Question

How do you simplify $\sin \left( {2 \cdot \arcsin \left( x \right)} \right)$ ?

Answer 1

Hint : The arcsine function is the inverse of the sine function. It returns the angle whose sine is a given number. For every trigonometry function, there is an inverse function that works in reverse. These inverse functions have the same name but with ‘arc’ in front.
The arcsine of $x$ is equal to the inverse sine function of $x$ when $- 1 < x < 1$
Follow the method given below to solve the given question.

Complete step-by-step answer :
It happens to be the inverse of cosine and can be used to solve more complicated right-angle problems. This lesson will define the arccosine in more detail and give some example problems.
Trigonometric functions are no different. They all have inverse operation.
The inverse of sine is arcsine
The inverse of cosine is arccosine
The inverse of tangent is arctangent
The inverse of secant is arcsecant
The inverse of cosecant is arccosecant
The inverse of cotangent is arccotangent
Since none of the six trigonometric functions are one-to-one, they must be restricted in order to have inverse functions. Therefore, the ranges of the inverse functions are proper subsets of the domain of the original functions.

Let $y = \arcsin x$ then
$x = \sin y$
$\sin \left( {2\arcsin x} \right) = \sin 2y \\\ = 2\sin y\cos y \\\ {\cos ^2}y + {\sin ^2}y = 1 \\\ {\cos ^2}y = 1 - {x^2} \\\ \Rightarrow \cos y = \sqrt {1 - {x^2}} \;$
Therefore,
$\sin \left( {2\arcsin x} \right) = 2x\sqrt {1 - {x^2}}$
So, the correct answer is “$2x\sqrt {1 - {x^2}}$ ”.

Note : In mathematics, the inverse trigonometry functions are the inverse functions of the trigonometric functions, specifically, inverse of sine, cosine, tangent, cotangent, secant, and cosecant functions.
These inverse trigonometric functions are used to obtain an angle from any of the angle’s trigonometric ratios.
These types of inverse trigonometric functions are widely applicable or used in engineering, navigation, physics and geometry.