Consider the following for the two (02) items that follow: Let y =...

Given $y = \sin^{-1}\left(x - \dfrac{4x^3}{27}\right)$

Substitute $x = 3\sin\theta$, which gives $\sin\theta = \dfrac{x}{3} or$\theta = \sin^{-1}\left(\dfrac{x}{3}\right)$<br><br>The expression becomes:<br><br>$y = \sin^{-1}\left(3\sin\theta - \dfrac{4(3\sin\theta)^3}{27}\right)$<br><br>$y = \sin^{-1}\left(3\sin\theta - \dfrac{4(27\sin^3\theta)}{27}\right)$<br><br>$y = \sin^{-1}(3\sin\theta - 4\sin^3\theta)$<br><br>Using the trigonometric identity$\sin(3\theta) = 3\sin\theta - 4\sin^3\theta$:<br><br>$y = \sin^{-1}(\sin(3\theta))$<br><br>$y = 3\theta$<br><br>Substituting back the value of$\theta$:<br><br>$y = 3\sin^{-1}\left(\dfrac{x}{3}\right)$<br><br>Answer:$3\sin^{-1}\left(\dfrac{x}{3}\right)