Problem 47 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-50: Solve the equation for \(x\) in terms of \(y\) if \(x\) is restricted to the given interval. $$ y=-3-\sin x ; \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$

Step-by-Step Solution

Answer

\(x = \arcsin(-3 - y)\) within \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 1: Isolate the trigonometric function

To solve for \(x\), start by isolating the trigonometric function. The given equation is \(y = -3 - \sin x\). Rearrange it to express \(\sin x\): \( \sin x = -3 - y\).

Step 2: Solve for x using arcsin

Use the inverse sine function to solve for \(x\). Since \(\sin x = -3 - y\), we can write \(x = \arcsin(-3 - y)\).

Step 3: Consider the interval restriction

\(x\) is restricted to \([-\frac{\pi}{2}, \frac{\pi}{2}]\). The arcsin function naturally gives values in this interval, so \(x = \arcsin(-3 - y)\) meets the interval requirement.

Key Concepts

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that make the equation true. In this case, we need to solve for \(x\) in terms of \(y\). To start, the given equation is \(y = -3 - \sin x\). Our goal is to express this equation as \(\sin x\) so that we can work with it more easily.

• First, we rearrange the equation as \(\sin x = -3 - y\). This step isolates the trigonometric function, making it easier to manipulate.
• Once \(\sin x\) is isolated, we can focus on finding \(x\) by applying the inverse function.

By isolating the trigonometric function, we prepare the equation for further steps. This forms the basis for solving the problem through inverse trigonometric functions.

Arcsin Function

The arcsin function is an inverse trigonometric function used to find an angle given its sine value. It is crucial when working to revert the trigonometric equation to solve for the desired angle \(x\). When we have \(\sin x = -3 - y\), we use the arcsine function denoted as \(x = \arcsin(-3 - y)\) to solve for \(x\).

• The arcsin function gives the angle whose sine is the specified number.
• This function is useful because it returns an angle within a specific range, which aligns perfectly with our interval requirements.

Beware of the domain: For \(\text{arcsin}(z)\) to make sense, \(z\) must be between \(-1\) and \(1\). Therefore, ensure \(-3 - y\) falls within this range, or the equation won't have a real solution.

Interval Restrictions

Interval restrictions are pivotal in ensuring the solutions to trigonometric equations comply with given constraints. The current problem specifies that \(x\) is constrained to the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\).This interval restriction is critical because:

• The arcsin function only outputs values within \([-\frac{\pi}{2}, \frac{\pi}{2}]\) by definition.
• This property allows us to use \(x = \arcsin(-3 - y)\) directly, knowing it will naturally satisfy the interval condition.

These interval restrictions must always be checked when solving trigonometric equations, ensuring that solutions are valid within the context given by the problem. Always verify that your final answer resides in the permissible range, as specified by the interval constraints.