Problem 46 - Algebra 2

Question

How many solutions does the equation \(x=\tan x\) have for \(0 \leq x<2 \pi ?\) Explain.

Step-by-Step Solution

Answer

The equation \(x = \tan(x)\) has three solutions in the interval \(0 \leq x < 2\pi\).

Step 1: Understand the Function \(\tan(x)\)

In the interval \(0 \leq x < 2\pi\), \(\tan(x)\) function repeats itself once. Hence, the graph of \(\tan(x)\) contains asymptotes (vertical lines where the function is undefined) at \(\pi/2\) and \(3\pi/2\).

Step 2: Understand the Function \(x\)

In the interval \(0 \leq x < 2\pi\), the function \(x\) is just a straight line passing through the origin (0,0).

Step 3: Graph Both Functions

By graphing both functions, we can visually observe the points where the functions intersect. There are three such intersection points within the interval.

Step 4: Number of Solutions

From the graph, it can be seen that the equation \(x = \tan(x)\) has three solutions in the interval \(0 \leq x < 2\pi\).

Key Concepts

Tangent Function

The tangent function, denoted by \( \tan(x) \), is a fundamental trigonometric function that arises when working with angles and triangles. It is defined as the ratio of the sine to the cosine function: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Unlike sine and cosine, the tangent function is not defined for all values of \( x \).
The function is undefined wherever the cosine of \( x \) equals zero, leading to vertical asymptotes at these points. In the interval \([0, 2\pi)\), these occur at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). The tangent function has a periodic nature with a period of \( \pi \), meaning it repeats every \( \pi \) radians.
Key characteristics of the tangent function include:

• Vertical asymptotes at \( x = (n+\frac{1}{2})\pi \), where \( n \) is an integer.
• Zero crossings (or roots) at multiples of \( \pi \) where \( \tan(x) = 0 \).
• Range of \((-\infty, \infty)\) unlike sine and cosine.
• Curve steepness and reversing direction due to its undefined points and zero crossings.

Understanding these aspects of the tangent function is crucial when solving equations involving it, as they affect where and how frequently solutions occur.

Graphical Solutions

Graphical solutions provide a visual method for solving equations by depicting where two functions, such as \( x \) and \( \tan(x) \), intersect. By plotting these functions on the same set of axes, we can find the solutions where they meet.
For the equation \( x = \tan(x) \), you would plot the linear function \( y = x \) alongside the trigonometric function \( y = \tan(x) \) over the interval \([0, 2\pi)\).
Here are the key steps to find graphical solutions:

• Plot the functions: Draw the line \( y = x \), which is a straight diagonal line through the origin, and the function \( y = \tan(x) \), which exhibits periodic behavior and asymptotes.
• Identify intersection points: Look for points where the two graphs cross each other. These are the x-values that make the equation true.

In this scenario, the graphical method reveals three intersection points in the specified interval, indicating three solutions to the equation \( x = \tan(x) \).

Intersection Points

Intersection points of graphs represent the solutions of equations set in the form \( f(x) = g(x) \). In our exercise, finding intersections of \( y = x \) and \( y = \tan(x) \) gives the solutions for \( x = \tan(x) \). These points are where the y-values of both functions are equal, meaning the equation holds true.
To determine these points, you sketch or compute the graphs of the functions in question. Observing them over \([0, 2\pi)\), you identify where they meet. Each intersection corresponds to a solution of the equation.
Key aspects to note about these intersection points:

• They occur at x-values where the graphs of both functions pass through each other.
• In the given range, you will find three of these points, consistent with the periodic and asymptotic behavior of the function \( \tan(x) \).
• Knowing where asymptotes lie helps in predicting possible intersections, avoiding undefined values.

These points provide a practical way to solve trigonometric equations, especially when algebraic manipulation is complex.