Trigonometry & Inverse Trigonometry Questions - JEE

Let $P = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\cos\theta\}$ and $Q = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\sin\theta\}$ be two sets. Then,

The value of $\cos A + \cos B + \cos C$ is

241. If $\dfrac{\cos x + \cos y + \cos z}{\cos(x+y+z)} = 2$ and $\dfrac{\sin x + \sin y + \sin z}{\sin(x+y+z)} = 2$, then the value of $\cos(x+y) + \cos(y+z) + \cos(z+x)$ is equal to: (where $x,\, y,\, z \in R$)

The value of $2ca\sin\!\left(\dfrac{A-B+C}{2}\right)$ equals

Find the value of $\cos\dfrac{\pi}{2^2} \cdot \cos\dfrac{\pi}{2^3} \cdots \cos\dfrac{\pi}{2^{10}} \cdot \sin\dfrac{\pi}{2^{10}}$.

If $\alpha$, $\beta$ are two values of $\theta$ obtained from the equation $a\cos\theta + b\sin\theta = c$ then the value of $\tan\frac{\alpha+\beta}{2}$ is

If $2\sin\theta + 1 = 0$ and $\sqrt{3}\tan\theta = 1$ then the most general value of $\theta$ is

Let a = sin α, b = cos α and c = √(1 + sin α cos α). In a triangle with sides a, b, c, what is the angle C?

94. The diameter of the circumcircle of a triangle with sides 5, 6 and 7 is

If $\frac{1+\sin 2x}{1-\sin 2x} = \tan^2(a+x)$ for all $x$ then the numerically smallest value of $a$ is

Calculate m = \sum_{k=1}^{17} \cos\left(\frac{k\pi}{9}\right) = \cos\left(\frac{\pi}{9}\right) + \cos\left(\frac{2\pi}{9}\right) + \cos\left(\frac{3\pi}{9}\right) + \ldots + \cos\left(\frac{17\pi}{9}\right), and find the value of (m^2 + m + 2).

The value of the expression \cos^2\left(\frac{\pi}{8}\right) + \cos^2\left(\frac{3\pi}{8}\right) + \cos^2\left(\frac{5\pi}{8}\right) + \cos^2\left(\frac{7\pi}{8}\right) is

If $\sin^{-1}\frac{1}{4} + \sin^{-1}\frac{3}{5} = \sin^{-1}x$, then the value of $x$ is

The value of $\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c}$ is

The minimum value of the expression $\dfrac{\sin^3\alpha + 6\sin^2\alpha + \sin\alpha + 2\cos^2\alpha - 8}{\sin\alpha - 1}$ is equal to:

If A + B = \frac{\pi}{3}, A, B > 0, then the maximum value of \tan A \cdot \tan B is

The value of $r_1 + r_2 + r_3 - 4R$ is

The value of $\dfrac{1 - \tan^2 15°}{1 + \tan^2 15°}$ is

The value of \frac{2\cos^3\left(\frac{\pi}{2}+x\right)\cot(3\pi+x)\sec(x-3\pi)\operatorname{cosec}\left(\frac{3\pi}{2}-x\right)}{\cot x\tan^2(x-\pi)\sin(x-2\pi)}\) is equal to

Given that $\sin x - \sin 2x + \sin 3x = 0$Find the number of values of $x$ that are possible.

In △ABC, If A − B = 120° and R = 8r, then the value of $\frac{1 + \cos C}{1 - \cos C}$ equals:(All symbols used have their usual meaning in a triangle)

In an isosceles triangle ABC, AB = AC. If the vertical angle ∠A is 20°, then a³ + b³ is equal to

If $\alpha = \dfrac{\pi}{3}$, find the value of $\dfrac{\cos 2\alpha + \sec\alpha + 3\sqrt{3}}{\tan\alpha}$.

The number of all possible 5-tuples (a_1, a_2, a_3, a_4, a_5) such that a_1 + a_2 \sin x + a_3 \cos x + a_4 \sin 2x + a_5 \cos 2x = 0 holds for all x is

Which are correct?(A) $\cot^{-1}x=\tan^{-1}(1/x)\ \forall x\in\mathbb{R}\setminus\{0\}$(B) $f(x)=\text{sgn}(e^x)$ is into(C) $f:\mathbb{R}^+\to\mathbb{R},\,f(x)=\sin x+x$ is odd(D) $f(x)=e^x/e^{[x]}$ is periodic

If $\sum_{m=1}^{6} \csc\left(\alpha + (m-1)\frac{\pi}{4}\right)\csc\left(\alpha + \frac{m\pi}{4}\right) = 4\sqrt{2}$, where $\alpha \in (0, \pi)$ then $\alpha$ can be:

If A, B, C, D are the angles of a quadrilateral, then $\frac{\sum \tan A}{\sum \cot A}$ is equal to

If $\cos\frac{p}{q} + \cos\frac{q}{q} = 0$, then the different values of $q$ are in AP, whose common difference is

176. If $\alpha = \sin\theta\,|\sin\theta|$ and $\beta = \cos\theta\,|\cos\theta|$ where $\theta \in \left[\dfrac{199\pi}{2},\, 100\pi\right]$, then:

Question nos. 687 to 689Column-1 represents a condition to form trigonometric equation. Column-2 represents the value of $\sin\theta + \cos\theta$ and Column-3 represents the general value of $\theta$ satisfying the trigonometric equation.Column-1Column-2Column-3(I) If $2^{\sin\theta}$, $\sqrt{2}$ and $2^{\cos\theta}$ are three terms of a decreasing G.P.(i) $\dfrac{\sqrt{3}+1}{2}$(P) $\theta = 2n\pi - \dfrac{\pi}{2}$(II) If $\cos\theta$, $\sec\theta$ and $\cot\theta$ are three positive numbers in H.P.(ii) $\sqrt{2}$(Q) $\theta = 2n\pi + \dfrac{\pi}{6}$(III) If $2\log\sec\theta$, $\log 2$ and $2\log\text{cosec}\,\theta$ are in A.P.(iii) $-1$(R) $\theta = 2n\pi + \dfrac{\pi}{2}$(IV) If G.M. of $(2+\sin\theta)$, $(3+\sin\theta)$ and $(4+\sin\theta)$ is equal to cube root of 6.(iv) $1$(S) $\theta = 2n\pi + \dfrac{\pi}{4}$688. Which of the following options is the only correct combination?

The maximum value of a \sin 2x + b \cos 2x for all real x is

If $0 \leq \theta \leq 2\pi$ and $2\sin^2 \theta - 5\sin \theta + 2 > 0$, then find the range of $\theta$.

If $\sum_{m=1}^{6} \csc\left(\alpha + (m-1)\dfrac{\pi}{4}\right)\csc\left(\alpha + \dfrac{m\pi}{4}\right) = 4\sqrt{2}$, where $\alpha \in (0, \pi)$ then $\alpha$ can be:

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

If $1 - \frac{\cos^2 A}{a^2} - 1 - \frac{\cos^2 B}{b^2} = \frac{2}{r_2} - \frac{2}{r_3}$, then the triangle is

The number of solutions of the equation $|\cot x| = \cot x + \frac{1}{\sin x}$, \(0

If |sin x + cos x| = |sin x| + |cos x|, x ∈ [0, 2π] then the solution set is

Let $\theta = \sin^{-1}\left(\dfrac{3\sin 2\alpha}{5 + 4\cos 2\alpha}\right)$. Then $\tan^{-1} x = \dfrac{\theta}{2}$ where $x$ equals:

If \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) , then

We have tan(15°) + tan(30°) = –p and tan(15°) + tan(30°) = –p. Given that tan(45°) = 1, find the value of (2 + q – p) where q – p = 1.

Find the value of $\cos\left(\frac{2\pi}{7}\right)\cos\left(\frac{4\pi}{7}\right)\cos\left(\frac{8\pi}{7}\right)$

Given $f_k(x) = \dfrac{1}{k}(\sin^k x + \cos^k x)$, find the value of $f_4(x) - f_6(x)$.

A vertical tower subtends an angle of $60°$ at a point on the same level as the foot of the tower. On moving $100$ m further from the first point in line with the tower, it subtends an angle of $30°$ at the point. If the height of the tower is $H$ m, then the value of $\frac{H}{\sqrt{3}}$ (in meters) is

If $\frac{\cos(\alpha+\gamma)}{\cos(\alpha-\gamma)} = \cos 2\beta$ then $\tan\alpha$, $\tan\beta$ and $\tan\gamma$ are in

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

The number of solutions of the equation $e^{\sin x} - e^{-\sin x} = 4$ is

Let a, b, c be three non-zero real numbers such that the equation acosx + 2bsinx = c, x ∈ [−π/2, π/2], has two distinct real roots α and β with α + β = π/3. Then, the value of b/a is __________________________.

Let $\alpha$ and $\beta$ respectively be the maximum and minimum values of the function $f(\theta)=4\left(\sin^4\!\left(\dfrac{7\pi}{2}-\theta\right)+\sin^4(11\pi+\theta)\right)-2\left(\sin^6\!\left(\dfrac{3\pi}{2}-\theta\right)+\sin^6(9\pi-\theta)\right)$, $\theta\in\mathbb{R}$. Then $\alpha+2\beta$ is equal to:

For $x \in (0, \pi)$, the equation $\sin x + 2 \sin 2x - \sin 3x = \frac{3}{5}$ has

If PQ be a vertical tower subtending angles α, β and γ at the points A, B and C respectively on the line in the horizontal plane through the foot D of tower and on the same side of it, then BC cot α - CA cot β + AB cot γ is equal to

Trigonometry & Inverse Trigonometry Questions (1013)