Trigonometry & Inverse Trigonometry Questions - JEE

The period of $f(\theta) = \sin^2\theta$ is:

Find number of solutions of the equation sin-1(|log₂₆(cos x) - 1|) + cos-1(|3 log₂₆(cos x) - 7|) = π/2, if x ∈ [0, 4π].

$\cot\dfrac{a+1}{a-b} + \cot\dfrac{b+1}{b-c} + \cot\dfrac{c+1}{c-a} = \underline{\quad}$.

ABC is a triangular park with $AB = AC = 100$ metres. 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 $\text{cosec}^{-1}(2\sqrt{2})$ respectively, then the height of the tower (in metres) is __________ (up to four decimal places).

The minimum value of $\frac{r_1 r_2}{r_3}$ in a triangle is (symbols have their usual meaning)

If $x$ be real, prove that $\frac{x^2 - 2x\cos\alpha + 1}{x^2 - 2x\cos\beta + 1}$ lies between $\sin^2\frac{\alpha}{2}\cdot\csc^2\frac{\beta}{2}$ and $\cos^2\frac{\alpha}{2}\cdot\sec^2\frac{\beta}{2}$.

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:

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 $\Delta ABC$]

The number of solutions of the equation $8\tan^2\theta + 9 = 6\sec\theta$ in the interval $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ is

If in a triangle ABC, $b\cos^2\frac{A}{2} + \cos^2\frac{B}{2} = \frac{3c}{2}$, then minimum value of $\frac{1}{5}\left(\frac{a+c}{2c-a} + \frac{b+c}{2c-b}\right)$ is equal to

If $\tan\alpha^2 = \tan(\alpha - \beta)\cdot\tan(\alpha + \beta)$, then which of the following is correct?(Given: $0 , \(\tan\alpha > 0$)

In a triangle $ABC$, if $A + C = 2B$ and $A + B + C = 180^\circ$ with $\sin A + \sin C = 2\sin^2 B$, find the value of some expression (answer 30).

If $a$, $b$, $g$, and $d$ are four solutions of the equation $\tan\left(\theta + \frac{\pi}{4}\right) = 3\tan 3\theta$, then $\tan a \tan b \tan g \tan d$ equals

If $|\sin x + \cos x| = |\sin x| + |\cos x|$ ($\sin x, \cos x \neq 0$), then in which quadrant does $x$ lie?

In a triangle ABC, let $\angle C = \pi/2$. If r is the inradius and R is the circumradius of the triangle ABC, then 2(r + R) equals

Suppose in $\triangle ABC$ with sides a, b, c the following equation holds true \[\frac{\cos A}{a} + k_1 = \frac{\cos B}{b} + k_2 = \frac{\cos C}{c} + k_3 = \frac{a^2 + b^2 + c^2}{8}\] If $abc = 4$, then the value of $k_1 k_2 k_3$ is:

If $\cos(\theta - \alpha)$, $\cos\theta$, $\cos(\theta + \alpha)$ are in HP, then $\cos\theta \sec\dfrac{\alpha}{2}$ is equal to

The number of values of $\theta$ in $\left[0, \dfrac{\pi}{2}\right]$ satisfying $2\cos\theta + \sin\theta = 1$ $\left(\theta \neq \dfrac{\pi}{2}\right)$ is

If $f(x) = \sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right) - 2\tan^{-1}x$ and $g(x) = \sin^{-1}\!\left(\dfrac{1-x^2}{1+x^2}\right) + 4\tan^{-1}x$, then range of $(f(x) - g(x))$ for $x \in (-\infty, -1]$ is:

If $x^2 + y^2 + z^2 = r^2$, then find the value of $\tan^{-1}\dfrac{yz}{rx} + \tan^{-1}\dfrac{zx}{ry} + \tan^{-1}\dfrac{xy}{rz}$.

From a point O on the ground, poles of equal heights are placed at equal distances $k$ apart along a straight line. The angle of elevation from O to the top of the 10th pole is $\alpha$. If the distance from O to the base of the first pole is $a$, the height $h$ of each pole is

In a $\triangle ABC$, $\dfrac{a}{b} = 2 + \sqrt{3}$ and $\angle C = 60^\circ$. The ordered pair $(\angle A,\, \angle B)$ is equal to

Let T1 be an isosceles triangle inscribed in a circle K. Let T2 be another isosceles triangle inscribed in K whose base is one of the equal sides of T1 and which overlaps the interior of T1. Similarly create isosceles triangles T3 from T2, T4 from T3 and so on. Do the triangles Tn approach an equilateral triangle as $n \to \infty$?

If an angle $A$ of a $\triangle ABC$ satisfies $5\cos A + 3 = 0$, then the roots of the quadratic equation, $9x^2 + 27x + 20 = 0$ are

In triangle ABC, a = 3, b = 4, c = 2. Point D and E trisect the side BC. If ∠DAE = θ, then cot 2θ is divisible by:

Let m and n be positive real numbers such that m + n = 3. If $\frac{m}{s} = \sin^2\theta$ and $\frac{n}{t} = \cos^2\theta$, then the minimum value of $s + t$ is:

If $E = (3\sqrt{5} - 4\cos x + \sqrt{13 - 12\sin x})$, find the minimum value of $E^2$.

The number of solutions of the equation $\sqrt{1 + \cos 2x} = \sqrt{2}\sin^{-1}(\sin x)$ for $x \in [-\pi, \pi]$ is:

The value of \[ I = \sum_{r=0}^{10} \frac{1}{4}\left(\cos\frac{3\pi r}{3} + 3\cos\frac{\pi r}{3}\right) \] is equal to ___.

If T(n) = cos²(30° − n°) − cos(30° − n°)cos(30° + n°) + cos²(30° + n°), find the value of $4\sum_{n=1}^{30} nT(n)$.

262. If $\sec^{-1}(x) + \tan^{-1}\sqrt{9y^2 - 1} + \sin^{-1}(x^2 + y^2) = \lambda$ has no solution, then exhaustive set of values of $\lambda$ is equal to:

If $(\sin^{-1}x)^2 + (\sin^{-1}y)^2 + 2\sin^{-1}x\sin^{-1}y = \pi^2$, then $x^2 + y^2$ is equal to:

The lengths of sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle.

If sum of all the solutions of the equation $8\cos x\left[\cos\left(\frac{\pi}{6}+x\right)\cdot\cos\left(\frac{\pi}{6}-x\right)-\frac{1}{2}\right]=1$ in $[0,\pi]$ is $k\pi$, then $k$ is equal to

Find the number of solutions of the equations $(\sin x - 1)^3 + (\cos x - 1)^3 + (\sin x)^3 = (2\sin x + \cos x - 2)^3$ in $[0, 2\pi]$.

If $3\sin P + 4\cos Q = 6$ and $4\sin Q + 3\cos P = 1$, then the angle $R$ in triangle $PQR$ is

In a triangle with sides $a, b, c$ where $s - a + s - b + s - c = 15$ (so $s = 15$) and the incircle touches side $BC$ at $Q$ and side $CA$ at $C'$ with $QC = s - c$. If $s - a = 3,\; s - b = 5,\; s - c = 7$, find the area of quadrilateral $QCRI$ (where $I$ is the incentre and $R$ is the point of tangency on $CA$).

Given the angle of elevation of a cloud from a point P which is 25 m above a lake is $30°$ and the angle of depression of the reflection of the cloud in the lake from P is $60°$. Find the height of the cloud from the surface (in metres).

Statement I: $y = \tan^{-1}(\tan x)$ and $y = \cos^{-1}(\cos x)$ are not the same functionStatement II: The range of $\tan^{-1}(\tan x)$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and the range of $\cos^{-1}(\cos x)$ is $[0, \pi]$

The number of solutions of the pair of equations 2 sin²θ − cos 2θ = 0 and 2 cos²θ − 3sin θ = 0 in the interval [0, 2π] is:

The value of $\sin\left[\tan^{-1}\left(\tan\dfrac{7\pi}{6}\right) + \cos^{-1}\left(\cos\dfrac{7\pi}{3}\right)\right]$ is

The range of values of $k$ for which the equation $2\cos 4x - \sin 4x + k = 0$ has at least one solution is $[l, m]$. Find the value of $9m + l$.

Total number of solutions of $\sin^4 x + \cos^4 x = \sin x \times \cos x$ in $[0, 2\pi]$ is equal to

The general solution of $e^x - 1 = 2(e^{\sin x} + e^{\cos x}) = 2$ is

If $\tan\frac{a}{2}$ and $\tan\frac{b}{2}$ are the roots of the equation $8x^2 - 26x + 15 = 0$, then $\cos(a + b)$ is equal to

The number of solutions of the equation $\sin^{-1}\left(x + \frac{2}{3}\right) + \cos^{-1}\left(x - \frac{2}{3}\right) = x^2$ for x ∈ [−1, 1], where [x] denotes the greatest integer less than or equal to x

$\sin 47° + \sin 61° - \sin 11° - \sin 25°$ is equal to

The value of $\cos\frac{\pi}{15}\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{8\pi}{15}$ is

If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression $\frac{a}{c}\sin 2C + \frac{c}{a}\sin 2A$ is

If cos(α + β) = 4/5, sin(α - β) = 5/13 and α, β lie between 0 and π/4, then tan 2α is equal to

Trigonometry & Inverse Trigonometry Questions (1013)