Trigonometry Questions - JEE

Let S be the set of all solutions of the equation \cos^{-1}(2x) - 2\cos^{-1}(\sqrt{1-x^2}) = \pi, \quad x \in \left[-\frac{1}{2}, \frac{1}{2}\right]. Then \sum_{x \in S} 2\sin^{-1}(x^2 - 1) \text{ is equal to}

Let $\theta \in \left(0, \frac{\pi}{4}\right)$ and $t_1 = (\tan \theta)^{\tan \theta}$, $t_2 = (\tan \theta)^{\cot \theta}$, $t_3 = (\cot \theta)^{\tan \theta}$ and $t_4 = (\cot \theta)^{\cot \theta}$, then:

In a $\triangle ABC$, $\angle B=\frac{\pi}{3}$ and $\angle C=\frac{\pi}{4}$, also $D$ divides $BC$ internally in the ratio $1:3$, then $\frac{\sin\angle BAD}{\sin\angle CAD}$ is equal to:

$\text{cosec}^{-1}(\cos x)$ exists if:

If $\cot x=\dfrac{5}{12}$ for some $x\in\left(\pi,\dfrac{3\pi}{2}\right)$, then $\sin7x\left(\cos\dfrac{13x}{2}+\sin\dfrac{13x}{2}\right)+\cos7x\left(\cos\dfrac{13x}{2}-\sin\dfrac{13x}{2}\right)$ is equal to

If $\alpha,\beta$ are roots of $x^2-3x+2=0$, then $\tan^{-1}\alpha+\tan^{-1}\beta=$

Values of $x$ satisfying $\sin^{-1}(x^2-5x+7)=2\tan^{-1}1$:

If $(x-a)\cos\theta + y\sin\theta = (x-a)\cos\phi + y\sin\phi = a$, $\tan\frac{\theta}{2} - \tan\frac{\phi}{2} = 2e$ and $\theta, \phi$ are unequal angles less than $360°$, then $y^2$ is equal to:

$f(x)=\cot^{-1}\!\sqrt{x(x+3)}+\cos^{-1}\!\sqrt{x^2+3x+1}$ is defined on set $S$. $S$ equals:

If $E = \cos^2 71° + \cos^2 49° + \cos 71° \cos 49°$, then the value of $10E$ is equal to

All $x$ satisfying $(\sin^{-1}x)^2-(\cos^{-1}x)^2>0$:

\tan^{-1}\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right) + \sec^{-1}\left(\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right) \text{ is equal to}

The area bounded by the curve $y = |\cos^{-1}(\sin x)| + |\frac{\pi}{2} - \cos^{-1}(\cos x)|$ and the $x$-axis, where $\frac{\pi}{2} \leq x \leq \pi$, is equal to

When the elevation of the sun changes from $45°$ to $30°$, the shadow of a tower increases by 60 units, then the height of the tower is

If $\cos \alpha + \cos \beta = a$, $\sin \alpha + \sin \beta = b$ and $\alpha - \beta = 2\theta$, then $\tan \frac{\alpha}{\tan \frac{\alpha}{2}} = \frac{a^2 + b^2 - 3}{\text{}}$

If $P$ be any interior point of the equilateral $\triangle ABC$ of side length $2$ units and also $x_a, x_b, x_c$ be the distances of $P$ from the sides $BC, CA, AB$ respectively, then $x_a + x_b + x_c =$

The value of $\cot^{-1}\left(2^2-\frac{1}{2}\right)+\cot^{-1}\left(2^3+\frac{1}{2^2}\right)+\cot^{-1}\left(2^4+\frac{1}{2^3}\right)+\ldots \infty$ is:

If $\cos^{-1}(2x^2-1)=2\pi-2\cos^{-1}x$, then:

A chimney of 20 m height standing on the top of a building subtends an angle whose tangent is $\frac{1}{4}$ at a distance of 70 m from the foot of the building, then the height of building is

The domain of $f(x)=\sin^{-1}(\log_2(x/3))$ is:

All $x$ satisfying $(\sin^{-1}x)^2-(\cos^{-1}x)^2>0$:

If $\alpha,\beta$ are roots of $x^2-3x+2=0$, then $\tan^{-1}\alpha+\tan^{-1}\beta=$

$\tan^{-1}\!\left(1-x^2-\dfrac{1}{x^2}\right)+\sin^{-1}\!\left(x^2+\dfrac{1}{x^2}-1\right)$, $x\ne 0$, equals:

If $\displaystyle\sum_{n=1}^{\infty} \cot^{-1}\!\left(2 + \frac{n(n+1)}{2}\right) = \tan^{-1} a$, then $a$ is equal to

A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45°. It flies off horizontally straight away from O. After 1 second, the elevation of the bird from O is reduced to 30°. The speed (in m/s) of the bird is

A tower is observed from three collinear points A, B, C on ground such that angles of elevation are $\alpha$, $2\alpha$, $3\alpha$. If $AB : BC = ?$

If $p = \cos 55°$, $q = \cos 65°$ and $r = \cos 175°$, then the value of $\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{pq}$ is equal to:

One root of $\tan^{-1}\!\cot\!\left(\dfrac{3x^2+3|x|+1}{x^2+|x|+1}\right)=\dfrac{\pi}{2}-\csc\!\cdot\!\csc^{-1}\!\left(\dfrac{3|x|+2}{|x|+1}\right)$ is $2\sin\theta$, $\theta\in\left(0,\dfrac{\pi}{2}\right)$. Value of $\tan\dfrac{7\theta}{9}\cdot\tan\dfrac{10\theta}{9}\cdot\tan\dfrac{13\theta}{9}$ is

If $\cos^{-1}\!\left(\dfrac{2}{3x}\right) + \cos^{-1}\!\left(\dfrac{3}{4x}\right) = \dfrac{\pi}{2}$ $\left(x > \dfrac{3}{4}\right)$, then $x$ is equal to:

If $\tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\pi/2$, then $xy+yz+zx=$

If the sum of values of $\theta$ in $(-3\pi, 3\pi)$ satisfying $\displaystyle\sum_{m=1}^{15}\sec\!\left(\theta+(m-1)\frac{\pi}{18}\right)\sec\!\left(\theta+m\frac{\pi}{18}\right)=(4+2\sqrt{3})\csc\frac{\pi}{18}$ is $\dfrac{k\pi}{10}$, then the value of $k$ is

Let $x=\sin1°$. The value of $\dfrac{1}{\cos0°\cos1°}+\dfrac{1}{\cos1°\cos2°}+\cdots+\dfrac{1}{\cos44°\cos45°}$ is

$\cot\!\left(\sum_{n=1}^{19}\cot^{-1}\!\left(1+\sum_{p=1}^{n}2p\right)\right)=$

Let $T(\theta) = \cos^2(30°-\theta) - \cos(30°-\theta)\cos(30°+\theta) + \cos^2(30°+\theta)$. Then the value of $4\displaystyle\sum_{\theta=1}^{30} \theta\, T(\theta)$ is

The upper four-fifth portion of a vertical tower subtends an angle $\tan^{-1}\frac{8}{21}$ at a point $A$ in the horizontal plane through its foot and at a distance 50 m from the foot. If the angle subtended by the lower one fifth of tower at point $A$ is $\beta$, then the height of the tower can be

In $\triangle ABC$, $\angle A = \tan^{-1}7$, $\angle C = \tan^{-1}\frac{4}{3}$. Let $D$ be an interior point on side $AC$ such that area of $\triangle ABD$ is twice the area of $\triangle BCD$. If $\angle ABD = \theta$, then the value of $\tan 2\theta$ is

The value of $\displaystyle\prod_{r=1}^{7} \cos\frac{r\pi}{15}$ is

The value of $\displaystyle\sum_{r=2}^{\infty} \cot^{-1}(r^2 - 5r + 7)$ is

Sum of all values of $\theta\in\left(0,\dfrac{\pi}{2}\right)$ satisfying $\sin^22\theta+\cos^42\theta=\dfrac{3}{4}$ is

The number of integral values of $\alpha$ for which the equation $\dfrac{16}{\tan x}+\dfrac{4}{4-\tan x}=\alpha$ does not have any solution is

Find the value of $\cos\left(\cos^{-1}\frac{\pi}{3} + \sin^{-1}\sin\frac{2\pi}{3}\right)$.

If $a=2$, then the sum of the infinite series $\cot^{-1}(2a^{-1}+a)+\cot^{-1}(2a^{-1}+3a)+\cot^{-1}(2a^{-1}+6a)+\cot^{-1}(2a^{-1}+10a)+\cdots$ is

If $\displaystyle\sum_{n=1}^{\infty} \cot^{-1}\!\left(2 + \frac{n(n+1)}{2}\right) = \tan^{-1} a$, then $a$ is equal to

A tower is observed from three collinear points A, B, C on ground such that angles of elevation are $\alpha$, $2\alpha$, $3\alpha$. If $AB : BC = ?$

If $\sum_{i=1}^{10}\sin^{-1}x_i=5\pi$, then $\sum_{i=1}^{10}x_i^2=$

The value of the angle $\tan^{-1}(\tan 65° - 2\tan 40°)$ in degrees is equal to

If $\cos^{-1}\frac{x}{a} - \sin^{-1}\frac{y}{b} = 0$ (where $a, b > 0$), then the maximum value of $b^2x^2 + a^2y^2 + 2abxy\sin\theta$ equals

The largest interval lying in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ for which $ f(x) = 4^{-x^2} + \cos^{-1}\left(\dfrac{x}{2}-1\right) + \log(\cos x) $ is defined, is:

If $ \sin^{-1}\dfrac{x}{5} + \sin^{-1}\dfrac{4}{5} = \dfrac{\pi}{2} $, then $ x $ equals:

The value of the expression $\dfrac{\sin 20°(4\cos 20°+1)}{\cos 20°\cdot\cos 30°}$ is

Trigonometry Questions (1127)