Problem 156 - Precalculus

Question

For what value of \(x\) does \(\sin x=\sin ^{-1} x ?\) Use a graphing calculator to approximate the answer.

Step-by-Step Solution

Answer

The value of \(x\) is 0.

Step 1: Understanding the Problem

The problem requires us to find a value of \(x\) for which the sine of \(x\) is equal to the inverse sine (arcsin) of \(x\). Mathematically, this is written as \( \sin(x) = \sin^{-1}(x) \).

Step 2: Domain Consideration

Recall that the domain of \(\sin(x)\) is all real numbers, while the domain of \(\sin^{-1}(x)\), also known as arcsin, is \(-1 \leq x \leq 1\). Therefore, \(x\) must be in the interval \([-1, 1]\).

Step 3: Understanding the Functions

The function \(\sin(x)\) periodically oscillates between -1 and 1, while \(\sin^{-1}(x)\) is a strictly increasing function from \([-\frac{\pi}{2}, \frac{\pi}{2}]\) that maps inputs from the interval \(-1\) to 1 onto the same interval. We are looking for points where these two graphs intersect.

Step 4: Graphing the Functions

Using a graphing calculator, plot both \(y = \sin(x)\) and \(y = \sin^{-1}(x)\) and look for intersection points within the domain \([-1, 1]\).

Step 5: Identify the Intersection Point

Upon graphing, observe the curves to identify where they intersect. The intersection is found at \(x = 0\).

Step 6: Verification through Calculation

Verify that \(\sin(0) = 0\) and \(\sin^{-1}(0) = 0\). Since both sides of the equation \(\sin(x) = \sin^{-1}(x)\) are equal at \(x = 0\), this solution is correct.

Key Concepts

Understanding the Sine Function

The sine function, often written as \( \sin(x) \), is one of the primary trigonometric functions used in mathematics. It relates to angles of a triangle, especially right triangles, and has a close connection to the unit circle. The function cycles smoothly and periodically between -1 and 1 as it maps angles, measured in radians or degrees, onto this range. This periodic behavior is repetitive, with a complete wave or cycle occurring every \(2\pi\) radians.

Key concepts to remember about the sine function include:

• The amplitude, which is the peak value the function can reach, is always 1.
• The period, which is the length of one complete wave cycle, is \(2\pi\) radians.
• The phase shift, which is any horizontal shift from the regular position.

The sine function’s periodicity makes it important in various applications, like modeling waves, oscillations, and even sound.

Using a Graphing Calculator

Graphing calculators are powerful tools for visualizing mathematical functions. They allow us to plot equations and observe characteristics such as intersections, maxima, minima, and symmetries. Here’s how you can apply a graphing calculator in this context:

• Enter the equations \( y = \sin(x) \) and \( y = \sin^{-1}(x) \) into the calculator. Make sure to use the correct inversions and functions available on the calculator.
• Adjust the viewing window to include the domain \([-1,1]\). This will focus the graph on the relevant section where intersections are possible.
• Analyze the graph to identify where both plots intersect within this domain.

The intersection provides a numerical solution visually, verifying where the functions are equal.

Domain Consideration

When working with functions, especially trigonometric and inverse trigonometric ones, understanding the domain is crucial. The domain defines the allowable inputs for which the function is defined and provides meaning. In our exercise, we’re considering the sine function's domain as all real numbers. However, the inverse sine, \( \sin^{-1}(x) \), has a more limited domain of \([-1, 1]\).

Key facts about domain consideration include:

• For \( \sin(x) \), any real number can be an input due to its periodic nature.
• For \( \sin^{-1}(x) \), only values between -1 and 1 can be input, as these are the only values the original sine function can output.
• This domain constraint limits the possible solutions to our equation \( \sin(x) = \sin^{-1}(x) \) to this range.

This concept helps in narrowing down the interval where function behavior, such as intersections, can be analyzed for solutions.

Understanding Function Intersection

Analyzing where functions intersect is a fundamental part of solving equations graphically. Intersection points of functions are the \((x, y)\) coordinates that satisfy both functions simultaneously. In our exercise, we’re looking for points where \( \sin(x) \) is equal to \( \sin^{-1}(x) \).

To find these intersections, follow these steps:

• Graph both functions, ensuring to account for any differences in their amplitude or range.
• Observe the graphs and locate points where both curves meet; these points are your intersections.
• At these points, verify the equality of the functions by checking their algebraic values.

In this problem, we found that the point \( x = 0 \) is where the functions intersect within the appropriate domain, confirming the solution to the equation. Understanding how graphs visually represent equations can be immensely beneficial in solving such problems.