Problem 106 - Precalculus with Limits

Question

In Exercises 103 - 106, find all solutions of the equation in the interval \( \left[0,2\pi\right) \). Use a graphing utility to graph the equation and verify the solutions. \( \sin^2 3x - \sin^2 x = 0 \)

Step-by-Step Solution

Answer

The solutions of the equation \( \sin^2 3x - \sin^2 x = 0 \) in the interval \( \left[0,2\pi\right) \) are \(0,\pi,2\pi\).

Step 1: Simplify the Equation

The equation can be rewritten as \( \sin^2 3x = \sin^2 x \). Therefore the equation can be reduced to: \( \sin 3x = \sin x \) or \( \sin 3x = - \sin x \)

Step 2: Solve the Equations

We solve these equations separately. For \( \sin 3x = \sin x \), the solutions are \(x = n\pi\). For \( \sin 3x = - \sin x \), we use the identity \( \sin a = -\sin b \Rightarrow a = -(n\pi + b) \) and get the solutions \( x = - n\pi/2 \) where \( n \) is an integer.

Step 3: Find solutions in the given interval

The interval is \( \left[0,2\pi\right) \). So, we find the values of n in the solution which lie in this interval. For \(x = n\pi\), the solutions within the interval are \(0,\pi,2\pi\). For \( x = - n\pi/2 \), there are no solutions within the interval.

Step 4: Graph the equation and verify the solutions

Graph the equation using a graphing utility. The x-intercepts of the graph coincide with the solutions. Verify that the x-intercepts are \(0,\pi,2\pi\).

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions such as the sine function is a useful method to visualize potential solutions. These functions are periodic, showing repetitive behavior over intervals, which helps in identifying all possible solutions. When you graph a trigonometric function, the x-axis typically represents the angle in radians, and the y-axis represents the function value, such as sine or cosine.

For the given equation, plotting each side of the equation separately, like \( \sin^2 3x \) and \(\sin^2 x\), allows you to visually compare them. Where the graphs intersect each other indicates solutions to the equation. These intersection points help confirm solutions obtained algebraically.

• Use graphing software or a graphing calculator for precision.
• Check various intervals as trigonometric functions repeat their values periodically.
• Ensure all potential solutions are covered by adjusting your view on the graphing tool.

Interval Notation

Interval notation is a commonly used method to describe a range of values within a certain domain. It is especially useful in trigonometry for specifying solution ranges within one period of the trigonometric function plot.

In interval notation, brackets indicate whether endpoints are included or excluded:

• \( [a, b] \) is a closed interval, including endpoints \(a\) and \(b\).
• \( (a, b) \) is an open interval, excluding endpoints \(a\) and \(b\).
• The given interval \( \left[0,2\pi\right) \) includes 0 and excludes \(2\pi\).

Understanding interval notation helps specify which part of the graph is relevant for finding solutions. This is crucial because trigonometric functions have infinite solutions outside a specified interval.

Trigonometric Identities

Trigonometric identities are equations that hold true for all permissible values of the variable involved. They are essential in simplifying and solving trigonometric equations. Knowing key identities can make solving trigonometric equations much easier.

For instance, in the exercise, the identity \(\sin^2 3x = \sin^2 x\) was simplified using identities like \(\sin 3x = \sin x\) and \(\sin 3x = -\sin x\).

Some common trigonometric identities include:

• Pythagorean Identities: \(\sin^2 x + \cos^2 x = 1\)
• Angle Sum and Difference: \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)
• Double Angle Formulas: \(\sin 2x = 2 \sin x \cos x\)

By applying these identities, you can transform complex equations into more manageable forms, allowing for easier solving.

Solution Verification

Solution verification is a pivotal step in mathematical problem-solving as it ensures that all obtained solutions are correct and applicable within the specified constraints. After solving an equation, it is imperative to verify the solutions graphically and algebraically.

For the given problem, graphical verification was achieved by using a graphing utility to plot the equation:

• Check that the plotted solutions match the expected x-intercepts on the graph.
• Ensure each solution lies within the specified interval \([0, 2\pi)\).
• If any solution doesn't match the graphical result, re-evaluate the steps to spot errors.
• Algebraically check each solution by substituting back into the original equation to see if it equals zero.

This comprehensive verification process guarantees the completeness and accuracy of your solutions, reinforcing the integrity of your results.