If sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}} | vedclass

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

Question

(A) Given, $\sin ^{-1}\left(x-\frac{x^{2}}{2}+\frac{x^{3}}{4}-\frac{x^{4}}{8}+\ldots\right)=\frac{\pi}{6}$.This is an infinite geometric series inside

Vedclass · Accepted Answer

$\frac{2}{3}$

Vedclass · Answer

(A) Given, $\sin ^{-1}\left(x-\frac{x^{2}}{2}+\frac{x^{3}}{4}-\frac{x^{4}}{8}+\ldots\right)=\frac{\pi}{6}$.This is an infinite geometric series inside the $\sin ^{-1}$ function with first term $a = x$ and common ratio $r = -\frac{x}{2}$.The sum of an infinite geometric series is given by $S_{\infty} = \frac{a}{1-r}$.Substituting the values, we get $S_{\infty} = \frac{x}{1 - (-\frac{x}{2})} = \frac{x}{1 + \frac{x}{2}} = \frac{2x}{2+x}$.So, $\sin ^{-1}\left(\frac{2x}{2+x}\right) = \frac{\pi}{6}$.Taking $\sin$ on both sides, we get $\frac{2x}{2+x} = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.Cross-multiplying, we get $4x = 2 + x$.$3x = 2$, which gives $x = \frac{2}{3}$.

If $\sin ^{-1}\left(x-\frac{x^{2}}{2}+\frac{x^{3}}{4}-\frac{x^{4}}{8}+\ldots\right)=\frac{\pi}{6},$ where $|x| < 2,$ then the value of $x$ is

Similar Questions

The value of $\sin^{-1}(\sin 10)$ is

If $\cos ^{-1} x = \sin ^{-1}(3x)$,then $x$ is equal to

$\sec^{-1}[\sec(-30^o)] = $ ....... $^o$

The value of $\cos ^{-1}\left(\cot \left(\frac{\pi}{2}\right)\right)+\cos ^{-1}\left(\sin \left(\frac{2 \pi}{3}\right)\right)$ is

If $\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x$,then $x$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module