Draw the graph of y = sin⁻¹(1 - sin(x))

To graph the function \( y = \sin^{-1}(1 - \sin(x)) \), we first need to understand the components involved.

Understanding the Function

The function consists of the inverse sine, or arcsine, which is defined for values between -1 and 1. The expression \( 1 - \sin(x) \) will determine the input for the arcsine function.

Range of \( \sin(x) \)

• The sine function oscillates between -1 and 1.
• Thus, \( 1 - \sin(x) \) will range from 0 to 2.

Valid Input for Arcsine

Since the arcsine function only accepts inputs from -1 to 1, we need to restrict our focus to when \( 1 - \sin(x) \) falls within this range. This occurs when:

• \( \sin(x) \leq 1 \) (always true)
• \( 1 - \sin(x) \leq 1 \) or \( \sin(x) \geq 0 \)

Key Points for the Graph

To plot the graph, consider the following:

• When \( \sin(x) = 0 \), \( y = \sin^{-1}(1) = \frac{\pi}{2} \).
• When \( \sin(x) = 1 \), \( y = \sin^{-1}(0) = 0 \).
• When \( \sin(x) = -1 \), \( y \) is undefined since \( 1 - (-1) = 2 \) is outside the valid range.

Graphing Steps

1. Plot the points where \( \sin(x) \) is non-negative (from 0 to \( \pi \)).

2. For each \( x \) in this range, calculate \( y \) using \( y = \sin^{-1}(1 - \sin(x)) \).

3. Connect the points smoothly, noting that the graph will be continuous and will approach the limits as \( x \) varies.

Final Thoughts

This function will create a wave-like pattern, reflecting the behavior of the sine function but transformed by the arcsine operation. Make sure to check the values at key points to ensure accuracy in your graph.