Problem 57 - Precalculus with Calculus Previews

Question

Find the first three \(x\) -intercepts of the graph of the given function on the positive \(x\) -axis. $$ f(x)=\sin x+\tan x $$

Step-by-Step Solution

Answer

The first three positive x-intercepts are \( x = \pi, 2\pi, 3\pi \).

Step 1: Understand the Problem

We need to find the points where the graph of the function \( f(x) = \sin x + \tan x \) intersects the x-axis on the positive x-axis. This means we need to solve the equation \( \sin x + \tan x = 0 \) for \( x > 0 \).

Step 2: Set the equation to zero

To find the x-intercepts, set the function equal to zero: \( \sin x + \tan x = 0 \). We will solve this equation for \( x \).

Step 3: Rearrange and Solve the Equation

Rearrange the equation to isolate one term: \( \sin x = -\tan x \). Given \( \tan x = \frac{\sin x}{\cos x} \), substitute \( \tan x \) in the equation, leading to \( \sin x = -\frac{\sin x}{\cos x} \).

Step 4: Simplification

Multiply both sides by \( \cos x \) to clear the fraction: \( \sin x \cos x = -\sin x \). Simplifying, we get \( \sin x (\cos x + 1) = 0 \).

Step 5: Solve for Solutions

This equation implies \( \sin x = 0 \) or \( \cos x = -1 \). For \( \sin x = 0 \), the solutions are \( x = n\pi \), where \( n \) is an integer. For \( \cos x = -1 \), the solutions are \( x = \pi + 2n\pi \).

Step 6: Identify the First Three Positive Solutions

List the smallest positive solutions: \( x = \pi, 2\pi \) from \( \sin x = 0 \), and \( x = 3\pi \) from \( \cos x = -1 \). All of them provide intercepts on the positive x-axis.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine and tangent are fundamental in understanding periodic phenomena. The function \( \sin x \) represents the sine of an angle \( x \), typically measured in radians, and derives from the ratio of the opposite side to the hypotenuse in a right triangle.
The graph of \( \sin x \) is a wave-like pattern, repeating every \( 2\pi \) interval. On the other hand, \( \tan x \), or the tangent function, is the ratio of \( \sin x \) to \( \cos x \), which leads to its unique properties and vertical asymptotes, as it becomes undefined where \( \cos x = 0 \).

• Both functions are periodic, meaning they repeat their values over regular intervals.
• Understanding the behavior of \( \sin x \) and \( \tan x \) is crucial for solving equations and graphing.
• As trig functions are built on circles (the unit circle), interpreting them involves angles and their respective radian measures.

Knowing how these functions behave aids in solving equations where they are involved. This means you can predict points where they intersect other functions or the axes, just like in finding where the graph touches the x-axis in our task.

Solving Equations

Solving equations involving trigonometric functions often entails manipulation to isolate terms and simplify expressions. For the equation \( \sin x + \tan x = 0 \), you need methods like substitution and factoring.
Start by rewriting \( \tan x \) using its identity: \( \tan x = \frac{\sin x}{\cos x} \). This substitution can simplify complex equations.
Once simplified, factor the resulting equations, like \( \sin x (\cos x + 1) = 0 \).

• Set each factor to zero individually to find the solutions: \( \sin x = 0 \) or \( \cos x = -1 \).
• These lead to simple forms: \( x = n\pi \) and \( x = \pi + 2n\pi \), where \( n \) is an integer.
• These equations reflect the periodic nature of sine and cosine functions, generating infinite solutions.

By determining the smallest positive \( x \)-intercepts, like \( x = \pi, 2\pi, 3\pi \), you solve for specific solutions suitable for scenarios on the positive x-axis.

Graphing Functions

Graphing a function like \( f(x) = \sin x + \tan x \) involves understanding both the shape and behavior of the individual parts. Each trigonometric function contributes differently to the graph.
The \( \sin x \) part provides a smooth, oscillating curve while \( \tan x \) introduces sharp rises and falls due to its asymptotic behavior.
When graphing, take these steps:

• Plot key points and asymptotes as guides.
• Understand the periodic intervals where these functions repeat: \( 2\pi \) for sine and \( \pi \) for tangent.
• Observe the points where the graph intersects the x-axis, known as x-intercepts, as these are where the function equals zero.

In this particular function, determining exact x-intercepts requires solving intersections by setting the function equal to zero. These intercepts \( x = \pi, 2\pi, 3\pi \) show where the graph crosses the x-axis. Graphing helps visualize these intersections, offering a clear picture of solutions in real contexts beyond pure computations.