Question
The equation $${\sin ^{ - 1}}\left( {3x - 4{x^3}} \right) = 3\,{\sin ^{ - 1}}\left( x \right)$$ is true for all values of $$x$$ lying in which one of the following intervals ?
A.
$$\left[ { - \frac{1}{2},\frac{1}{2}} \right]$$
B.
$$\left[ {\frac{1}{2},1} \right]$$
C.
$$\left[ { - 1, - \frac{1}{2}} \right]$$
D.
$$\left[ { - 1,1} \right]$$
Answer :
$$\left[ { - 1,1} \right]$$
Solution :
Let, $${\sin ^{ - 1}}x = \theta $$
$$ \Rightarrow x = \sin \theta $$
$$\eqalign{
& {\sin ^{ - 1}}\left( {3\sin \theta - 4\,{{\sin }^3}\theta } \right) = {\sin ^{ - 1}}\sin 3\theta \cr
& = 3\theta = 3\,{\sin ^{ - 1}}x \cr} $$
Equation $${\sin ^{ - 1}}\left( {3x - 4{x^3}} \right) = 3\,{\sin ^{ - 1}}x$$ is true for all values of $$x$$ lying in the interval $$\left[ { - 1,1} \right].$$