Problem 96 - 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 \(f(x)=2 \sin x+\cos 2 x\)

Step-by-Step Solution

Answer

The maximum and minimum points of \(f(x)=2 \sin x+\cos 2 x\) can be approximated using a graphing utility. By then solving the given trigonometric equations across the interval [0,2π], it can be verified that these x-coordinates correspond to solutions of the equations.

Step 1: Plot the function

Use a graphing utility to plot the function \(f(x)=2 \sin x+\cos 2 x\) in the interval [0,2π].

Step 2: Approximate Maximum and Minimum

By observing the graph, approximate the maximum and minimum points (to four decimal places) within the interval [0,2π]. Both the x-coordinates and y-coordinates of these points will be needed for further verification.

Step 3: Solve Trigonometric Equations

Next, solve each of the following trigonometric equations: \(2 \cos 2 x=0\), \(-2 \sin 2 x=0\), \(2 \sin x \cos x-\sin x=0\), \(-2 \sin x \cos x-\cos x=0\), \(\cos x-\sin x=0\), and \(2 \cos x-4 \sin x \cos x=0\). Remember that the interval is between [0,2π].

Step 4: Verify Solutions

Verify that the x-coordinates of the maximum and minimum points of the graph of \(f(x)\) are among the solutions obtained from solving the above equations. If a solution does not match an x-coordinate, then check the calculation again or check if the solution does not fall within the [0,2π] interval.

Key Concepts

Solving Trigonometric Equations

Trigonometric equations are mathematical statements that express a relationship involving the trigonometric functions—sine (sin), cosine (cos), tangent (tan), etc. Solving these equations means finding all angles that make the equation true. Solving a trigonometric equation often involves the following steps:

• Isolating the trigonometric function, if necessary.
• Using algebraic skills to simplify the equation.
• Consulting trigonometric identities to further simplify the equation.
• Applying the unit circle or inverse trigonometric functions to find specific angle solutions.
• Considering the periodic nature of trigonometric functions to find all possible solutions within a given interval.

When working through the solutions, utmost care should be taken to remain within the given interval, which commonly is the period of the function. For instance, in the given problem, all solutions must be in the interval \[0,2\pi\].

Maximum and Minimum Points of Trigonometric Functions

Trigonometric functions oscillate and create waves, which means they naturally have maximum (highest point) and minimum (lowest point) values. These points can be understood graphically as the peaks and troughs of the wave.

Mathematically, these points occur where the derivative of the function equals zero. However, if you are graphing the function without calculus, you can approximate them by:

• Plotting the function across its period
• Observing where the function reaches its highest and lowest points within the interval
• Using graphing utilities to better estimate these points, as they provide visual cues and exact coordinates when possible

In the context of the given exercise, recognizing these points is crucial since their x-coordinates should correspond with solutions to the trigonometric equations derived from calculus.

Using Graphing Utilities in Trigonometry

Graphing utilities, such as graphing calculators or computer software, can be invaluable tools in trigonometry. They help with visualizing the behavior of trigonometric functions and with identifying key features such as intercepts, maximum and minimum points, and points of inflection.

Software can also assist in solving trigonometric equations by providing numerical solutions when algebraic solutions might be cumbersome. To effectively use graphing utilities, you should be able to:

• Input the function correctly.
• Set an appropriate viewing window to display significant features of the function.
• Use zoom and trace features to find and verify the approximate values of maximum, minimum, and intercepts.

These utilities complement analytical approaches, offering both a visual understanding and a means to check the accuracy of your solutions.

Trigonometry in Calculus

Trigonometry plays a crucial role in calculus, particularly when dealing with periodic functions or wave motion. Integrals and derivatives of trigonometric functions are common in many areas of calculus. Derivatives help in finding the slope of the tangent to a curve at any point, which leads to locating the maximum and minimum points of trigonometric functions.

When faced with a trigonometric function in calculus, the following steps may apply:

• Finding the derivative of the function to determine when the slopes of tangent lines are zero, which signifies potential maxima or minima.
• Checking the second derivative, if needed, to confirm whether these are maxima, minima, or points of inflection.
• Solving related trigonometric equations that arise from setting the first derivative to zero, which gives critical points.

For example, in our problem, calculus is used to determine the trigonometric equations whose solutions should coincide with the x-coordinates of the maximum and minimum points.