If $\sin h^{-1}\left(\sqrt{8}\right)+\sin h^{-1}\left(\sqrt{24}\right)=\alpha ,$ then $\sin h\alpha =$

Important Questions on Hyperbolic Functions

MEDIUM

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

The domain of the function $f\left(x\right)={\sec }^{-1}\left(3x-4\right)+\tan h^{-1}\left(\frac{x+3}{5}\right)$ is

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

$\cot h^{-1}\left(2\right)+\mathrm{cosec}{h}^{-1}(-2\sqrt{2})=$

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

Let $k>0$ and $t={\mathrm{sech}}^{-1}\left(\frac{1}{2}\right)-{\mathrm{cosech}}^{-1}\left(\frac{3}{k}\right).$ If $3e^t=2+\sqrt{3},$ then $k=$

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

$\tan h^{-1}\left(\frac{1}{3}\right)+\cot h^{-1}\left(2\right)=$

MEDIUM

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

$\sin h^{-1}\left(-2\right)+\mathrm{cosec}{h}^{-1}\left(-2\right)+\cot h^{-1}\left(-2\right)=$

MEDIUM

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

$e^{\left(\sec h^{-1}\frac{1}{2}+\tan h^{-1}\frac{1}{2}+\sin h^{-1}\frac{1}{2}\right)}=$

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If $\tan h^{-1}x=alog\left(\frac{1+x}{1-x}\right),\left|x\right|<1,$ then a is equal to

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If ${\mathrm{sech}}^{-1}\left(\frac{1}{2}\right)-{cosech}^{-1}\left(\frac{3}{4}\right)={\log }_{e}k,$ then

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If $\sinh x=\frac{3}{4}$ and $\cosh y=\frac{5}{3}$ then $x+y=$

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

$\tan h^{-1}\frac{1}{2}+\cot h^{-1}3=$

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

The imaginary part of ${\tan }^{-1}\left(\cos \theta +i\sin \theta \right)$ is

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If $\sinh \left(2{\tanh }^{-1}x\right)=\frac{11}{60},$ then $x=$

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

For $0<x\le \pi ,\sin h^{-1}\left(\cot x\right)$ is equal to

MEDIUM

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

${\sinh }^{-1}\left(2^{\frac{3}{2}}\right)$ is equal to
 

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

${coth}^{-1}3+{\tanh }^{-1}\frac{1}{3}-{cosech}^{-1}\left(-\sqrt{3}\right)=$

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

If $\tan h^{-1}\left(x+iy\right)=\frac{1}{2}\tan h^{-1}\left(\frac{2x}{1+x^2+y^2}\right)+\frac{i}{2}{\tan }^{-1}\left(\frac{2y}{1-x^2-y^2}\right)$, where $x, y\in R$, then $\tan h^{-1}\left(iy\right)=$

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

$\sin h^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$ is equal to

HARD

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

$\sec h^{-1}(\sin \theta )$ is equal to
 

EASY

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

$2 \tan h^{-1}\left(\frac{1}{2}\right)$ is equal to

MEDIUM

Mathematics>Trigonometry>Hyperbolic Functions>Basics of Inverse Hyperbolic Functions

$x=\log \left(\frac{1}{y}+\sqrt{1+\frac{1}{y^2}}\right)\Rightarrow y$ is equal to