Problem 99 - Contemporary Precalculus

Question

Solve the equation graphically. $$\sin ^{3} x+2 \sin ^{2} x-3 \cos x+2=0$$

Step-by-Step Solution

Answer

Answer: The approximate solutions are the x-coordinates of the intersection points of the graph of the function \(f(x) = \sin ^{3} x+2 \sin ^{2} x-3 \cos x+2\) with the x-axis over the range \(0 \le x \le 2\pi\). They can be found by sketching the graph of the function using a table of values or graphing software and noting the points where the graph intersects the x-axis. Make sure to verify the solutions by substituting them back into the original equation.

Step 1: Rewrite the given equation as a single function

To find the intersection points of the equation, we rewrite the equation as a single function of x in terms of sine and cosine functions. $$f(x) = \sin ^{3} x+2 \sin ^{2} x-3 \cos x+2$$

Step 2: Find the range of x

As f(x) depends on the sine and cosine functions, the graph would be periodic in nature. The periodicity of sine and cosine functions is \(2\pi\). So, we will consider the range of x as \(0 \le x \le 2\pi\).

Step 3: Create a table of values for f(x)

Calculate the values of f(x) for different values of x in the considered range (0, \(\frac \pi 6\) , \(\frac \pi 4\), \(\frac \pi 3\), \(\frac \pi 2\), \(\frac{2\pi}{3}\), \(\frac{3\pi}{4}\), \(\frac{5\pi}{6}\), \(\pi\), \(\frac{7\pi}{6}\), \(\frac{5\pi}{4}\), \(\frac{4\pi}{3}\), \(\frac{3\pi}{2}\), \(\frac{5\pi}{3}\), \(\frac{7\pi}{4}\), \(\frac{11\pi}{6}\), \(2\pi\)).

Step 4: Plot the graph of f(x)

Based on the table of values, plot the graph of f(x) against x over the chosen range. Make sure to label the axes and indicate the scale for easy interpretation.

Step 5: Identify the intersection points

See where the graph of f(x) intersects the x-axis. These are the points where f(x) equals 0. Note the x-coordinates of these intersection points.

Step 6: Verify the solutions

Substitute the x-coordinates of the intersection points back into the original equation to verify if they satisfy the equation. If they do, then these are the solutions (roots) to the equation. Note: Graphs made in the classroom or using software can give approximate solutions. To find the exact solutions, use numerical methods such as the bisection method or Newton's method to refine the approximation.

Key Concepts

Graphical Method in Trigonometry

The graphical method in trigonometry involves visually representing trigonometric equations on a set of axes to find their solutions. This process transforms abstract mathematical expressions into a more intuitive visual form, which can be particularly helpful when dealing with complex trigonometric functions like
in this exercise. To solve the given equation graphically, one would plot the function . To achieve this, youthematic details as needed. The graphical approach provides an initial approximation of the roots where the graph intersects the x-axis.

Periodicity of Trigonometric Functions

Trigonometric functions, including sine and cosine, exhibit periodicity, which means they repeat their values in regular intervals. The fundamental period of both sine and cosine functions is . This intrinsic property plays a vital role when solving trigonometric equations graphically. In the context of the given exercise, understanding that a single curve within the interval of can help predict the function's behavior beyond this interval. Since the graph repeats after it implies the solutions to the given function will repeat with the same periodicity. Recognizing this allows students to focus on finding roots within one period, which simplifies the graphing process significantly.

Trigonometric Function Intersections

Intersections of trigonometric functions with the x-axis represent the angles for which the function values are zero, which are the roots or solutions to the equation. The graph of generated in Step 4 of the solution process tells us precisely where these intersections occur. Each point of intersection means that the trigonometric equation equals zero at that x-value. To solve trigonometric equations graphically, it’s essential to accurately plot the function and to clearly identify these points of intersection. Students should be reminded to verify their graphical solutions algebraically due to the approximate nature of graphical representations. This verification process, as outlined in Step 6, ensures the graphical solutions are indeed valid for the original equation, thus bridging the gap between visual understanding and analytical calculation.