Solve each equation for x, where x is restricted to the given interval.
y = sin x ―2 , for x in [―π/2. π/2]

Verified step by step guidance

1. Step 1: Start with the given equation: \( y = \sin x - 2 \).
2. Step 2: Rearrange the equation to solve for \( \sin x \): \( \sin x = y + 2 \).
3. Step 3: Determine the range of \( \sin x \) which is \([-1, 1]\). Therefore, \( y + 2 \) must also be within this range.
4. Step 4: Solve the inequality \(-1 \leq y + 2 \leq 1\) to find the possible values of \( y \).
5. Step 5: Use the inverse sine function to solve for \( x \) within the interval \([-\pi/2, \pi/2]\) for the valid values of \( y \).

Key Concepts

Sine Function

The sine function, denoted as sin(x), is a fundamental trigonometric function that relates the angle x (measured in radians) to the ratio of the length of the opposite side to the hypotenuse in a right triangle. It oscillates between -1 and 1, making it crucial for understanding periodic behavior in trigonometric equations.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable (in this case, x) that satisfy the equation. This often requires using inverse trigonometric functions, identities, and understanding the periodic nature of trigonometric functions to find all possible solutions within a specified interval.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [−π/2, π/2] indicates that x can take any value from −π/2 to π/2, inclusive. Understanding this notation is essential for determining the valid solutions to the equation within the specified bounds.