Trigonometry Questions (1127)

The value of $\cos^{-1} x + \cos^{-1}\left(\frac{x}{2} + \frac{1}{2}\sqrt{3-3x^2}\right)$ is equal to: $\left(\frac{1}{2} \leq x \leq 1\right)$

If \( \cos^{-1}\!\left(\dfrac{2}{3x}\right) + \cos^{-1}\!\left(\dfrac{3}{4x}\right) = \dfrac{\pi}{2} \left(x > \dfrac{3}{4}\right) \), then \( x \) is equal to

Range of f(x) = \sin^{-1}\log_{[x]} + \log(\sin^{-1}[x]), where [] denotes GIF is

Find the maximum value of \(3\cos\theta + 5\sin\left(\theta - \dfrac{\pi}{6}\right)\).

Ex. 65: Let $f(x) = ab \sin x + b\sqrt{1 - a^2} \cos x + c$, where $|a| 0$ then

The value of \(\tan\left(\cos^{-1}\left(\frac{4}{5}\right) + \tan^{-1}\left(\frac{2}{3}\right)\right)\) is

$\tan^{-1}(\tan \theta) = \theta$, for all $\theta$ belonging to

Solve: tan⁻¹((x+1)/(x-1)) + tan⁻¹((x-1)/x) = tan⁻¹(-7)

Considering only the principal values of inverse functions, the set \( A = \left\{x \geq 0;\, \tan^{-1}(2x) + \tan^{-1}(3x) = \dfrac{\pi}{4}\right\} \)

\(\cos[\tan^{-1}\{\sin(\cot^{-1}x)\}]=\)

934. Find the number of integers not in the domain of \(f(x) = \cos^{-1}\!\left(\dfrac{2-x}{2x}\right)\).

Let $S_1$ is the complete solution set of the inequality $\cos^{-1}(x) > \cos^{-1}\left(x^2\right)$ and $S_2$ is the complete solution set of the inequality $\left(\cos^{-1} x^2\right) > 0$, then $S_1 \cap S_2$ is

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan^{-1}(x)+\tan^{-1}(2x)=\dfrac{\pi}{4}$ is:

Value of $\sin32°\cdot\sin88°\cdot\sin152°$ equals

If $\tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2+x+1} = \frac{\pi}{2}$, find $x$.

If \(\tan^{-1} y : \tan^{-1} x = 4:1\), express \(y\) as an algebraic function of \(x\). Hence or otherwise prove that \(22\frac{1}{2}\) is a root of the equation \(x^4 + 1 = 6x^2\).

The range of the function \( f(x) = \sin^{-1}\!\left(\log_2 \dfrac{x^2}{2}\right) \) is:

$\cos\!\left(\sin^{-1}\dfrac{3}{5}+\sin^{-1}\dfrac{5}{13}+\sin^{-1}\dfrac{33}{65}\right)$ is equal to:

Find the value of \(\cos^{-1}(\cos 13)\).

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2\sin^{-1}x+3\cos^{-1}x=\dfrac{2\pi}{5}$, is

The value of \(\tan^{-1}\left(\frac{x\cos\theta}{1-x\sin\theta}\right) - \cot^{-1}\left(\frac{\cos\theta}{x-\sin\theta}\right)\) is

Find the value of \(\tan^{-1}\left(\dfrac{1}{2}\tan 2A\right) + \tan^{-1}(\cot A) + \tan^{-1}(\cot^3 A)\) for \(0

Given expression = \(1 + 2^2 + 1 + 3^2 + \text{cosec}\left(\tan^{-1}\dfrac{4}{3} + \tan^{-1}\dfrac{4}{3}\right)\). Find the value of the expression.

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