Problem 96 - Contemporary Precalculus

Question

Solve the equation graphically. $$\tan x=3 \cos x$$

Step-by-Step Solution

Answer

Answer: To find the solutions to the equation \(\tan x = 3\cos x\) graphically, follow these steps: 1. Rewrite the equation in terms of a single trigonometric function, such as \(\sin x = 3\cos^2 x\). 2. Sketch the graphs of \(\sin x\) and \(3\cos^2 x\), taking note of their periodic behavior and maximum and minimum values. 3. Identify the intersection points between the two graphs, which represent the solutions to the given equation. 4. Confirm the solutions algebraically by substituting the x-values into the original equation and checking if both sides are equal.

Step 1: Rewrite the equation in terms of a single trigonometric function

Using the relationship between sine, cosine, and tangent, we can rewrite the given equation as follows: $$\tan x=3 \cos x \iff \frac{\sin x}{\cos x}=3\cos x$$ Now, multiply both sides by \(\cos x\) to eliminate the fraction: $$\sin x = 3\cos^2 x$$

Step 2: Sketch the graphs of \(\sin x\) and \(3\cos^2 x\)

First, sketch the graph of \(\sin x\). It's a periodic function with a period of \(2\pi\) and has a maximum value of 1 and a minimum value of -1. Next, sketch the graph of \(3\cos^2 x\). Since \(\cos^2 x\) is always positive or zero, \(3\cos^2 x\) will also always be positive or zero. The maximum value of \(\cos^2 x\) is 1, so the maximum value of \(3\cos^2 x\) will be 3. The graph will also be periodic with a period of \(\pi\).

Step 3: Identify the intersection points

The intersection points of the two graphs represent the points where \(\sin x = 3\cos^2 x\). Observe the graphs and find where they intersect. Remember that the sine function has a period of \(2\pi\) and the \(3\cos^2 x\) function has a period of \(\pi\). The intersection points will occur periodically and be solutions to the given equation \(\tan x=3 \cos x\).

Step 4: Confirm the solutions algebraically

For the intersection points identified in step 3, check if they truly satisfy the given equation. Substitute the x-values into the equation and see if both sides are equal. If they are, then the solution is valid. In summary, solving the equation graphically involves rewriting the equation in terms of a single trigonometric function, sketching the graphs, identifying the intersection points, and confirming the solutions algebraically.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial in simplifying and solving trigonometric equations. These identities are equations that are true for all values of the variables involved. One basic set includes the Pythagorean identities, such as \( \sin^2x + \cos^2x = 1 \) and \( 1 + \tan^2x = \sec^2x \), which come from the Pythagorean Theorem in a right triangle.

In the given exercise, the tangent function is involved, which can be expressed using sine and cosine as \( \tan x = \frac{\sin x}{\cos x} \). This expression shows the interrelated nature of the trigonometric functions and illustrates how complex equations can be broken down using identities. By leveraging these relationships, as seen in the exercise, we can rewrite the original equation \( \tan x=3 \cos x \) in terms of sine and cosine, making it easier to graph and solve.

Graphing Periodic Functions

Graphing is a powerful tool for understanding and solving periodic functions such as the trigonometric functions sine and cosine. These functions have a repeating pattern over a certain interval known as the period. For the \( \sin x \) function, the period is \( 2\pi \) and for the \( \cos x \) function, it’s the same. However, when \( \cos x \) is squared as in \( 3\cos^2 x \), its graph still repeats the same values but over half the original period of cosine, thus \( \pi \).

When sketching these graphs, it’s important to note key characteristics such as amplitude (the height of the peaks and depth of the troughs from the centerline), phase shift (horizontal shift from the usual starting point), and the period. Understanding how to graph these functions accurately is essential in visually identifying solutions to trigonometric equations, as the points of intersection often represent the solutions.

Trigonometric Functions Intersection

The intersection of trigonometric functions on a graph provides a visual representation of the solutions to the equation. With the graph of \( \sin x \) and \( 3\cos^2 x \) plotted on the same axes, their points of intersection are the x-values where \( \sin x = 3\cos^2 x \), satisfying the given equation. Due to the periodic nature of these functions, there may be multiple points of intersection, indicating several solutions within the span of a common period.

These intersections are critical; not only do they give a visual confirmation of solutions, but they also provide insight into the behavior of the functions involved. When graphs intersect at these points, students can list the x-values, often within one period of the function, to find potential solutions. It is important to then verify these solutions algebraically to ensure they satisfy the original equation in the entire domain of the trigonometric function.