Trigonometric Ratios and Equations — JEE Main PYQs

Let $P = \{\theta \in [0, 4\pi] : \tan^2\theta \neq 1\}$ and $S = \{a \in \mathbb{Z} : 2(\cos^8\theta - \sin^8\theta)\sec 2\theta = a^2, \theta \in P\}$. Then $n(S)$ is :

If $A = \frac{\sin 3^\circ}{\cos 9^\circ} + \frac{\sin 9^\circ}{\cos 27^\circ} + \frac{\sin 27^\circ}{\cos 81^\circ}$ and $B = \tan 81^\circ - \tan 3^\circ$, then $\frac{B}{A}$ is equal to ______.

Let $\tan A, \tan B$, where $A, B \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$, be the roots of the quadratic equation $x^2 - 2x - 5 = 0$. Then $20\sin^2\left( \frac{A+B}{2} \right)$ is equal to:

The sum of all the integral values of $p$ such that the equation $3\sin^2 x + 12\cos x - 3 = p$, $x \in \mathbb{R}$, has at least one solution, is:

If $S = \left\{\theta \in [-\pi, \pi] : \cos\theta \cos\dfrac{5\theta}{2} = \cos 7\theta \cos\dfrac{7\theta}{2}\right\}$, then $n(S)$ is equal to __________.

Let $S=\left\{\theta \in (-2\pi, 2\pi) : \cos\theta+1=\sqrt{3}\sin\theta\right\}$.

Then $\sum_{\theta \in S}\theta$ is equal to:

If $\theta \in [-2\pi, 2\pi]$, then the number of solutions of $2\sqrt{2}\cos^2\theta + \left(2 - \sqrt{6}\right)\cos\theta - \sqrt{3} = 0$, is equal to :

If $\theta \in \left[-\frac{7\pi}{6}, \frac{4\pi}{3}\right]$, then the number of solutions of $\sqrt{3}\,\mathrm{cosec}^2\theta-2(\sqrt{3}-1)\,\mathrm{cosec}\,\theta-4=0$, is equal to :

The number of solutions of the equation $2x + 3\tan x = \pi$, $x \in [-2\pi, 2\pi] - \left\{\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}\right\}$ is:

The number of solutions of the equation $\left(4 - \sqrt{3}\right)\sin x - 2\sqrt{3}\cos^2 x = -\dfrac{4}{1 + \sqrt{3}}$, $x \in \left[-2\pi, \dfrac{5\pi}{2}\right]$ is

If $10\sin^4\theta + 15\cos^4\theta = 6$, then the value of $\frac{27\,\mathrm{cosec}^6\theta + 8\sec^6\theta}{16\sec^8\theta}$ is

If for $\theta \in \left[-\frac{\pi}{3}, 0\right]$, the points $(x, y) = \left(3\tan\left(\theta + \frac{\pi}{3}\right), 2\tan\left(\theta + \frac{\pi}{6}\right)\right)$ lie on $xy + \alpha x + \beta y + \gamma = 0$, then $\alpha^2 + \beta^2 + \gamma^2$ is equal to

The number of solutions of the equation $\cos 2\theta \cos \dfrac{\theta}{2} + \cos \dfrac{5\theta}{2} = 2 \cos^3 \dfrac{5\theta}{2}$ in $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ is :

The sum of all values of $\theta \in [0, 2\pi]$ satisfying $2\sin^2\theta = \cos 2\theta$ and $2\cos^2\theta = 3\sin\theta$ is

The value of $(\sin 70°)(\cot 10° \cot 70° - 1)$ is

Let the range of the function $f(x) = 6 + 16\cos x \cdot \cos\left(\dfrac{\pi}{3} - x\right) \cdot \cos\left(\dfrac{\pi}{3} + x\right) \cdot \sin 3x \cdot \cos 6x$, $x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3x + 4y + 12 = 0$ is :

If $\sum_{r=1}^{13} \left\{\frac{1}{\sin\left(\frac{\pi}{4} + (r-1)\frac{\pi}{6}\right) \sin\left(\frac{\pi}{4} + \frac{r\pi}{6}\right)}\right\} = a\sqrt{3} + b$, $a,\ b \in \mathbf{Z}$, then $a^2 + b^2$ is equal to :

Suppose $\theta \in \left[0, \dfrac{\pi}{4}\right]$ is a solution of $4\cos\theta - 3\sin\theta = 1$. Then $\cos\theta$ is equal to :

The number of solutions of $\sin^2 x + (2 + 2x - x^2)\sin x - 3(x - 1)^2 = 0$, where $-\pi \leq x \leq \pi$, is ________.

In a triangle $ABC$, $BC=7$, $AC=8$, $AB=\alpha \in \mathbf{N}$ and $\cos A=\dfrac{2}{3}$. If $49\cos(3C)+42=\dfrac{\mathrm{m}}{\mathrm{n}}$, where $\gcd(\mathrm{m,n})=1$, then $\mathrm{m+n}$ is equal to ________.

If $\sin x=-\frac{3}{5}$, where $\pi<x<\frac{3\pi}{2}$, then $80(\tan^2 x-\cos x)$ is equal to

If the value of $\dfrac{3\cos 36^\circ + 5\sin 18^\circ}{5\cos 36^\circ - 3\sin 18^\circ}$ is $\dfrac{a\sqrt{5} - b}{c}$, where $a$, $b$, $c$ are natural numbers and $\gcd(a, c) = 1$, then $a + b + c$ is equal to :

Let $|\cos\theta\,\cos(60-\theta)\,\cos(60+\theta)|\le\frac{1}{8}$, $\theta\in[0, 2\pi]$. Then, the sum of all $\theta\in[0, 2\pi]$, where $\cos 3\theta$ attains its maximum value, is :

Let the range of the function $f(x)=\frac{1}{2+\sin 3x+\cos 3x}, x \in \mathbb{R}$ be $[a, b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac{\alpha}{\beta}$ is equal to

If $\tan A = \dfrac{1}{\sqrt{x(x^2 + x + 1)}}$, $\tan B = \dfrac{\sqrt{x}}{\sqrt{x^2 + x + 1}}$ and

$\tan C = \left(x^{-3} + x^{-2} + x^{-1}\right)^{1/2}$, $0 < A, B, C < \dfrac{\pi}{2}$, then $A + B$ is equal to :

The number of solutions of the equation $4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0$; $x \in [-2\pi, 2\pi]$ is :

Let the set of all $a \in \mathbf{R}$ such that the equation $\cos 2x + a \sin x = 2a - 7$ has a solution be $[p, q]$ and $r = \tan 9° - \tan 27° - \dfrac{1}{\cot 63°} + \tan 81°$, then $pqr$ is equal to ________.

If $\alpha, -\dfrac{\pi}{2} < \alpha < \dfrac{\pi}{2}$ is the solution of $4\cos\theta + 5\sin\theta = 1$, then the value of $\tan\alpha$ is

If $2\sin^3 x+\sin 2x\cos x+4\sin x-4=0$ has exactly 3 solutions in the interval $\left[0, \frac{n\pi}{2}\right]$, $n\in\mathbf{N}$, then the roots of the equation $x^2+nx+(n-3)=0$ belong to :

For $\alpha, \beta \in (0, \pi/2)$, let $3\sin(\alpha+\beta)=2\sin(\alpha-\beta)$ and a real number $k$ be such that $\tan\alpha = k\tan\beta$. Then, the value of $k$ is equal to

The number of solutions, of the equation $e^{\sin x}-2 e^{-\sin x}=2$, is :