If ^ - 1 3 5 + ^ - 1 ( 12 13 ) = ^ - 1 C, then C = - Vidyadip

MCQ

If ${\sin ^{ - 1}}\frac{3}{5} + {\cos ^{ - 1}}\left( {\frac{{12}}{{13}}} \right) = {\sin ^{ - 1}}C,$ then $ C =$

A
$\frac{{65}}{{56}}$
B
$\frac{{24}}{{65}}$
C
$\frac{{16}}{{65}}$
✓
$\frac{{56}}{{65}}$

✓

Answer

Correct option: D.

$\frac{{56}}{{65}}$

d
(d) Given ${\sin ^{ - 1}}C = {\sin ^{ - 1}}\frac{3}{5} + {\cos ^{ - 1}}\frac{{12}}{{13}}$

$ \Rightarrow C = \sin \left( {{{\sin }^{ - 1}}\frac{3}{5} + {{\cos }^{ - 1}}\frac{{12}}{{13}}} \right)$

using, $\sin (A + B) = \sin A\,\,\cos B + \cos A\,\sin B$

$ \Rightarrow \,\,C = \frac{3}{5} \times \frac{{12}}{{13}} + \sqrt {1 - \frac{9}{{25}}} \sqrt {1 - \frac{{144}}{{169}}} $

$ \Rightarrow C = \frac{{56}}{{65}}$.

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