sin inverse (3 sin x-4 sin cube x)
well,
sin(3x) = 3sinx - 4sin^3x
so that gives you
sin^-1(sin(3x)) = 3x
To find the value of \(\sin^{-1}(3\sin x - 4\sin^3 x)\), we need to use the identity for the inverse sine function.
The inverse sine function, \(\sin^{-1}\), returns the angle (in radians) whose sine is equal to the input value. In other words, if \(\sin^{-1}(a) = b\), then \(\sin(b) = a\).
Now, let's simplify the expression: \(3\sin x - 4\sin^3 x\).
First, we can factor out \(\sin x\) from both terms:
\(3\sin x - 4\sin^3 x = \sin x(3 - 4\sin^2 x)\).
Next, we can use the Pythagorean identity, \(\sin^2 x + \cos^2 x = 1\), to substitute \(\cos^2 x\) for \(1 - \sin^2 x\):
\(\sin x(3 - 4\sin^2 x) = \sin x(3 - 4(1 - \cos^2 x))\).
Simplifying further:
\(\sin x(3 - 4 + 4\cos^2 x) = \sin x(-1 + 4\cos^2 x)\).
Now, we have an expression in terms of \(\cos x\). To find the angle whose sine equals this expression, we need to convert it to an equation using the inverse sine function:
\(\sin^{-1}(3\sin x - 4\sin^3 x) = \sin^{-1}(-1 + 4\cos^2 x)\).
At this point, we cannot simplify the expression any further without specific values for \(x\) or \(\cos x\). It is currently in a form where it represents the angle whose sine is equal to the given expression. If you have specific values for \(x\) or \(\cos x\), you can evaluate the expression using a calculator or mathematical software that provides the inverse sine function.