Trigonometry Questions - JEE

Let incircle of radius $4$ units of a triangle $ABC$ touches the side $BC$ at $D$. If $BD = 6, DC = 8$ and $\Delta$ be the area of triangle, then $\sqrt[4]{\Delta - 3}$ = _______.

The total number of solutions of $\tan\{x\} = \cot\{x\}$ ; where $\{x\}$ denotes the fractional part of $x$ in $[0, 2\pi)$ is _______.

If $\sin x + \sin^2 x + \sin^3 x = 1$, then $\cos^6 x - 4\cos^4 x + 8\cos^2 x$ = _______.

If $\tan\left(\frac{2\pi}{3} - x\right) = \frac{\sin\frac{2\pi}{3} - \sin x}{\cos\frac{2\pi}{3} - \cos x}$ where $0 < x < \frac{3\pi}{2}$, and the values of $x$ are $x_1$ and $x_2$, then the value of $\frac{12}{\pi}|x_2 - x_1|$ is

If $10\sin^4 u + 15\cos^4 u = 6$ and the value of $9\cos\sec^4 u + 8\sec^4 u$ is $S$, then find the value of $\frac{S}{25}$

If $\sum_{r=1}^{q}\frac{\tan 2^{r-1}}{\cos 2^r} = \tan p^n - \tan q$, then find the value of $(p + q)$

Given that for $a, b, c, d \in \mathbb{R}$, if $a\sec(200°) - c\tan(200°) = d$ and $b\sec(200°) + d\tan(200°) = c$, then find the value of $\left(\frac{a^2 + b^2 + c^2 + d^2}{bd - ac}\right)\sin 20°$

If the value of $\cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{6\pi}{7} + \cos\frac{7\pi}{7} = -\frac{l}{2}$. Find the value of $l$

The complete set of values of $x$ satisfying $\frac{2\sin 6x}{\sin x - 1} < 0$ and $\sec^2 x - 2\sqrt{2}\tan x \leq 0$ in $\left[0, \frac{\pi}{2}\right)$ is $[a, b) \cup (c, d]$, then find the value of $\left(\frac{cd}{ab}\right)$

If the sum of all values of $\theta$, $0 \leq \theta \leq 2\pi$ satisfying the equation $(8\cos 40 - 3)(\cot \theta + \tan \theta - 2)(\cot \theta + \tan \theta + 2) = 12$ is $k\pi$, then $k$ is equal to:

Find number of solutions of the equation $\sin^{-1}(\log_2(\cos x)) - 1) + \cos^{-1}(3\log_2^2(\cos x) - 7)) = \frac{\pi}{2}$, if $x \in [0, 4\pi]$.

The two adjacent sides of a cyclic quadrilateral are $2, 5$ and the angle between them is $60°$. If the area of the quadrilateral is $4\sqrt{3}$, then the remaining two sides are:

$\cos^{-1} l + \cos^{-1} m + \cos^{-1} n$ is equal to

Find the value of a + b + c where OA = r cot(π/4 − q/2) = 2r cot(q/2), and tan(q/2) = (−3 ± √17)/2.

Let $\cos(\alpha+\beta)=\dfrac{4}{5}$ and let $\sin(\alpha-\beta)=\dfrac{5}{13}$, where $0 \le \alpha,\ \beta \le \dfrac{\pi}{4}$, then $\tan 2\alpha =$

Minimum possible area of the triangle is :

If $\alpha = \frac{1}{3}\sin^{-1}\left(\frac{2x}{1+x^2}\right) + \frac{1}{3}\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ where $x \geq \frac{4}{3}$, then the value of $\dfrac{\cos 2\alpha + \sec\alpha + 3\sqrt{3}}{\sqrt{3}}$ is equal to:

Let $0 \leq \alpha, \beta, \gamma, \delta \leq \pi$ where $\beta$ and $\gamma$ are not complementary such that$2\cos \alpha + 6\cos \beta + 7\cos \gamma + 9\cos \delta = 0$ and $2\sin \alpha - 6\sin \beta + 7\sin \gamma - 9\sin \delta = 0$If $\frac{\cos(\alpha + \delta)}{\cos(\beta + \gamma)} = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive numbers, then the value of $(m + n)$ is equal to:

The angle of elevation of a cloud from a point $250$ m above a lake is $15°$ and angle of depression of its reflection in the lake is $45°$. The height of the cloud is

If 1 + \sin 3x + \cos 3x = \frac{3}{2}\sin 2x, then x is

The value of $\cos\left(\cos^{-1}\left(\frac{1}{8}\right)\right)$ is equal to

If $\cot^{-1}\left(\frac{n}{p}\right) = \frac{p}{6}$, $n \in \mathbb{N}$, then the maximum value of $\tan^{-1}x + \cot^{-1}\left(\frac{1}{y}\right) = \sin^{-1}\left(\frac{3}{10}\right)$ is $'n'$ is

The number of positive integral solutions of $\tan^{-1}x + \cot^{-1}\left(\frac{1}{y}\right) = \sin^{-1}\left(\frac{3}{10}\right)$ is

The value of $\cos\left(\cos^{-1}\left(\cos\left(\sin^{-1}\left(\frac{63}{8}\right)\right)\right)\right)$ is

$\sin^{-1}(\sin \theta) = \theta$, for all $\theta$ belonging to

$\csc^{-1}(\csc \theta) = \theta$, for all $\theta$ belonging to

Which of the following is greatest?

If $r_1 = r + r_2 + r_3$, then the triangle is

The general value of $\theta$ satisfying $\sin^2\theta + \sin\theta = 2$ is

Given, $\cot^{-1}\left(\frac{n^2 - 10n + 21.6}{\pi}\right) > \frac{\pi}{4}$ and $\cos^{-1}(x^2) = \frac{\pi}{6}$, where $n \in \mathbb{N}$. Find the minimum value of $n$.

If $\sin(\cot^{-1}(x+1)) = \cos(\tan^{-1} x)$, then the value of x is

159. A continuous even periodic function $f$ with period 8 is such that $f(0) = 0$, $f(1) = -2$, $f(2) = 1$, $f(3) = 2$, $f(4) = 3$. Then the value of $\tan^{-1}(\tan(f(-5) + f(20)) + \cos^{-1}(f(-10) + f(17)))$ is equal to:

If $\sin^{-1}\left(\frac{x}{5}\right) + \text{cosec}^{-1}\left(\frac{5}{4}\right) = \frac{\pi}{2}$, then $x$ is

The number of distinct solutions of the equation $\cos^2 2x + \cos 4x + \sin 4x + \cos 6x + \sin 6x = \frac{5}{4}$ in the interval $[0, 2\pi]$ is

Considering the principal values of inverse trigonometric functions, the value of $\tan\!\left(2\sin^{-1}\!\left(\dfrac{2}{\sqrt{13}}\right)-2\cos^{-1}\!\left(\dfrac{3}{\sqrt{10}}\right)\right)$ is equal to:

If the domain of the function $f(x)=\sin^{-1}\!\left(\dfrac{1}{x^2-2x-2}\right)$ is $(-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty)$, then $\alpha+\beta+\gamma+\delta$ is equal to

If the domain of the function $f(x)=\cos^{-1}\!\left(\dfrac{2x-5}{11-3x}\right)+\sin^{-1}(2x^2-3x+1)$ is the interval $[\alpha,\beta]$, then $\alpha+2\beta$ is equal to:

The number of values of $x$, for which $\tan^{-1}\!\left(\dfrac{1}{x}\right) = \pi + \tan^{-1} x$, \(0

The value of the expression $\frac{2\cos^4 + \cos^4^3 - \cos^4 \alpha + \cdots - \cos^4}{2\cos^4 + \cos^4^3 - \cos^4 - \cdots - \cos^4 + 1}$ is equal to

$\sin \alpha + \sin \beta + \sin \gamma$ can be equal to

96. In the △ABC, A > B. If the measures of A and B satisfy the equation $3\sin x - 4\sin^3 x - k = 0$, \(0

Given $\sin^4\alpha + 4\cos^4\beta + 2 = 4\sqrt{2}\sin\alpha\cos\beta$where $\alpha, \beta \in [0, \pi]$, find the value of $\cos(\alpha+\beta) - \cos(\alpha-\beta)$.

The value of $\cot^{-1}\left(\frac{xy+1}{x-y}\right) + \cot^{-1}\left(\frac{yz+1}{y-z}\right) + \cot^{-1}\left(\frac{zx+1}{z-x}\right)$ is

Value of $\tan^{-1}\left(\frac{\sin 2}{1 - \cos 2}\right)$ is

If $x^2 + y^2 + z^2 = r^2$, then $\tan^{-1}\left(\frac{xy}{zr}\right) + \tan^{-1}\left(\frac{yz}{xr}\right) + \tan^{-1}\left(\frac{xz}{yr}\right)$ is equal to

If in a $\triangle ABC$, $\angle A = \tan^{-1}2$ and $\angle B = \tan^{-1}3$, then $\angle C$ is equal to

In a triangle $ PQR $, $ \angle R = \dfrac{\pi}{2} $. If $ \tan\left(\dfrac{P}{2}\right) $ and $ \tan\left(\dfrac{Q}{2}\right) $ are roots of $ ax^2 + bx + c = 0 $ (where $ a \neq 0 $), then which of the following is true?

The inradius of $\triangle ABC$ is $100\sqrt{3}$ and the circumradius is $200\sqrt{3}$. Consider the line perpendicular to plane $ABC$ through the circumcenter of $\triangle ABC$. Note that $P, Q, O$ must lie on that line to be equidistant from each of the triangle's vertices. Also, note that since $P, Q, O$ are collinear, and $OP = OQ$, we must have $O$ is the midpoint of $PQ$. Now, Let $K$ be the circumcenter of $\triangle ABC$, and $L$ be the foot of the altitude from $A$ to $BC$. We must have $\tan(\angle KLP + \angle QLK) = \tan(120°)$. Setting $KP = x$ and $KQ = y$, assuming WLOG $x > y$, we must have \[\tan(120°) = -\sqrt{3} = \frac{\dfrac{x+y}{100\sqrt{3}}}{\dfrac{30000 - xy}{30000}}.\] Thus $100(x+y) = xy - 30000$. Also, $\left(\dfrac{x+y}{2}\right)^2 = \left(\dfrac{x-y}{2}\right)^2 + 120000$ by the Pythagorean theorem, so $xy = 120000$, and substituting, $90000 = 100(x+y)$, or $x + y = 900$. The desired answer is $\dfrac{x+y}{2}$.

The number of solutions of the equation $2\sin^3\alpha - 7\sin^2\alpha + 7\sin\alpha - 2 = 0$ in $[0, 2\pi]$ is

Sum of series $\displaystyle\sum_{r=1}^{n} \sin^{-1}\left[\dfrac{2r+1}{r(r+1)\left(\sqrt{r^2+2r} + \sqrt{r^2-1}\right)}\right]$ is

Trigonometry Questions (1127)