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What does $2sin^{-1} \frac{3}{5}$ reduce to?

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Toolbox:
  • \( sin^{-1}x=tan^{-1}\frac{x}{\sqrt{1-x^2}}\)
  • \( 2tan^{-1}x=tan^{-1}\frac{2x}{1-x^2} |x| < 1\)
Given $2sin^{-1} \frac{3}{5}$, Let $x =$ \(\frac{3}{5}\), \( \rightarrow \large \frac{x}{\sqrt{1-x^2}}=\frac{\frac{3}{5}}{\sqrt{1-\frac{9}{25}}}=\frac{3}{4}\)
\(\Rightarrow\:sin^{-1}\frac{3}{5}=tan^{-1}\frac{3}{4}\)
\(\Rightarrow 2sin^{-1}\frac{3}{5}=2tan^{-1}\frac{3}{4}\)
Given \( 2tan^{-1}x=tan^{-1}\frac{2x}{1-x^2}\), we need to evaluate \(2tan^{-1}\frac{3}{4}\)
Let $x =\frac{3}{4} \rightarrow \large \frac{2x}{1-x^2}=\frac{2.\frac{3}{4}}{1-\frac{9}{16}}=\frac{6}{4}.\frac{16}{7}=\frac{24}{7}$
\( \Rightarrow\:2tan^{-1}\frac{3}{4} = tan^{-1}\frac{24}{7} \)
answered Mar 14, 2013 by balaji.thirumalai
 
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