Inverse Trigonometry Questions - JEE

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

If \cos(\theta - \alpha) = a\ and \cos(\theta - \beta) = b\, then \sin^2(\alpha - \beta) + 2ab\cos(\alpha - \beta)\ is equal to

Circumradius of an isosceles △ABC with ∠A = ∠B is 4 times its inradius, then cos A is root of the equation:

Let $\overrightarrow{AB}=-2\hat{i}+\hat{j}+3\hat{k}$, $\overrightarrow{CB}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$, $\overrightarrow{CA}=4\hat{i}+3\hat{j}+\delta\hat{k}$. If $\delta>0$ and area of $\triangle ABC=5\sqrt{6}$, then $\overrightarrow{CB}\cdot\overrightarrow{CA}$ is equal to

The difference between the greatest and the least possible value of the expression $3 - \cos a + \sin^2 a$ is

If \sec x \cos 5x = -1 and 0 , then x is equal to

If $\alpha$ satisfies the equation $2\sqrt{2}\tan^3 x - 54\sqrt{2}\cot^3 x = 19$, then possible value of $(2\tan^2\alpha + \sqrt{2}\tan\alpha)$ can be equal to:

From the top $A$ of a vertical wall $AB$ of height 30 m, the angles of depression of the top $P$ and bottom $Q$ of a vertical tower $PQ$ are $15°$ and $60°$ respectively, $B$ and $Q$ are on the same horizontal level. If $C$ is a point on $AB$ such that $CB=PQ$, then the area (in m²) of the quadrilateral $BCPQ$ is equal to

The maximum value of $5\sin\theta + 3\sin(\theta - \alpha)$ is 7, then the set of all possible values of $\alpha$ is

Suppose that the side lengths of a triangle are three consecutive integers and one of the angles is twice another. The number of such triangles is are

In a triangle ABC, (a + b + c)(b + c - a) = kbc, then which of the following is true?

If A + B + C = 180°, then find the value of cos A}{sin B sin C} + cos B}{sin C sin A} + cos C}{sin A sin B}.

In a △ABC if $9(a^2 + b^2) = 17c^2$ then the value of the expression $\frac{\cot A + \cot B}{\cot C}$ is:

Let 10 vertical poles standing at equal distances on a straight line, subtend the same angle of elevation $\alpha$ at a point $O$ on this line and all the poles are on the same side of $O$. If the height of the longest pole is '$h$' and the distance of the foot of the smallest pole from $O$ is '$a$'; then the distance between two consecutive poles, is

If $A$, $B$, $C$ are in arithmetic progression and $B = \frac{\pi}{4}$, then $A \tan B \tan C =$

If 0 x

Ex. 11: In a triangle ABC, if $4 \cos A \cos B + 4\sin A \sin B \sin C = 4$, then triangle ABC is

If $x$, $y$ and $z$ are real numbers that satisfy the three equations \[\begin{cases} \tan(x) + \tan(y) + \tan(z) = 6 - (\cot(x) + \cot(y) + \cot(z)) \\ \tan^2(x) + \tan^2(y) + \tan^2(z) = 6 - (\cot^2(x) + \cot^2(y) + \cot^2(z)) \\ \tan^3(x) + \tan^3(y) + \tan^3(z) = 6 - (\cot^3(x) + \cot^3(y) + \cot^3(z)) \end{cases}\] Find the value of the expression $\left(\dfrac{\tan(x)}{\tan(y)} + \dfrac{\tan(y)}{\tan(z)} + \dfrac{\tan(z)}{\tan(x)} + 3\tan(x)\tan(y)\tan(z)\right)$.

The value of $(1 + \cos \frac{2\pi}{7})(1 + \cos \frac{4\pi}{7})(1 + \cos \frac{6\pi}{7})$ is equal to

\frac{\cos 2x - 3\cos x + 1}{(\cot 2x - \cot x)\sin(x - \pi)} = 0 holds if

If \sin^{100} \theta - \cos^{100} \theta = 1, then \theta is

If for some \alpha, \beta; \alpha \le \beta, \alpha + \beta - 8 and sec (tan 2 -1 2 \alpha) + cosec (cot -1 \beta) - 36 , then \alpha + \beta is_______. 2

If \alpha, \beta, \gamma \in \left(0, \frac{\pi}{2}\right)\, then the value of \frac{\sin(\alpha + \beta + \gamma)}{\sin\alpha + \sin\beta + \sin\gamma}\ is

Given the equation $8\cos x\left[\cos\left(\dfrac{\pi}{6}+x\right)\cdot\cos\left(\dfrac{\pi}{6}-x\right)-\dfrac{1}{2}\right]=1$, find the sum of all solutions in $[0, \pi]$ and express in the form $k\pi$. What is $k$?

In the interval $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ the equation $\log_{\sin}(\cos 2\theta) = 2$ has

The equation $\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x$ has solution

If $\sin x + \sin^2 x = 1$, $x \in \left(0, \dfrac{\pi}{2}\right)$, then $\left(\cos^{12} x + \tan^{12} x\right) + 3\left(\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x\right) + \left(\cos^6 x + \tan^6 x\right)$ is equal to

For a constant $k$, the two roots of the quadratic equation $3x^2 - x + k = 0$ are $\sin\theta$ and $\cos\theta$. The value of $54(\sin^3\theta + \cos 3\theta)$ is:

If in a △ABC, $\frac{a^2 - b^2}{a^2 + b^2} = \frac{\sin(A-B)}{\sin(A+B)}$, then the triangle is

Range of the function sin–1(fog(x)) is

In triangle $ABC$ if $\dfrac{[\Delta ABC]}{R} = 4$, then the value of $a\cos A + b\cos B + c\cos C$ is:[Note: $R$ is the circumradius of triangle $ABC$ and $[\Delta ABC]$ is the area of $\Delta ABC$]

97. In a △ABC, AD is the bisector of the angle A meeting BC at D. If I be the incentre of the triangle then AD : DI is equal to

If in a right angled triangle the greatest side is $a$, then $\tan\left(\frac{C}{2}\right) =$

Let $f(x) = \max \{\sin x, \cos x\}$. Then the number of roots of the equation $f(x) = \frac{1}{\sqrt{2}}$ in $(0, 2\pi)$ is

$\tan 40° + 2\tan 10°$ is equal to

A is the orthocentre of △ABC and D is reflection point of A w.r.t. perpendicular bisector of BC, then orthocentre of △DBC is:

The median $AD$ of a triangle $ABC$ is bisected at $E$ and $BE$ meets $AC$ at $F$; then $AF:AC =$

Which are correct for relation $R$ on $\mathbb{R}$?

If \pi \le x \le 3\pi , then cos -1 ( 12 cos x + 5 sin x) is equal to 2 4 13 13

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x² - c² = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is (JEE Main 2019)

If \alpha > \beta > \gamma > 0, then the expression cot 2 2 2 (1+\beta ) (1+\gamma ) (1+\alpha ) -1 {\beta + (\alpha-\beta) } + cot -1 {\gamma + (\beta-\gamma) }+ cot -1 {\alpha + (\gamma-\alpha) } is equal to :

The expression $\frac{(a + b + c)(b + c - a)(c + a - b)(a + b - c)}{4b^2c^2}$ is equal to:(where symbols used have usual meanings)

$ABC$ is a triangle. Forces $\vec{P}$, $\vec{Q}$, $\vec{R}$ acting along $IA$, $IB$ and $IC$ respectively are in equilibrium, where $I$ is the incentre of $\triangle ABC$. Then $P : Q : R$ is

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

The angular depressions of the top and the foot of a tower, as seen from the top of a second tower which is $150$ meters high and standing on the same level as the first, are $13°$ and $\tan^{-1}\left(\frac{5}{6}\right)$ respectively. If the distance between their tops is $d$, then

If $\tan\theta + \tan\left(\frac{\pi}{3} + \theta\right) + \tan\left(\frac{2\pi}{3} + \theta\right) = k\tan 3\theta$ then $k$ is equal to

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

In triangle $ABD$, using the sine rule, if $BD = \sqrt{p^2+q^2}$ and $\angle ABD = \theta$, $\angle ADB = \alpha$, then $AB$ equals

Let M be the greatest and m be the least value of $\sqrt{\sin^{-1} x} + \sqrt{\cos^{-1} x}$, then find the value of $(M/m)^4$.

Let set $A$ denote the solutions of $\cos^{-1}(4x^3-3x)=\tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)$. Then

Inverse Trigonometry Questions (1043)