in which $\alpha +\beta$ lies, respectively are

The number of solutions of the equation $\mathrm{cos}(x+\frac{\pi }{3})\mathrm{cos}(\frac{\pi }{3}-x)=\frac{1}{4}{\mathrm{cos}}^{2}2x$, $x\in [-3\pi ,3\pi ]$ is:

For $k\in \mathbb{R}$, let the solutions of the equation $\mathrm{cos}({\mathrm{sin}}^{-1}(x\mathrm{cot}({\mathrm{tan}}^{-1}(\mathrm{cos}({\mathrm{sin}}^{-1}x)))))=k,0<|x|<\frac{1}{\sqrt{2}}$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation ${x}^{2}-bx-5=0$ are $\frac{1}{{\alpha }^{2}}+\frac{1}{{\beta }^{2}}$ and $\frac{\alpha }{\beta }$, then $\frac{b}{{k}^{2}}$ is equal to ______.

Let $x=\mathrm{sin}(2{\mathrm{tan}}^{-1}\alpha )$ and $y=\mathrm{sin}(\frac{1}{2}{\mathrm{tan}}^{-1}\frac{4}{3})$. If $S={\alpha \in \mathbb{R}:{y}^{2}=1-x}$, then $\underset{\alpha \in S}{\sum }16{\alpha }^{3}$ is equal to _______.

If $0<x<\frac{1}{\sqrt{2}}$ and $\frac{{\mathrm{sin}}^{-1}x}{\alpha }=\frac{{\mathrm{cos}}^{-1}x}{\beta }$, then a value of $\mathrm{sin}(\frac{2\pi \alpha }{\alpha +\beta })$ is

$\mathrm{tan}(2{\mathrm{tan}}^{-1}\frac{1}{5}+{\mathrm{sec}}^{-1}\frac{\sqrt{5}}{2}+2{\mathrm{tan}}^{-1}\frac{1}{8})$ is equal to:

The value of ${\mathrm{tan}}^{-1}[\frac{\mathrm{cos}(\frac{15\pi }{4})-1}{\mathrm{sin}(\frac{\pi }{4})}]$ is equal to

The value of $\mathrm{cot}(\sum _{n=1}^{50}{\mathrm{tan}}^{-1}(\frac{1}{1+n+{n}^{2}}))$ is

${\mathrm{sin}}^{-1}(\mathrm{sin}\frac{2\pi }{3})+{\mathrm{cos}}^{-1}(\mathrm{cos}\frac{7\pi }{6})+{\mathrm{tan}}^{-1}(\mathrm{tan}\frac{3\pi }{4})$ is equal to

Let $x\times y={x}^{2}+{y}^{3}$ and $(x\times 1)\times 1=x\times (1\times 1)$. Then a value of $2{\mathrm{sin}}^{-1}(\frac{{x}^{4}+{x}^{2}-2}{{x}^{4}+{x}^{2}+2})$ is

The set of all values of $k$ for which ${({\mathrm{tan}}^{-1}x)}^{3}+{({\mathrm{cot}}^{-1}x)}^{3}=k{\pi }^{3},x\in R$, is the interval

The value of $\mathrm{cos}(\frac{2\pi }{7})+\mathrm{cos}(\frac{4\pi }{7})+\mathrm{cos}(\frac{6\pi }{7})$ is equal to

If the inverse trigonometric functions take principal values, then ${\mathrm{cos}}^{-1}(\frac{3}{10}\mathrm{cos}({\mathrm{tan}}^{-1}(\frac{4}{3}))+\frac{2}{5}\mathrm{sin}({\mathrm{tan}}^{-1}(\frac{4}{3})))$ is equal to

The value of $2\mathrm{sin}12^{\circ}-\mathrm{sin}72^{\circ}$ is

The value of $\underset{n\rightarrow \infty }{\mathrm{lim}}6\mathrm{tan}{\sum _{r=1}^{n}{\mathrm{tan}}^{-1}(\frac{1}{{r}^{2}+3r+3})}$ is equal to

Let $S={\theta \in [0,2\pi ]:{8}^{2{\mathrm{sin}}^{2}\theta }+{8}^{2{\mathrm{cos}}^{2}\theta }=16}$. Then $n(S)+\underset{\theta \in S}{\sum }(\mathrm{sec}(\frac{\pi }{4}+2\theta )cosec(\frac{\pi }{4}+2\theta ))$ is equal to:

$50\mathrm{tan}(3{\mathrm{tan}}^{-1}(\frac{1}{2})+2{\mathrm{cos}}^{-1}(\frac{1}{\sqrt{5}}))+$$4\sqrt{2}\mathrm{tan}(\frac{1}{2}{\mathrm{tan}}^{-1}(2\sqrt{2}))$ is equal to ______.

The number of elements in the set $S={x\in \mathbb{R}:2\mathrm{cos}(\frac{{x}^{2}+x}{6})={4}^{x}+{4}^{-x}}$ is