Problem 37 - Algebra and Trigonometry

Question

\(35-38=(a)\) Graph \(f\) and \(g\) in the given viewing rectangle and find the intersection points graphically, rounded to two decimal places. (b) Find the intersection points of \(f\) and \(g\) algebraically. Give exact answers. $$ f(x)=\tan x, g(x)=\sqrt{3} ; \quad-\frac{\pi}{2}, \frac{\pi}{2} | \text { by }[-10,10] $$

Step-by-Step Solution

Answer

Graphically, the intersection at approximately 1.05. Algebraically, exact intersection point is \(x = \frac{\pi}{3}\).

Step 1: Understand the Problem

We are given two functions: \( f(x) = \tan x \) and \( g(x) = \sqrt{3} \). We need to find their intersection points within the specified viewing rectangle \((-\frac{\pi}{2}, \frac{\pi}{2})\) and domain \([-10,10]\). The intersection points represent the \(x\) values where the two functions equal each other.

Step 2: Visualize the Problem

Graphically plot \( f(x) = \tan x \) and \( g(x) = \sqrt{3} \) in the viewing rectangle using a graphing tool or calculator. \( \tan x \) is a periodic function with vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \), and \( g(x) = \sqrt{3} \) is a horizontal line. Find where the graph of \( f(x) \) intersects with the line \( g(x) \).

Step 3: Identify Intersection Points Graphically

Upon graphing, we can see that \( f(x) = \tan x \) intersects \( g(x) = \sqrt{3} \) at several points within the interval. Graphically determine the approximation of these points and round them to two decimal places.

Step 4: Use the Tangent Function Identity

We know that \( \tan \left(\frac{\pi}{3}\right) = \sqrt{3} \). Since \( \tan x \) is periodic with a period of \( \pi \), the primary angle for \( \tan x = \sqrt{3} \) is \( x = \frac{\pi}{3} \), and since \( \tan x \) is symmetric, we also consider \( x = -\frac{\pi}{3} \).

Step 5: Algebraically Solve for Intersection Points

Set \( \tan x = \sqrt{3} \) and solve for \( x \). \( \tan x = \sqrt{3} \) implies \( x = \frac{\pi}{3} + n\pi \) for integer \( n \). Consider the range \( -\frac{\pi}{2} < x < \frac{\pi}{2}\). This gives the solutions \( x = \frac{\pi}{3} \). The next consecutive point in the periodic cycle is not within this range.

Key Concepts

Intersection Points

Intersection points are places on a graph where two functions cross each other. In this exercise, the intersection occurs between the function \( f(x) = \tan x \) and the constant function \( g(x) = \sqrt{3} \). To determine these points, you need to find the \( x \)-values at which \( f(x) \) and \( g(x) \) are equal.

Think of the intersection as where the path of the tangent graph meets the steady line of \( \sqrt{3} \). Visualizing such intersections is crucial because it helps in understanding the values of \( x \) that satisfy both equations.

The intersection points calculated both graphically and algebraically provide a confirmation that your results are accurate. When finding these points, rounding them to two decimal places can help in making the data more manageable and visually clear.

Always remember, finding intersection points graphically is a great way to verify your algebraic solutions.

Tangent Function

The tangent function, \( \tan x \), is one of the fundamental trigonometric functions. It is periodic and has a period of \( \pi \), meaning it repeats its values every \( \pi \) units along the x-axis.

• It is an odd function, so it is symmetrical around the origin.
• \( \tan x \) has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \), meaning it goes to infinity at these points.

The behavior of the tangent function is crucial to understand because the points where it becomes undefined or 'blows up' are often critical in solving trigonometric equations like finding intersections.

In our problem, we determine that \( \tan x = \sqrt{3} \) is solvable by identifying known tangent values, specifically noting that \( \tan(\frac{\pi}{3}) = \sqrt{3} \).
Using periodic properties, we find all potential intersection points within the valid range.

Solving Trigonometric Equations

Solving trigonometric equations involves finding values of \( x \) that satisfy a given trigonometric equation. In this problem, we solve \( \tan x = \sqrt{3} \).

To solve:

• Identify known angle solutions: Recognize that one known value where \( \tan x = \sqrt{3} \) is \( \frac{\pi}{3} \).
• Use the periodicity of the tangent function: Since \( \tan(x) \) repeats every \( \pi \), any solution can be expressed as \( x = \frac{\pi}{3} + n\pi \), where \( n \) is an integer.
• Apply constraints: Consider the interval \( -\frac{\pi}{2} < x < \frac{\pi}{2} \) in this exercise to filter the solutions.

Typically, trigonometric equations might have multiple solutions within a specific interval due to the periodic nature of trigonometric functions. Understanding which solutions fall within the desired range is critical to solving these equations correctly.

Graphical Solutions

Graphical solutions provide a visual insight into mathematical problems. By graphing the functions \( f(x) = \tan x \) and \( g(x) = \sqrt{3} \), you can visually identify intersection points.

Here's how you can approach such a task:

• Use graphing tools: Plot both functions within the specified domain and range.
• Identify where they meet exactly: The intersection appears as points where the curve and line cross.
• Approximate these points visually to understand where the solutions lie.

Graphical solutions are particularly powerful because they offer an immediate understanding. They show transformations, interactions, and behaviors of functions that could be complex to understand algebraically.

The critical part is to use graphical results to verify your algebraic findings, especially in a problem with multiple potential solutions or ambiguities.