If x=-frac{1}{2},then sinh ^{-1} x+operatorna | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given $x = -\frac{1}{2}$.We know that $\sinh^{-1} x = \ln(x + \sqrt{x^2+1})$ and $\operatorname{cosech}^{-1} x = \ln\left(\frac{1}{x} + \sqrt{\fra

Vedclass · Accepted Answer

$\log _e\left(\frac{7-3 \sqrt{5}}{2}\right)$

Vedclass · Answer

(A) Given $x = -\frac{1}{2}$.We know that $\sinh^{-1} x = \ln(x + \sqrt{x^2+1})$ and $\operatorname{cosech}^{-1} x = \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2}+1}\right)$.Substituting $x = -\frac{1}{2}$:$\sinh^{-1}\left(-\frac{1}{2}\right) = \ln\left(-\frac{1}{2} + \sqrt{\frac{1}{4}+1}\right) = \ln\left(\frac{\sqrt{5}-1}{2}\right)$.$\operatorname{cosech}^{-1}\left(-\frac{1}{2}\right) = \ln\left(-2 + \sqrt{4+1}\right) = \ln(\sqrt{5}-2)$.Adding these:$\sinh^{-1} x + \operatorname{cosech}^{-1} x = \ln\left(\frac{\sqrt{5}-1}{2}\right) + \ln(\sqrt{5}-2) = \ln\left(\frac{(\sqrt{5}-1)(\sqrt{5}-2)}{2}\right)$.$= \ln\left(\frac{5 - 2\sqrt{5} - \sqrt{5} + 2}{2}\right) = \ln\left(\frac{7-3\sqrt{5}}{2}\right)$.

If $x=-\frac{1}{2}$,then $\sinh ^{-1} x+\operatorname{cosech}^{-1} x=$

Similar Questions

If $\tan \theta + \sec \theta = e^x$,then $\cos \theta$ equals

If $\frac{\cos (\theta_1+\theta_2)}{\cos (\theta_1-\theta_2)}+\frac{\cos (\theta_3-\theta_4)}{\cos (\theta_3+\theta_4)}=0$,then $\cot \theta_1 \cdot \cot \theta_2 \cdot \cot \theta_3 \cdot \cot \theta_4=$

If $x=\log _e\left[\cot \left(\frac{\pi}{4}+\theta\right)\right]$ and $\theta \in\left(\frac{-\pi}{4}, \frac{\pi}{4}\right)$,then consider the following statements:
$I$ : $\cosh x=\sec 2 \theta$
$II$ : $\sinh x=-\tan 2 \theta$
Then which one of the following options is true?

If $\theta$ is eliminated from the equations $x = a \cos(\theta - \alpha)$ and $y = b \cos(\theta - \beta)$,then $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{2xy}{ab} \cos(\alpha - \beta)$ is equal to

If $\sin A+\sin B=\sqrt{3}(\cos B-\cos A)$,then $\sin 3A+\sin 3B$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module