\(\rm \sin^{-1}\frac{4}{5}-\sin^{-1}\frac{5}{13}=?\)

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AAI ATC Junior Executive 21 Feb 2023 Shift 1 Official Paper
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  1. \(\rm \sin^{-1}\frac{53}{65}\)
  2. \(\rm \sin^{-1}\frac{33}{63}\)
  3. \(\rm \sin^{-1}\frac{33}{65}\)
  4. \(\rm \sin^{-1}\frac{65}{33}\)

Answer (Detailed Solution Below)

Option 3 : \(\rm \sin^{-1}\frac{33}{65}\)
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Detailed Solution

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Given:

\(\rm \sin^{-1}\frac{4}{5}-\sin^{-1}\frac{5}{13}\)

Concept:

Use formula

\(\rm sin^{1}x-sin^{-1}y=sin^{-1}[x\sqrt{1-(y)^2}-y\sqrt{1-(x)^2\ }\ ]\)

Calculation:

\(\rm \sin^{-1}\frac{4}{5}-\sin^{-1}\frac{5}{13}\)

\(\rm =sin^{-1}[\frac{4}{5}\sqrt{1-(\frac{5}{13})^2}-\frac{5}{13}\sqrt{1-(\frac{4}{5})^2\ }\ ]\)

\(\rm =sin^{-1}[\frac{48}{65}-\frac{15}{65}]\)

\(\rm =sin^{-1}\frac{33}{65}\)

Hence the option (3) is correct.

 

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