Trigonometry Questions (1127)

The general solution of sin 2θ sec θ + √3 tan θ = 0.

Two parallel chords are drawn on the same side of the centre of a circle of radius R. It is found that they subtend an angle of θ and 2θ at the centre of the circle. The perpendicular distance between the chords is

Which of the following is the least?(a) \(\sin 3\)(b) \(\sin 2\)(c) \(\sin 1\)(d) \(\sin 7\)

In a triangle ABC, if \(\tan\frac{A}{2}\tan\frac{C}{2} = \frac{1}{3}\) and \(ac = 4\), then the least value of b is:(notation have their usual meaning)

Statement I: \(\tan 50° - \tan 30° - \tan 20° = \tan 50° \tan 30° \tan 20°\)Statement II: If \(x = y + z\), then \(\tan x - \tan y - \tan z = \tan x \tan y \tan z\)

In a triangle ABC, if \(\tan A = 2 \sin 2C\) and \(3 \cos A = 2 \sin B \sin C\) then possible values of C is/are:

The minimum value of the function f(x) = sin x/√(1 - cos² x) + cos x/√(1 - sin² x) + tan x/√(sec² x - 1) + cot x/√(cosec² x - 1) whenever it is defined is

Angles A, B and C of a △ABC are in AP. If \(\frac{b}{c} = \frac{\sqrt{3}}{2}\), then ∠A is equal to

The complete set of values of x satisfying the inequality sin-1(sin 5) > x2 - 4x is ( 2 - √(1 - 2π), 2 + √(1 - 2π) ), then l =

If \(\sec A \tan B + \tan A \sec B = 91\), then the value of \((\sec A \sec B + \tan A \tan B)^2\) is equal to:

Let ABC be a right triangle with $\angle BAC = \frac{\pi}{2}$, then $\left(\frac{r_2}{2R^2} + \frac{r}{R}\right)$ is equal to: (where symbols used have usual meaning in a triangle)

In triangle ABC, if \(2a^2b^2 + 2b^2c^2 = a^4 + b^4 + c^4\), then angle B is equal to

All positions of point P for which triangle DEF is isosceles lie on

If \(x + \frac{1}{x} = 2\), the principal value of \(\sin^{-1}x\) is

If \(\dfrac{1}{16}(\cos 36^\circ \sin 54^\circ)^2 - \left(\dfrac{1}{4}\sin 36^\circ \sin 36^\circ\right)^2 \equiv \dfrac{\sqrt{a}-b}{c}\), find \((a + b + c)\).

In a right angled triangle ABC, the bisector of the right angle C divides AB into segments x and y. If \(\tan\frac{A-B}{2} = t\), then x : y is equal to

If in a $\triangle ABC$, $\sum\sin 3A = 0$, then at least one angle of $\triangle ABC$ is:

The number of values of \(x\) in the interval \([0, 3\pi]\) satisfying the equation \(2\sin^2 x + 5\sin x - 3 = 0\) is

The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin(x + y)$ and $|x| + |y| = 1$ 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)Number of point(s) where the graphs of the two functions, y = f(x) and y = g(x) intersects in [0, π], is:

Find the number of solutions of the equations \(2\sin^2 x + \sin^2 2x = 2\) and \(\sin 2x + \cos 2x = \tan x\) in \([0, 4\pi]\) satisfying the condition \(2\cos 2x + \sin x \le 2\).

If triangle DEF is equilateral, then P

The value of x in \left(0, \frac{π}{2}\right) satisfying the equation \frac{\sqrt{5}-1}{4} \cdot \frac{1}{\sin x} + \frac{\sqrt{10+2\sqrt{5}}}{4} \cdot \frac{1}{\cos x} = 2 is

If $\sin x + \cos x + \tan x + \cot y = 4$, where $x, y \in [0, \frac{\pi}{2}]$, then $\tan(\frac{y}{2})$ is a root of the equation:

In triangle ABC, the ratio \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) is always equal to (All symbols used have usual meaning in a triangle.)

If \sin^{-1}: [-1,1] \to \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] and \cos^{-1}: [-1,1] \to [0, \pi] be two bijective functions, respectively inverses of bijective functions \sin: \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \to [-1,1] and \cos: [0,\pi] \to [-1,1], then \sin^{-1}x + \cos^{-1}x is

703. Let \(f(x) = \cos^{-1}\!\left(\sqrt{\sin^{-1}\!\left(\sec\!\left(\ln\!\left(\dfrac{2x^2+3x-2}{x^2-3x+2}\right)\right)\right)}\right)\). Find the value of \(1 + \left(\displaystyle\sum \alpha_i^2\right)\) where \(\alpha_i\) represents the integers in the range of \(f(x)\). If there are no integers in the range of \(f(x)\), then enter your answer as zero.

If the sum of all values of \(\theta\), \(0 \le \theta \le 2\pi\) satisfying the equation\((8\cos^4 \theta - 3)(\cot \theta + \tan \theta - 2)(\cot \theta + \tan \theta + 2) = 12\)is \(k\pi\), then \(k\) is equal to:

Given the equations with solutions in the interval (0, π/2):(i) sin a = 1/2, so cos a = √3/2(ii) cos β = 1/3, so sin β = √(8/9)(iii) sin γ = cos γ = 1/√2(iv) cos γ = 1, sin γ = 0Find: cos a + cos β + cos γ

In a triangle ABC, BC = 3, AC = 4 and AB = 5. The value of $\sin A + \sin 2B + \sin 3C$ equals

If the sum of all solutions of the equation \(3\cot^2 \theta + 10\cot \theta + 3 = 0\) in \([0, 2\pi]\) is \(k\pi\) where \(k \in \mathbb{I}\), then find the value of \(k\).

If $\Delta$ be area of incircle of a triangle $ABC$ and $A_1, A_2, A_3$ be the area of excircles then find the least value of $$\frac{A_1A_2A_3}{729A^3}$$

If $x\sin\theta = y\sin\left(\theta+\frac{2\pi}{3}\right) = z\sin\left(\theta+\frac{4\pi}{3}\right)$, then $\sum xy=$

Statement I: \(\csc^{-1}\left(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right) > \sec^{-1}\left(\frac{1}{2} + \frac{1}{2}\right)\)Statement II: \(\csc^{-1}x > \sec^{-1}x\) if \(1

If \(\cot \frac{A}{2} = \frac{a+b+c}{4\Delta}\), then △ABC is

If inside a big circle exactly n (n > 3) small circles, each of radius r, can be drawn in such a way that each small circle touches the big circle and also touches both its adjacent small circles, then the radius of big circle is

Find \(\tan\frac{A}{2}\) where notations have their usual meaning.

In a triangle ABC, medians AD and BE are drawn. If $AD = 4$; $\angle DAB = \frac{\pi}{6}$ and $\angle ABE = \frac{\pi}{3}$ then the area of the triangle ABC is:

In any triangle ABC, the value of $\frac{r_1 + r_2}{1 + \cos C}$ is equal to (where notation have their usual meaning):

The radius of the circle passing through the incentre $I$ of $\triangle ABC$ and through the end points of $BC$ is given by:

In a $\triangle ABC$; inscribed circle with centre $I$ touches sides $AB, AC$ and $BC$ at $D, E, F$ respectively. Let area of quadrilateral $ADIE$ is $5$ square units and area of quadrilateral $BFID$ is $10$ square units. Find the value of $$\frac{\cos\left(\frac{C}{2}\right)}{\sin\left(\frac{A-B}{2}\right)}$$

Which one of the following function contains only one integer in its range?[Note: sgn(k) denotes the signum function of k.](a) \(f(x) = |\cos\frac{1-x^2}{x}|\)

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 =