Problem 101 - Algebra and Trigonometry

Question

How can the graph of \(y=\sin ^{-1} x\) be obtained from the graph of the restricted sine function?

Step-by-Step Solution

Answer

The graph of \(y=\sin^{-1}x\) can be obtained from the graph of the restricted sine function by reflecting the graph of the sine function about the line \(y=x\). This reflects the property that the two functions are inverses of each other.

Step 1: Understand the Graph of the Sine Function

The graph of the sine function, between -90 degrees and +90 degrees or, in radians, between \(-\pi/2\) and \(+\pi/2\), varies between -1 and 1. The graph is symmetric with respect to the y-axis, meaning that the left part of the graph is a mirror image of the right part.

Step 2: Visualize the Reflection About the Line \(y=x\)

Since the graph of the inverse function is the reflection of the graph of the main function over the line \(y = x\), then the primary task is to perform this reflection. In this graph, vertical lines will become horizontal and vice versa. This will change the orientation of the graph, demonstrating the influence of the inverse operation.

Step 3: Graph of the Inverse Sine Function

The final result, after performing the reflection, corresponds to the graph of \(y = \sin^{-1} x\), also known as the arcsine function. This time, the graph is symmetric with respect to the line \(y=x\), and you'll see that the domain and range roles are swapped. This function will vary between \(-\pi/2\) and \(+\pi/2\) for \(x\) in the interval from -1 to 1.