What is y equal to?
Consider the following for the two (02) items that follow : Let y=\sin^{-1}(x-\frac{4x^{3}}{27}).
- A. \sin^{-1}x
- B. \sin^{-1}(\frac{x}{3})
- C. 3 \sin^{-1}x
- D. 3 \sin^{-1}(\frac{x}{3}) ✓
Correct Answer: D. 3 \sin^{-1}(\frac{x}{3})
Explanation
Let x = 3\sin\theta, then \theta = \sin^{-1}(x/3). Substituting this into the expression gives x - \frac{4x^3}{27} = 3\sin\theta - \frac{4(27\sin^3\theta)}{27} = 3\sin\theta - 4\sin^3\theta = \sin(3\theta). Therefore, y = \sin^{-1}(\sin 3\theta) = 3\theta = 3\sin^{-1}(x/3).