Trigonometry & Inverse Trigonometry Questions (1013)

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 $(9\mu + \lambda)$

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: