Question

Find the value of
${\cos }^{-1}\frac{3}{5}+{\cos }^{-1}\frac{5}{13}.$

• A. $\pi -{\sin }^{-1}\frac{56}{65}$
• B. ${\sin }^{-1}\frac{56}{65}$
• C. ${\cos }^{-1}\frac{33}{65}$
• D. $\pi -{\cos }^{-1}\frac{33}{65}$

Answer 1

Answer: (A), (D)

$\pi -{\cos }^{-1}\frac{33}{65}$

${\cos }^{-1}\frac{3}{5}+{\cos }^{-1}\frac{5}{13}$
$={\cos }^{-1}\left(\frac{3}{5}\times \frac{5}{13}-\sqrt{1-{\left(\frac{3}{5}\right)}^2}\times \sqrt{1-{\left(\frac{5}{13}\right)}^2}\right)$
$={\cos }^{-1}\left(\frac{15}{65}-\frac{4}{5}\times \frac{12}{13}\right)$
$={\cos }^{-1}\left(\frac{15}{65}-\frac{48}{65}\right)$
$={\cos }^{-1}\left(-\frac{33}{65}\right)$
$=\pi -{\cos }^{-1}\frac{33}{65}$
$=\pi -{\sin }^{-1}\frac{56}{65}$