Inverse Trigonometric Functions Question 111
Question: If $ {{\sin }^{-1}}a+{{\sin }^{-1}}b+{{\sin }^{-1}}c=\pi , $ then find the value of $ a\sqrt{1-a^{2}}+b\sqrt{1-b^{2}}+c\sqrt{1-c^{2}}. $
Options:
A) $ abc $
B) $ a+b+c $
C) $ \frac{1}{a}\times \frac{1}{b}\times \frac{1}{c} $
D) $ 2abc $
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Answer:
Correct Answer: D
Let $ {{\sin }^{-1}}a=x $
$ \therefore a=\sin x $
$ {{\sin }^{-1}}b=y $
$ \therefore b=\sin y;{{\sin }^{-1}}c=z $
$ \therefore c=\sin z $
$ \therefore a\sqrt{1-a^{2}}+b\sqrt{1-b^{2}}+c\sqrt{1-c^{2}} $
$ =\sin x\cos x+\sin y\cos y+sinzcosz $
$ =(1/2)(sin2x+sin2y+sin2z)=(1/2)(4sinxsinysinz) $
$ =2\sin x\sin y\sin z=2abc $
BETA
