Trigonometry Questions (1127)

Let $f(x) = \max \{\sin x, \cos x\}$. Then the number of roots of the equation $f(x) = \frac{1}{\sqrt{2}}$ in $(0, 2\pi)$ is

\(\tan 40° + 2\tan 10°\) is equal to

A is the orthocentre of △ABC and D is reflection point of A w.r.t. perpendicular bisector of BC, then orthocentre of △DBC is:

The median $AD$ of a triangle $ABC$ is bisected at $E$ and $BE$ meets $AC$ at $F$; then $AF:AC =$

Which are correct for relation \(R\) on \(\mathbb{R}\)?

If \pi \le x \le 3\pi , then cos -1 ( 12 cos x + 5 sin x) is equal to 2 4 13 13

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x² - c² = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is (JEE Main 2019)

If \alpha > \beta > \gamma > 0, then the expression cot 2 2 2 (1+\beta ) (1+\gamma ) (1+\alpha ) -1 {\beta + (\alpha-\beta) } + cot -1 {\gamma + (\beta-\gamma) }+ cot -1 {\alpha + (\gamma-\alpha) } is equal to :

The expression \(\frac{(a + b + c)(b + c - a)(c + a - b)(a + b - c)}{4b^2c^2}\) is equal to:(where symbols used have usual meanings)

\(ABC\) is a triangle. Forces \(\vec{P}\), \(\vec{Q}\), \(\vec{R}\) acting along \(IA\), \(IB\) and \(IC\) respectively are in equilibrium, where \(I\) is the incentre of \(\triangle ABC\). Then \(P : Q : R\) is

The value of $(\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)$ is

The angular depressions of the top and the foot of a tower, as seen from the top of a second tower which is $150$ meters high and standing on the same level as the first, are $13°$ and $\tan^{-1}\left(\frac{5}{6}\right)$ respectively. If the distance between their tops is $d$, then

If \(\tan\theta + \tan\left(\frac{\pi}{3} + \theta\right) + \tan\left(\frac{2\pi}{3} + \theta\right) = k\tan 3\theta\) then \(k\) is equal to

If $\displaystyle\sum_{r=1}^{13} \left\{\frac{1}{\sin\!\left(\tfrac{\pi}{4}+(r-1)\tfrac{\pi}{6}\right)\sin\!\left(\tfrac{\pi}{4}+r\tfrac{\pi}{6}\right)}\right\} = a\sqrt{3} + b,\ a, b \in \mathbf{Z}$, then $a^2 + b^2$ is equal to

In triangle \(ABD\), using the sine rule, if \(BD = \sqrt{p^2+q^2}\) and \(\angle ABD = \theta\), \(\angle ADB = \alpha\), then \(AB\) equals

Let M be the greatest and m be the least value of \(\sqrt{\sin^{-1} x} + \sqrt{\cos^{-1} x}\), then find the value of \((M/m)^4\).

Let set $A$ denote the solutions of $\cos^{-1}(4x^3-3x)=\tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)$. Then

95. In the △ABC, AB = 5 cm, AC = 12 cm and BC = 13 cm then the distance of A from the side BC is (in cm)

Sum of all values of $\theta\in\left(0,\dfrac{\pi}{2}\right)$ satisfying $\sin^22\theta+\cos^42\theta=\dfrac{3}{4}$ is

$PQR$ is a triangular park with $PQ = PR = 200$ m. A TV tower stands at the mid-point of $QR$. If the angles of elevation of the top of the tower from $P$, $Q$ and $R$ are $45°$, $30°$ and $30°$ respectively, then the height of the tower (in meters) is

175. If \(A\) lies in the fourth quadrant and \(3\tan A + 4 = 0\), then \(5\sin 2A + 2\sin A + 4\cos A\) is equal to:

810. In \(\triangle ABC\) if inradius \(r = 1\), circumradius \(R = 3\) and semiperimeter \(s = 7\), then find the value of \((a^2 + b^2 + c^2)\), where \(a, b, c\) are the sides of triangle \(ABC\).

If a root of the equation $n^2\sin^2 x - 2\sin x - (2n+1) = 0$ lies in $\left[\dfrac{\pi}{2}, \pi\right]$, then the minimum positive integer value of $n$ is

Let $S = \{\theta \in [0, 2\pi): \tan(\pi\cos\theta) + \tan(\pi\sin\theta) = 0\}$. Then $\displaystyle\sum_{\theta \in S} \sin^2\left(\theta + \frac{\pi}{4}\right)$ is equal to

If $\tan 15° + \frac{1}{\tan 75°} + \frac{1}{\tan 105°} + \tan 195° = 2a$, then the value of $\left(a + \frac{1}{a}\right)$ is:

The number of integral values of $\alpha$ for which the equation $\dfrac{16}{\tan x}+\dfrac{4}{4-\tan x}=\alpha$ does not have any solution is

158. 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:

If |k| = 5 and 0°

The equation sin x + sin y + sin z = -3 for 0

Ex. 24: Statement I In a triangle ABC, \(\cos^2\dfrac{A}{2}\) has the value equal to \(\dfrac{s(s-a)}{abc}\)Statement II In a triangle ABC, \(\cos\dfrac{A}{2} = \sqrt{\dfrac{(s-b)(s-c)}{bc}}\), \(\cos\dfrac{B}{2} = \sqrt{\dfrac{(s-a)(s-c)}{ac}}\), \(\cos\dfrac{C}{2} = \sqrt{\dfrac{(s-a)(s-b)}{ab}}\)

101. Two straight roads intersect at 30°. From the junction, two persons A and B start walking at the same time, one on each road. A walks at the rate of 5 km/h. At the end of 3 hours they are 9 km apart. If B walks at uniform rate then the speed of B is

Ex. 45. If $a$, $P$, $y$ are acute angles and $\cos \theta = \frac{\sin P}{\sin a}$, $\cos \theta = \frac{\sin y}{\sin a}$ and $\cos(\theta - \phi) = \sin P \sin y$, then the value of $\tan^2 a - \tan^2 P - \tan^2 y$ is equal to

171. The least positive value of \(x\) satisfying the equation \(\dfrac{\sin x}{\cos 3x} + \dfrac{\sin 3x}{\cos 9x} + \dfrac{\sin 9x}{\cos 27x} = 0\) is:

152. 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:

276. If \(\dfrac{1}{2}\sin^{-1}\!\left(\dfrac{3\sin 2\alpha}{5+4\cos 2\alpha}\right) = \tan^{-1} x\), then the possible value of \(x\) is:

169. If \(\sin\alpha + \sin\beta + \sin\gamma = -3\), \(\alpha, \beta, \gamma \in (0, 2\pi)\), then \(\cos 2\alpha + \cos 4\beta + \cos 6\gamma\) is equal to:

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.

175. If \(A\) lies in the fourth quadrant and \(3\tan A + 4 = 0\), then \(5\sin 2A + 2\sin A + 4\cos A\) is equal to:

The angles A, B and C of a triangle ABC are in arithmetic progression. If \(2b^2 = 3c^2\) then the angle A is:

The general solution of $\cos 3x\cdot\cos^3 x+\sin 3x\cdot\sin^3 x=0$ is

Let $|\cos\theta\cos(60^\circ-\theta)\cos(60^\circ+\theta)|\leq\dfrac{1}{8}$, $\theta\in[0,2\pi]$. Then, the sum of all $\theta\in[0,2\pi]$, where $\cos3\theta$ attains its maximum value, is:

The angle of elevation of the top $P$ of a tower from the feet of one person standing due south of the tower is $45°$ and from the feet of another person standing due west of the tower is $30°$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to

If $S=\left\{x\in\mathbb{R}:\sin^{-1}\!\left(\dfrac{x+1}{\sqrt{x^2+2x+2}}\right)-\sin^{-1}\!\left(\dfrac{x}{\sqrt{x^2+1}}\right)=\dfrac{\pi}{4}\right\}$, then $\displaystyle\sum_{x\in S}\left(\sin\frac{(x^2+x+5)\pi}{2}-\cos\frac{(x^2+x+5)\pi}{2}\right)$ is equal to _________.

For $x\in(-1,1]$, the number of solutions of the equation $\sin^{-1}x=2\tan^{-1}x$ is equal to

In $\triangle ABC$, if $\cos A+2\cos B+\cos C=2$ and the sides opposite to $A$ and $C$ are $3$ and $7$ respectively, then $\cos A-\cos C$ is equal to

The value of $\tan 9°-\tan 27°-\tan 63°+\tan 81°$ is _____.

In △ABC, a = 4, b = 12 and B = 60°, then the value of sin A is

The principal value of \(\tan^{-1}(-\sqrt{3})\) is ______.

If \sum , then a + b is equal to : 13 1 2 2 { } = a\sqrt3 + b, a, b \in Z r=1 \pi \pi \pi r\pi sin( +(r-1) ) sin( + ) 4 6 4 6

The value of (sin 70 ) (cot 10 cot 70 - 1) is ∘ ∘ ∘