Question

To prove that $3{\sin ^{ - 1}}x = {\sin ^{ - 1}}\left( {3x - 4{x^3}} \right)$, using a range $x \in \left[ { - \dfrac{1}{2},\dfrac{1}{2}} \right]$.

Answer 1

Hint: In this problem, using the trigonometric function range table to define value and reverse trigonometric method for confirming equal on both sides. We're taking one side of it to solve this problem. And make the other side as the final solution.

Formula used: To put $x = \sin \theta $
Then ${\sin ^{ - 1}}x = \theta $
The given trigonometric function of $\sin 3x = 3\sin x - 4{\sin ^3}x$ is derived to get $\left( {{{\sin }^{ - 1}}\left( {\sin x} \right) = x} \right)$

Complete step-by-step answer:
Given that,
The value of L.H.S is $3{\sin ^{ - 1}}x$
the value of R.H.S is ${\sin ^{ - 1}}\left( {3x - 4{x^3}} \right)$
Let take R.H.S values,
R.H.S
${\sin ^{ - 1}}\left( {3x - 4{x^3}} \right)$…………$(1)$
To put $x = \sin \theta $
To substituting a value $x$ in given equation
We get,
$\Rightarrow$${\sin ^{ - 1}}\left( {3\sin \theta - 4{{\left( {\sin \theta } \right)}^3}} \right)$
Now we using a trigonometric formula, and the range of the function is $\dfrac{1}{2} \leqslant - 4{x^3} \leqslant - \dfrac{1}{2}$
We know that,
The trigonometric value of ${\sin ^{ - 1}}\left( {3\sin \theta - 4\left( {{{\sin }^3}\theta } \right)} \right)$ the equation is given by
$\Rightarrow$${\sin ^{ - 1}}\left( {\sin 3\theta } \right)$………..$(2)$
On simplifying the value,
Again, we use trigonometric formula in equation $(2)$
We get the value of R.H.S is $3\theta $
By using the above formula, we apply for $\theta $ value
To solving a given $3\theta $
Therefore,
 $\Rightarrow$$3\theta = $$3{\sin ^{ - 1}}x$
Hence proved that the given L.H.S is equal to R.H.S
$\Rightarrow$$3{\sin ^{ - 1}}x = \sin \left( {3x - 4{x^3}} \right)$

Note: Alternative method
Using the trigonometric range formula,
Therefore,
$\Rightarrow$$\sin 3\theta = 3x - 4{x^{^3}}$
On simplifying,
$\Rightarrow$$3\theta = {\sin ^{ - 1}}\left( {3x - 4{x^3}} \right)$
Using trigonometric formula,
$x = \sin \theta $
$\Rightarrow$${\sin ^{ - 1}}x = \theta $
To apply the above formula,
We get,
$\Rightarrow$$3{\sin ^{ - 1}}x = {\sin ^{ - 1}}\left( {3x - 4{x^3}} \right)$
Hence the solution is proved.