Trigonometry Questions (1127)

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 =

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 $(9\mu + \lambda)$