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STD 11 - 3. trignometrical ratios,functions and identities — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 3. trignometrical ratios,functions and identities4 Marks
MCQ
If $\sin \theta = \frac{{12}}{{13}},(0 < \theta < \frac{\pi }{2})$ and $\cos \phi = - \frac{3}{5},\left( {\pi < \phi < \frac{{3\pi }}{2}} \right)$. Then $\sin (\theta + \phi )$will be
  • A
    $\frac{{ - 56}}{{61}}$
  • ✓
    $\frac{{ - 56}}{{65}}$
  • C
    $\frac{1}{{65}}$
  • D
    $-56$
✓

Answer

Correct option: B.
$\frac{{ - 56}}{{65}}$
b
(b) We have $\sin \theta = \frac{{12}}{{13}}$ 

$\cos \theta = \sqrt {1 - {{\sin }^2}\theta } = \sqrt {1 - {{\left( {\frac{{12}}{{13}}} \right)}^2}} = \frac{5}{{13}}$ 

and $\cos \phi = \frac{{ - 3}}{5},\sin \phi = \sqrt {1 - \frac{9}{{25}}} = \frac{{ - 4}}{5}$, 

                              $\left[ \because {\pi  < \phi < \frac{{3\pi }}{2}} \right]$

Now, $\sin (\theta + \phi ) = \sin \theta .\cos \phi + \cos \theta .\sin \phi $ 

$ = \left( {\frac{{12}}{{13}}} \right)\,\left( {\frac{{ - 3}}{5}} \right) + \left( {\frac{5}{{13}}} \right)\,\left( {\frac{{ - 4}}{5}} \right)$

$= \frac{{ - 36}}{{65}} - \frac{{20}}{{65}}$$ = \frac{{ - 56}}{{65}}$.

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