Trigonometry & Inverse Trigonometry Questions - JEE

The ratio in which the curve $y = \left[\sin\frac{2x}{4} + \cos\frac{x}{4}\right]$, where [·] denote greatest integer function divides the curve S1 is:

If A represents the area of acute angled triangle ABC, then $\sqrt{a^2b^2 - 4A^2} + \sqrt{b^2c^2 - 4A^2} + \sqrt{c^2a^2 - 4A^2}$ is equal to

Each side of an equilateral triangle subtends an angle of 60° at the top of a tower h m high located at the centre of the triangle. If a is the length of each side of the triangle, then

The number of points in interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ where the graphs of the curves $y = \cos x$ and $y = \sin 3x$, with $-\frac{\pi}{2} \le x \le \frac{\pi}{2}$, intersect is:

The value of $\dfrac{\sqrt{3}\,\text{cosec}\,20°-\sec20°}{\cos20°\cos40°\cos60°\cos80°}$ is equal to

In a triangle ABC; AD, BE and CF are the altitudes and R is the circumradius, then the radius of the circle DEF is

Two men on the opposite sides of a tower measure the angles of elevation of the top of the tower as $45°$ and $30°$ respectively. If the height of the tower is $40$ m, then the distance between the men is

If 2 tan-1(1/5) - sin-1(3/5) = -cos-1(9l/65), then l =

Let a, b, c be sides of a triangle ABC and D denotes its area. If $a = 2$, $D = \sqrt{3}$, and $a\cos C + \sqrt{3}a\sin C - b - c = 0$, then find the value of $(b + c)$.

Solution of equation $\cot^{-1}x + \sin^{-1}\frac{1}{\sqrt{1+x^2}} = \frac{7\pi}{1}$ is

In $\triangle ABC$, if circumradius $'R'$ and inradius $'r'$ are connected by relation $R^2 - 4Rr + 8r^2 - 12r + 9 = 0$, then the greatest integer which is less than the semiperimeter of $\triangle ABC$ is:

Consider f, g and h be three real valued functions defined on ℝ.Let f(x) = sin 3x + cos x, g(x) = cos 3x + sin x and h(x) = f²(x) + g²(x)General solution of the equation h(x) = 4, is:

In triangle ABC, if cos(a) = 1/2 = 1/4 from triangle OED, and θ = π - 2a, find the area of triangle ABC where BD = 2cot(θ/2) = 2cot(π/2 - a) = 2tan(a) = 2√15 and AC = 3.

In a triangle the length of two larger sides are 10 and 9 respectively. If the angles are in A.P., the length of third side can be:

If in a $\triangle ABC$, $a = 5$, $b = 4$ and $\cos(A - B) = \frac{31}{32}$, then the third side $c$ is equal to

The range of the function f(x) = tan−1x + ½ sin−1x is:

28. If the equation $\sum_{n=0}^{10} \text{arc cot}\left(\frac{1+2^{2n+1}}{2^n}\right) = \text{arc cot}\frac{a}{b}$, where $a$ and $b$ are coprime positive integers. The value of $\log_2\left(\frac{b+a}{a-b}\right)$, is:

10. For a triangle ABC with $\cot A + \cot B + \cot C = \cot \theta$, find $\sin(A - \theta)\sin(B - \theta)\sin(C - \theta)$:

The range of the function f(x) = sec−1(x) + tan−1(x) is

Statement I: If $a, b, c \in \mathbb{R}$ and not all equal, then $\frac{bc + ca + ab}{a^2 + b^2 + c^2} Statement II: \(\sec \theta 1$

Let $\cos(\alpha+\beta)=-\dfrac{1}{10}$ and $\sin(\alpha-\beta)=\dfrac{3}{8}$, where $0<\alpha<\dfrac{\pi}{3}$ and $0<\beta<\dfrac{\pi}{4}$. If $\tan2\alpha=\dfrac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}$, $r,s\in\mathbb{N}$, then $r+s$ is equal to _____.

If ∑∞n=0 2 cot-1(n² + n + 4)/2 = kπ, then find the value of k.

In the given figure, if AB = AC, ∠BAD = 30° and AE = AD, then x is equal to

In a triangle ABC, \angle C = \frac{\pi}{4}, a = \sqrt{2} and b = \sqrt{2 + \sqrt{2}}. Find the sum of digits in the measure of angle A (in degrees).

Let $\triangle ABC$ be inscribed in a circle having radius unity. The three internal bisectors of the angles $A, B$ and $C$ are extended to intersect the circumcircle of $\triangle ABC$ at $A_1, B_1$ and $C_1$ respectively. Find $$\frac{AA_1\cos\frac{A}{2} + BB_1\cos\frac{B}{2} + CC_1\cos\frac{C}{2}}{\sin A + \sin B + \sin C}$$

From point $ D $, 40 m away from the base $ A $ of a vertical tower $ BC $ of height $ h $, the angle of elevation of the top $ C $ is $ 30^\circ $. From point $ B $ (at the base of the tower), the angle of elevation of $ C $ is $ 60^\circ $, and $ B $ is at a horizontal distance $ x $ from $ A $. Find $ x $ (in metres).

ABC is a triangular park with AB = AC = 100 m. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at A and B are $\cot^{-1}(3\sqrt{2})$ and $\csc^{-1}(2\sqrt{2})$ respectively, then the height of the tower (in m) is (JEE Main 2019)

If OA = r cot(π/4 - q/2) = 2r cot(q/2), and tan(q/2) = t, find a + b + c where (1+t²)/(1-t) = t and tan(q/2) = (√17-3)/2.

If $\frac{\sin \theta}{a} + \frac{\cos \theta}{b} = 1$, then $\frac{\sin^3 \theta}{a^3} + \frac{\cos^3 \theta}{b^3}$ is

If $x + \sin y = 2014$ and $x + 2014\cos y = 2013, 0 \leq y \leq \frac{\pi}{2}$, then find the value of $[x + y] - 2005$ (where $[.]$ denotes greatest integer function)

The range of value's of $k$ for which the equation $2\cos^4 x - \sin^4 x + k = 0$ has atleast one solution is $[\lambda, \mu]$. Find the value of $(9\mu + \lambda)$

Given $5\cos A + 3 = 0$, the roots of the equation $9x^2 + 27x + 20 = 0$ are:

The value of $\cos\left(\frac{\pi}{14}\right)\cos\left(\frac{3\pi}{14}\right)\cos\left(\frac{5\pi}{14}\right)$ is

Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \displaystyle\sum_{k=1}^{n} \sin k$, where $k$ represents radian measure. Find the index of the 100th term for which \(a_n

If $\frac{1}{a+c} + \frac{1}{b+c} = \frac{1}{a+b+c}$, then $\angle C$ is

If \(0 \leq x

If the angles A, B and C of triangle ABC are in arithmetic progression and a, b, c represents length of sides opposite to angles A, B and C respectively, then the value of $\dfrac{a+c}{\sqrt{(a^2 - ac + c^2)}}$ is:

The number of solutions of $\sin x \cdot \tan 4x = \cos x$ in $\left(0, \pi\right)$ is:

If $\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi$, then $x^2 + y^2 + z^2 + 2xyz$ equals

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

If $A + B = \frac{\pi}{3}$, $(\cot A - 1)(\cot B - 1)$ is equal to

The number of solutions of |cos x| > 1 in (0, 2013π) is

If $c^4 - 2(a^2 + b^2)c^2 + a^4 + a^2b^2 + b^4 = 0$, then the angle $C$ is

Assertion (A): The value of tan 3α · cot α cannot lie between 3 and 1/3.Reason (R): In a triangle ABC, the maximum value of sin(A/2) sin(B/2) sin(C/2) is 1/8.

Is \(|\tan x + \cot x|

The possible value(s) of \theta satisfying the equation \sin 2\theta \tan\theta + \cos 2\theta \cot\theta - \sin 2\theta = 1 + \tan\theta + \cot\theta where \theta \in [0, \pi] is/are:

The equation whose roots are $\tan^2\left(\frac{\pi}{7}\right)$, $\tan^2\left(\frac{3\pi}{7}\right)$, $\tan^2\left(\frac{5\pi}{7}\right)$ is

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

Number of solutions of cos²\left(\frac{\pi}{4}\right)(\sin x + \sqrt{2}\cos 2x) = 0 in the interval x ∈ [-2π, 2π].

Assertion (A): The minimum value of a² tan²θ + b² cot²θ is 2ab.Reason (R): For positive real numbers AM ≥ GM.

Trigonometry & Inverse Trigonometry Questions (1013)