Question

Find the value of
$co{t}^{-1}\frac{6}{8}+{\sin }^{-1}\frac{5}{13}.$

• A. ${\cos }^{-1}\frac{16}{65}$
• B. ${\sin }^{-1}\frac{63}{65}$
• C. ${\sin }^{-1}\frac{126}{130}$
• D. None

Answer 1

Answer: (A), (B), (C)

${\sin }^{-1}\frac{126}{130}$

$co{t}^{-1}\frac{6}{8}+{\sin }^{-1}\frac{5}{13}$
$={\sin }^{-1}\frac{8}{10}+{\sin }^{-1}\frac{5}{13}$
$={\sin }^{-1}\left(\frac{8}{10}\times \sqrt{1-{\left(\frac{5}{13}\right)}^2}+\frac{5}{13}\times \sqrt{1-{\left(\frac{8}{10}\right)}^2}\right)={\sin }^{-1}\left(\frac{8}{10}\times \frac{12}{13}+\frac{5}{13}\times \frac{6}{10}\right)={\sin }^{-1}\left(\frac{96}{130}+\frac{30}{130}\right)={\sin }^{-1}\left(\frac{126}{130}\right)={\sin }^{-1}\left(\frac{63}{65}\right)={\cos }^{-1}\left(\frac{16}{65}\right)$