Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function and approximate the maximum and minimum points (to four decimal places) of the graph in the interval \([0,2 \pi],\) and (b) solve the trigonometric equation and verify that the \(x\) -coordinates of the maximum and minimum points of \(f\) are among its solutions. (The trigonometric equation is found using calculus.) Trigonometric Equation \(2 \cos 2 x=0\) \(-2 \sin 2 x=0\) \(2 \sin x \cos x-\sin x=0\) \(-2 \sin x \cos x-\cos x=0\) \(\cos x-\sin x=0\) \(2 \cos x-4 \sin x \cos x=0\) Function \(f(x)=\cos 2 x\)

Step-by-Step Solution

Answer

Graphing the function \(f(x) = \cos 2x\) and approximating the maxima and minima would likely give values close to points of intersection on x-axis for the function. The accurate solution would depend on the accurate solution of given equations and corresponding maxima and minima. Though remember, the approximated values from the graph might exhibit a slight difference due to the round-off or truncation errors in approximation.

Step 1: Graph the Function

Using a graphing utility, graph \(f(x) = \cos 2x\) within the interval of \([0,2\pi]\). Inspect the graph to identify maximum and minimum points.

Step 2: Approximate the Extreme Points

On the graph of \(f(x) = \cos 2x\), look for the peak and valley points, which correspond to the maximum and minimum points of the function respectively. Approximate these points to four decimal places.

Step 3: Solve the Trigonometric Equations

The exercise provided several trigonometric equations. One by one, solve each equation using trigonometric identities and rules of algebra.

Step 4: Verify Solutions Against the Extreme Points

Upon getting solutions from step 3, verify them with the approximated extreme points from step 2. The x-coordinates of the maximum and minimum points of the function should match the solutions to the equations.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions like \( f(x) = \cos 2x \) is an essential skill in understanding the behavior of these functions. The function \( \cos 2x \) is a transformation of the basic cosine curve, where the frequency is doubled. This means that within the interval from \([0, 2\pi] \), the function completes two full cycles instead of one, due to the factor of 2.

• Amplitudes of the trigonometric function remain constant unless specified otherwise. For \( \cos 2x \), this is 1.

To graph \( \cos 2x \), use graphing utilities such as graphing calculators or online software. These tools help to visualize how the graph oscillates, moving between its maximum and minimum points.

Finding Maximum and Minimum Points

To find the maximum and minimum points of \( f(x) = \cos 2x \), once graphed, look for the highest peaks and lowest troughs. These points are called extreme points.

• The maximum value of \( \cos 2x \) is 1, occurring whenever \( \cos 2x \) is at its peak.
• The minimum value is -1, found at the deepest parts of the wave.

Approximate the x-coordinate of these points to four decimal places, especially when using a graphing utility that supports precise measurements. The cycle of the cosine function repeats every \( \pi \) radians here due to the doubling frequency, showcasing more frequent peaks and valleys within the standard interval \([0, 2\pi]\).

Trigonometric Equations

Solving trigonometric equations can sometimes be challenging due to their periodic nature and the need for using identities. For example, with \(2 \cos 2x = 0\), simplifying leads to \( \cos 2x = 0\). Set \(2x = \frac{\pi}{2} + k\pi \), where \(k\) is an integer, for the solutions.

• Equations like \( \cos x - \sin x = 0 \) imply \( \cos x = \sin x \), useful if transformed with identities \( \tan x = 1 \), thus \( x = \frac{\pi}{4} + k\pi \).

Explore trigonometric relationships to perform these transformations, using identities like \( \sin^2 x + \cos^2 x = 1 \) to aid in manipulation and solving of equations.

Using Graphing Utility

A graphing utility is a powerful tool that can visually explore functions such as \(\cos 2x\). By inputting the function, observe its overlaid graph on the regular coordinate plane, providing insight into the characteristic peaks and troughs.

• Zoom and pan features help view specific intervals or areas of interest.
• Tools for setting grid lines or intervals allow for more precise inspection of the graph's behavior.

Students can utilize these utilities to quickly verify solutions from analytical methods by checking against the visual output. It also allows for comparison with calculated x-coordinates of maximum and minimum points, aiding in cross-verification of solution accuracy.