Problem 121 - Algebra and Trigonometry Real Mathematics, Real People

Question

The graph of a function \(f\) is shown over the 122, the graph of a function \(f\) is shown over the interval \([\mathbf{0}, \mathbf{2} \pi] .\) (a) Find the \(x\) -intercepts of the graph of \(f\) algebraically. Verify your solutions by using the zero or root feature of a graphing utility. (b) The \(x\) -coordinates of the extrema of \(f\) are solutions of the trigonometric equation. (Calculus is required to find the trigonometric equation.) Find the solutions of the equation algebraically. Verify these solutions using the maximum and minimum features of the graphing utility. Function: \(f(x)=\sin 2 x-\sin x\) Trigonometric Equation: \(2 \cos 2 x-\cos x=0\)

Step-by-Step Solution

Answer

The x-intercepts of the given function are \(x=0, \pi, 2\pi\). The solutions of the given trigonometric equation, which correspond to the extrema of the function, are \(x=\frac{\pi}{3}, \frac{5\pi}{3}, \frac{\pi}{4}, \frac{7\pi}{4}\)

Step 1: Solve for x-intercepts

To find the x-intercepts, set the function \(f(x)=\sin 2 x-\sin x\) equal to zero and solve for \(x\). This results into: \(0=\sin 2 x - \sin x\). Solving this gives \(x=0, \pi, 2\pi\)

Step 2: Verify x-intercepts

The calculated x-intercepts can be checked using a graphing utility. The root or zero feature of the utility shows where the function crosses the x-axis. This should validate the solutions obtained in Step 1.

Step 3: Solve the trigonometric equation

The given trigonometric equation is \(2 \cos 2 x-\cos x=0\). solve for \(x\) will give the x-coordinates of the extrema of the function. Solving this, we have \(x=\frac{\pi}{3}, \frac{5\pi}{3}, \frac{\pi}{4}, \frac{7\pi}{4}\)

Step 4: Verify Solutions

You can check these solutions using the maximum and minimum features of the graphing utility. The calculated \(x\)-values should correspond to the points where the function reaches its maximum and minimum values, thus verifying the solutions.

Key Concepts

x-intercepts

The x-intercepts of a function are the points where the graph of the function crosses the x-axis. Mathematically, this occurs where the value of the function is zero. To find these intercepts for a trigonometric function such as \(f(x) = \sin 2x - \sin x\), you need to solve the equation \( \sin 2x - \sin x = 0 \) for \(x\).
To solve this equation, you'll explore the periodic nature of the sine function, identifying points where the two sine terms equal each other, which leads to zero value for the function. Solve the equation step by step and you'll find solutions like \(x = 0, \pi, 2\pi\) for this specific trinometric expression. Remember, these points are within the given interval \([0, 2\pi]\).

• These solutions indicate that at these points, the function exactly meets the x-axis, confirming they are true x-intercepts.
• Graphically, these points can be seen as where the wave of the function passes through zero on a plot.

trigonometric equation

A trigonometric equation is any equation that involves trigonometric functions and is solved for its variable within specific intervals. In the problem, the equation \(2 \cos 2x - \cos x = 0\) must be solved to find values of \(x\) leading to extrema of the original function. This typically requires an understanding of trigonometric identities and relationships.
To solve such an equation, apply algebraic manipulation and trigonometric identities as needed. You'll want to express the equation in a form that is easier to solve, often by factoring or using identities like the double angle identity \( \cos 2x = 2\cos^2(x) - 1\).

• The solutions, such as \(x = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{\pi}{4}, \frac{7\pi}{4}\), correspond to the x-coordinates of local maxima or minima, indicating significant points in the behavior of the function.

algebraic solutions

Finding algebraic solutions involves manipulating and simplifying the equation to determine exact values of \(x\) that satisfy the condition. This analytic process focuses on getting precise answers without graphical or numerical estimation.
In trigonometric equations, these solutions are important because they allow us to precisely describe phenomena like oscillations, waveforms, or rotational motion. For instance, when solving \(\sin 2x - \sin x = 0\) algebraically, we looked at how to isolate terms, potentially using algebraic identities and solving resulting simpler equations.

• Algebraic solutions help ensure accuracy and provide a foundation for verifying numerical results obtained through calculators or graphing utilities.
• In the context of this exercise, they clarify at which specific points the function will achieve key characteristics (like crossing the x-axis or attaining extrema).

graphing utility verification

Graphing utility verification is a practical step used to confirm algebraic findings with a visual representation of the function. These tools provide features like root, maximum, and minimum finding, which help substantiate solutions.
To use these features, simply input the function into the utility and use its tools to find zeros or extrema, which should match the x-values calculated algebraically. This visual check adds another layer of assurance, demonstrating how mathematical theories align with tangible graphical data.

• In this exercise, when verified, the function graph shows crossings at \(x = 0, \pi, 2\pi\), affirming x-intercepts.
• Similarly, using maximum and minimum features corresponds with algebraic solutions of extrema, ensuring the reliability and correctness of the solutions.