Problem 60 - Algebra and Trigonometry

Question

\(57-62\) . Use a graphing device to find the solutions of the equation, correct to two decimal places. $$ \sin x=x^{3} $$

Step-by-Step Solution

Answer

The solutions are approximately \( x \approx 0.87 \) and \( x \approx -0.87 \).

Step 1: Understanding the Problem

We need to find the solutions for the equation \( \sin x = x^3 \). We should be aware that this means we want to find the values of \( x \) where the graph of \( \sin x \) intersects the graph of \( x^3 \). A graphing device can assist in visualizing and identifying these points.

Step 2: Graphing the Functions

Plot the function \( f(x) = \sin x \) and the function \( g(x) = x^3 \) on the same set of axes. \( \sin x \) will oscillate between -1 and 1, while \( x^3 \) will be a cubic curve that starts from the origin and moves through both the positive and negative directions.

Step 3: Finding the Intersection Points

Using the graph, identify where \( f(x) = \sin x \) and \( g(x) = x^3 \) intersect. Look for points in different intervals because the sine function will have periodic intersections with \( x^3 \). These intersection points represent the solutions to the equation \( \sin x = x^3 \).

Step 4: Estimating Solutions

Zoom into the intersection points on the graph to estimate the \( x \)-values. Use the graphing tool's ability to pinpoint these intersections as closely as possible. Adjust the settings on your graphing device to increase accuracy.

Step 5: Correct to Two Decimal Places

After finding the initial estimates, refine them to determine the \( x \)-values correct to two decimal places. Ensure the precision of your graphing tool is set to provide two decimal place accuracy.

Step 6: Solutions

Upon examining the graph closely and submitting the estimated values through the graphing tool's solver, we find the solutions to the equation \( \sin x = x^3 \) to be approximately \( x \approx 0.87 \) and \( x \approx -0.87 \), correct to two decimal places.

Key Concepts

Intersection Points

Understanding intersection points is key to solving the equation \( \sin x = x^3 \). An intersection point is where two graphs meet on a coordinate plane. For our problem, it's where the sine function and the cubic function cross each other. These points are significant because they represent the solutions to our equation. To find these intersections precisely:

• Graph both functions using a graphing tool.
• Look for where the plots touch or cross each other.
• Make sure to examine multiple intervals, as functions like sine are periodic and might intersect the cubic curve more than once.

Identifying intersection points visually on the graph is an essential step to uncovering the \( x \)-values that satisfy \( \sin x = x^3 \). By focusing on these points, we can effectively solve our equation.

Sine Function

The sine function, written as \( f(x) = \sin x \), is a periodic trigonometric function.Key properties include:

• It oscillates between -1 and 1.
• The period is \( 2\pi \), meaning it repeats every \( 2\pi \) units.

These characteristics make it a wave-like curve that continuously ripples across the graph.When graphing \( \sin x \), observe how the wave interacts with other functions. In our equation \( \sin x = x^3 \), we need to find where this wave-like pattern intersects with the cubic function. Remember, these intersections will occur at the same \( x \) values where both function outputs are equal. Understanding the behavior of the sine function helps us predict where possible intersection points may lie.

Cubic Function

The cubic function for our problem is \( g(x) = x^3 \). It is a polynomial function characterized by its distinct curviness.Features of a cubic function:

• Its curve passes through the origin (0,0).
• It extends towards infinity as \( x \) increases or decreases.
• The curve is symmetric about the origin, so it reflects over both axes.

When graphed, \( x^3 \) will rise steeply in the positive direction and descend steeply in the negative direction, creating an S-shaped curve.In our equation, solving \( \sin x = x^3 \) means finding common values on this S-curve and the sine curve. Visualization aids in understanding where their paths cross, leading us to the solutions of the intersection points.

Graphing Tool

A graphing tool is an essential component for solving the equation \( \sin x = x^3 \). It allows us to visualize complex functions like the sine and cubic functions.Advantages of a graphing tool:

• Plots functions across defined intervals, providing a clear visual representation.
• Offers zooming capabilities, which help pinpoint intersection points with accuracy.
• Can often solve equations or locate specific points like intersections, aiding in precise solution finding.

To use a graphing tool effectively in this context:

• Input both \( \sin x \) and \( x^3 \) to view their graphs.
• Use the zoom feature to focus on specific areas where intersections occur.
• Utilize built-in tools or settings to hone in on intersections to two decimal places for accurate results.

Enhancing familiarity with your graphing tool can significantly improve your efficiency in solving similar equations and identifying crucial solution points.