Trigonometric Identities Question 43
Question: If $ \sin \theta =\frac{12}{13},(0<\theta <\frac{\pi }{2}) $ and $ \cos \varphi =-\frac{3}{5},( \pi <\varphi <\frac{3\pi }{2} ) $ . Then $ \sin (\theta +\varphi ) $ will be
[Orissa JEE 2004]
Options:
A) $ \frac{-56}{61} $
B) $ \frac{-56}{65} $
C) $ \frac{1}{65} $
D) -56
Show Answer
Answer:
Correct Answer: B
Solution:
We have $ \sin \theta =\frac{12}{13} $ $ \cos \theta =\sqrt{1-{{\sin }^{2}}\theta }=\sqrt{1-{{( \frac{12}{13} )}^{2}}}=\frac{5}{13} $ and $ \cos \varphi =\frac{-3}{5},\sin \varphi =\sqrt{1-\frac{9}{25}}=\frac{-4}{5} $ , $ [ \because \pi <\varphi <\frac{3\pi }{2} ] $ Now, $ \sin (\theta +\varphi )=\sin \theta .\cos \varphi +\cos \theta .\sin \varphi $ $ =( \frac{12}{13} ),( \frac{-3}{5} )+( \frac{5}{13} ),( \frac{-4}{5} )=\frac{-36}{65}-\frac{20}{65} $ $ =\frac{-56}{65} $ .
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