Problem 100 - Algebra and Trigonometry

Question

Graph each equation in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right] b y\) \([-3,3,1]\) viewing rectangle. Then a. Describe the graph using another equation, and b. Verify that the two equations are equivalent. $$ y=\frac{2 \tan \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}} $$

Step-by-Step Solution

Answer

The graph of the given function is similar to that of \(y=\tan x\). Therefore the function can be equivalently described with the equation \(y= \tan x\). After manipulating the given equation using identities, we can verify that the original equation is indeed equivalent to \(y=\tan x\).

Step 1: Plotting Points

Plot the points of the function \(y=\frac{2 \tan \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}\) within the \(x\) range of \(-2\pi\) to \(2\pi\) and for \(y\) in the range \(-3\) to \(3\). In plotting these points, observe the trend and draw the curve line accordingly. Remember that tan function is undefined at odd multiples of \(\pi/2\) so make sure to show the vertical asymptotes on your graph.

Step 2: Recognize the periodicity and Symmetry

Looking at the graph, recognize that the graph of the function repeats itself after every pi, similar to the standard tangent function. Also notice that the function is symmetrical around the y-axis, this is a property of even functions.

Step 3: Describe the Graph using Another Equation

The graph looks like a tangent function with period \(2\pi\) and amplitude 1. The graph can be equivalently described using the equation \(y=\tan x\). This equation has the same properties as the original - it repeats every \(2\pi\) and is undefined at odd multiples of \(\pi/2\). The amplitude is also the same, reaching a maximum of 1 and a minimum of -1.

Step 4: Verify the Two Equations are Equivalent

To verify the equivalence of the two equations \(y=\frac{2 \tan \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}\) and \(y= \tan x\), use the identity \(\tan x = \frac {\sin x}{\cos x}\), and the double-angle identities \(\sin 2u = 2 \sin u \cos u\) and \(\cos 2u = \cos^2u - \sin^2u\). Apply these identities and simplify to show that it indeed matches the given two equations.

Key Concepts

Graphing Equations

When graphing an equation, especially a trigonometric one like the tangent function, it's important to first understand the domain and range. You are tasked with graphing within the interval - \([-2\pi, 2\pi]\) for the x-values, and \([-3, 3]\) for the y-values.This defines the window in which we'll see the graph's behavior.Begin by plotting key points within this window. For the tangent function, this includes noting where it is undefined.

• Typically, this occurs at odd multiples of \(\frac{\pi}{2}\).
• These points manifest as vertical asymptotes on the graph.
• Between these asymptotes, plot enough points to detect the curve's trend.

Once these elements are marked, draw curves that correspond with the behavior of the graph. Keep in mind that graphing is about capturing the pattern, shape, and features of the equation.

Tangent Function

The tangent function, often referred to as \(\tan x\), is one of the primary trigonometric functions. It has distinct characteristics:

• Its graph exhibits a repeating wave-like pattern due to its periodicity.
• Each section between vertical asymptotes looks similar, forming a regular repeating pattern.
• The tangent function ranges from \(-\infty\) to \(\infty\) but has no local extrema such as peaks or troughs, unlike sine and cosine functions.

In the context of this exercise, it's expressed as \(y=\frac{2 \tan \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}\).This re-formulation showcases similar properties. Notably, it can be simplified to resemble \(y = \tan x\), thanks to trigonometric identities and equivalence rules.

Periodicity

Periodicity is a key feature of trigonometric functions. For the tangent function, this means:

• It repeats its shape every \(\pi\) units along the x-axis.
• This periodic nature is evident between vertical asymptotes.

In this exercise, the modified equation shows periodicity in the pattern of the graph.

• The function \(\frac{2 \tan \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}\) shares this periodic trait.
• Despite appearances, the graph still repeats every \(2\pi\) due to the inherent properties of tangent and the transformations applied.

Understanding periodicity helps in graphing and predicting behavior beyond the given interval.

Symmetry

Symmetry is an important attribute in trigonometry. It makes equations more predictable by indicating balance or equivalence in structure.For tangent functions:

• The standard \(y = \tan x\) is symmetric about the origin. It exhibits odd symmetry such that \(\tan(-x) = -\tan(x)\).
• However, in transformations, modified functions like the one in this problem show even symmetry centered on the y-axis.

This function remains consistent with the property of even functions when described graphically. Understanding symmetry is crucial for verifying equivalence between different trigonometric expressions. This includes using identities to simplify and check if two expressions, like those presented, truly mirror one another across various transformations.