Trigonometry Questions (1127)

The value of \cos^{-1}\left(\cot\left(\sin^{-1}\sqrt{\frac{1-x^2}{4}}\right)\right) + \sec^{-1}\left(\sqrt{1+x^2}\right)

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\!\left((\sec^{-1}x)^2+(\operatorname{cosec}^{-1}x)^2\right)$ is:

The value of \(\sin\left(\cos^{-1}\dfrac{1}{2} + \sin^{-1}\dfrac{\sqrt{3}}{2}\right)\) is ______.

If $\sum_{r=1}^n\left(\dfrac{\tan 2^{r-1}}{\cos 2^r}\right)=\tan p^n-\tan q$, then find the value of $(p+q)$

Let $x=\sin1°$. The value of $\dfrac{1}{\cos0°\cos1°}+\dfrac{1}{\cos1°\cos2°}+\cdots+\dfrac{1}{\cos44°\cos45°}$ is

The number of solutions of $\sin^2 x+(2+2x-x^2)\sin x-3(x-1)^2=0$ in $\left[0,\frac{\pi}{2}\right]$ is $\alpha$, and in $[-2\pi,2\pi]$ is $\beta$. Then $\alpha+\beta=$

In a triangle ABC the expression \(a\cos B\cos C + b\cos C\cos A + c\cos A\cos B\) equals to:

The numerical value of \(\cos[\tan^{-1}(-3) + \cot^{-1}(-3)]\) = ______.

Number of real solutions of $\sin(\pi x^2/4)=0$ and $\tan(\pi x/2)=$ ... solve the system $\sin(\pi x^2/4)=0$, $\cos(\pi x/2)=0$ simultaneously. Number of solutions in $[-2,2]$.

If $\displaystyle\sum_{n=1}^{\infty} \cot^{-1}\!\left(2 + \frac{n(n+1)}{2}\right) = \tan^{-1} a$, then $a$ is equal to

Let $T(\theta) = \cos^2(30°-\theta) - \cos(30°-\theta)\cos(30°+\theta) + \cos^2(30°+\theta)$. Then the value of $4\displaystyle\sum_{\theta=1}^{30} \theta\, T(\theta)$ is

If $2(\sin A-\sin^3 A)=\cos B$ and $2(\cos A+\cos^3 A)=\sin B$; $0<A,B<\pi/2$, then $\cos B=\sqrt{\frac{m}{n}}$ where $m,n$ are co-prime and $m+n$ is

Let $p=\tan\left(\frac{5\pi}{9}-\cos\left(2\sin^{-1}\frac{1}{\sqrt{5}}\right)\right)$, $q=\sin^{-1}\left(\sin\frac{2\pi}{3}\right)+\cos^{-1}\left(\cos\frac{7\pi}{6}\right)$. Then the quadratic equation whose roots are $p$, $\sec q$ is

$\cos 1°\cos 2°\cos 3°\cdots\cos 179°=$

The value of the expression $\dfrac{\sin 20°(4\cos 20°+1)}{\cos 20°\cdot\cos 30°}$ is

In a $\triangle ABC$, if $a=2x$, $b=2y$ and $C=120°$, the area of triangle is

If the sum of values of $\theta$ in $(-3\pi, 3\pi)$ satisfying $\displaystyle\sum_{m=1}^{15}\sec\!\left(\theta+(m-1)\frac{\pi}{18}\right)\sec\!\left(\theta+m\frac{\pi}{18}\right)=(4+2\sqrt{3})\csc\frac{\pi}{18}$ is $\dfrac{k\pi}{10}$, then the value of $k$ is

The value of $x \in \left(0,\dfrac{\pi}{2}\right)$ satisfying $\dfrac{\sqrt{5}-1}{\sin x} + \dfrac{\sqrt{10+2\sqrt{5}}}{\cos x} = 8$ is

If $x=\cos1°\cos2°\cos3°\cdots\cos89°$ and $y=\cos2°\cos6°\cos10°\cdots\cos86°$, then $\dfrac{2}{7}\log_2\!\left(\dfrac{y}{x}\right)$ is equal to

The value of $4\cos\dfrac{\pi}{10} - 3\sec\dfrac{\pi}{10} - 2\tan\dfrac{\pi}{10}$ is equal to

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 $A+B+C=\pi$ and $\tan A+\tan B+\tan C=k$, then $k$ can NOT be

Since f(x) is onto, the range of f(x) equals co-domain. The range of f(x) = cos−1(4x2 + 3x) is \(\left[\frac{\pi}{2}, \pi - \cos^{-1}\frac{9}{16}\right]\). What is the answer? (Integer answer: 25)

The maximum value of $\cos^2\theta+\cos^2(\theta+\pi/3)-\cos\theta\cos(\theta+\pi/3)$ is

Find the value of \(\sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ\).

If $\theta\in\mathbb{R}$, then the range of $f(\theta)=\begin{vmatrix}1&\cos\theta&1\\-\cos\theta&1&\cos\theta\\-1&-\cos\theta&1\end{vmatrix}$ is

In a triangle \(ABC\), \(a = 4\), \(b = 3\), \(\angle A = 60°\), then \(c\) is the root of the equation

Number of ordered pairs $(x,y)$ satisfying $\dfrac{4^{\sin x}\cdot16^{\sin y}}{(1+16^{\sin x})(1+256^{\sin y})}=\dfrac{1}{4}$ and $3^{1+\sqrt{\cos^2x}}+3^{1+\cos y}=10$; $x,y\in[0,2\pi]$ is

If $\cos^{-1}\!\sqrt{p}+\cos^{-1}\!\sqrt{1-p}+\cos^{-1}\!\sqrt{1-q}=\dfrac{3\pi}{4}$, then $q$ is

The domain of the function \( f(x) = \sin^{-1}\left[\log_3\left(\dfrac{x}{3}\right)\right] \) is:

If \(\cos\theta + \sec\theta = 2\) then \(\cos^n\theta + \sec^n\theta\) is equal to

If \(\sin(\alpha + \beta) = 1\) and \(\sin(\alpha - \beta) = \dfrac{1}{2}\), then \(\tan(\alpha + 2\beta) \cdot \tan(2\alpha + \beta)\) is equal to:

From a point on the ground, the angle of elevation of the top of a tower is \( \tan^{-1}\left(\dfrac{3}{5}\right) \). The tower is 40 m away from the point. A flag is hoisted at the top of the tower and the angle of elevation of the bottom of the flag from the same point is \( \alpha \) where \( \tan\alpha = \dfrac{3}{5} \). If \( \tan(\alpha + \beta) = \dfrac{x}{40} \) (where \( \beta \) is the angle subtended by the flag at the point on the ground), find the height \( x \) of the flag (in metres).

Let $\frac{5}{6}\cos^{-1}\sqrt{\dfrac{3}{3+\pi^2}}+\frac{1}{3}\sin^{-1}\dfrac{2\sqrt{3}\pi}{3+\pi^2}+\frac{1}{6}\tan^{-1}\dfrac{\sqrt{3}}{\pi}=a$ and $\cos^{-1}\!\left[\frac{13}{40}\cos\!\left(\cot^{-1}\frac{5}{12}\right)+\frac{13}{32}\sin\!\left(\cos^{-1}\frac{5}{13}\right)\right]=b$. Then $\csc\!\left(\displaystyle\int_b^a\left[\frac{\tan x}{\sqrt{3}}\right]dx\right)$ is ($[\cdot]$ = GIF)

The principal value of \(\cos^{-1}\left(\cos\dfrac{7\pi}{4}\right)\) is ______.

If \(\cos\theta - \sin\theta = \cos\alpha - \sin\alpha\), then the value of \(|\theta + \alpha|\) is:

If the angles of elevation of the top of a tower from three collinear points \(A\), \(B\) and \(C\), on a line leading to the foot of the tower, are 30°, 45° and 60° respectively, then the ratio \(AB : BC\), is

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

If \(k = \displaystyle\sum_{r=0}^{10} \cos^3\dfrac{\pi r}{3}\), then the value of \(\dfrac{16}{k^2}\) is

One root of $\tan^{-1}\!\cot\!\left(\dfrac{3x^2+3|x|+1}{x^2+|x|+1}\right)=\dfrac{\pi}{2}-\csc\!\cdot\!\csc^{-1}\!\left(\dfrac{3|x|+2}{|x|+1}\right)$ is $2\sin\theta$, $\theta\in\left(0,\dfrac{\pi}{2}\right)$. Value of $\tan\dfrac{7\theta}{9}\cdot\tan\dfrac{10\theta}{9}\cdot\tan\dfrac{13\theta}{9}$ is

The maximum value of the function $f(x) = \dfrac{4\cot^{-1}x}{\pi} - \dfrac{\pi}{4\cot^{-1}(-x)}$ occurs at $x$ equal to

A tower is observed from three collinear points A, B, C on ground such that angles of elevation are $\alpha$, $2\alpha$, $3\alpha$. If $AB : BC = ?$

Let \(f_k(x) = \dfrac{1}{k}(\sin^k x + \cos^k x)\) where \(x \in \mathbb{R}\) and \(k \geq 1\). Then \(f_4(x) - f_6(x)\) equals:

989. Let \(T(n) = \cos^2(30° - n°) - \cos(30° - n°)\cos(30° + n°) + \cos^2(30° + n°)\). Find the value of \(4\displaystyle\sum_{n=1}^{30} nT(n)\).

The value of $\displaystyle\prod_{r=1}^{7} \cos\frac{r\pi}{15}$ is

If $A+B+C=\pi$, find the maximum value of $\sin A\sin B\sin C$

The angle of elevation of the top of a vertical tower from a point \(A\), due east of it is 45°. The angle of elevation of the top of the same tower from a point \(B\), due south of \(A\) is 30°. If the distance between \(A\) and \(B\) is \(54\sqrt{2}\) m, then the height of the tower (in metres) is:

The general solution of \(\sin x + \cos x = 1\) is given by

If $y = \tan^{-1}\dfrac{4x}{1+5x^2} + \tan^{-1}\dfrac{2+3x}{3-2x}$, find $\dfrac{dy}{dx} = \dfrac{\alpha}{1+25x^2}$. Find $\alpha$.

The given equation \(3\tan^{-1}(2-\sqrt{3}) - \tan^{-1}\left(\dfrac{1}{x}\right) = \tan^{-1}\left(\dfrac{1}{2}\right)\) is solved. Find the value of \(x\).