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Chapter 6: Problem 103

Graph the functions. $$y=\frac{1-\tan ^{2}\left(\frac{x}{2}\right)}{1+\tan ^{2}\left(\frac{x}{2}\right)}$$

Short Answer

Expert verified
The graph of the function \( y = \frac{1-\tan^{2}(\frac{x}{2})}{1+\tan^{2}(\frac{x}{2})} \) is identical to \( y = \cos(x) \).

Step by step solution

01

Apply Trigonometric Identity

Recognize that the function given is a form of a trigonometric identity related to the tangent function. The expression can be rewritten using the identity for the cosine of a double angle: \( \cos(x) = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \). This identity indicates that the function \( y = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \) simplifies to \( y = \cos(x) \).
02

Identify Basic Characteristics of the Function

Recognize that \( y = \cos(x) \) is a standard cosine function with period \( 2\pi \), amplitude 1, and no phase shift or vertical displacement. It oscillates between -1 and 1 along the x-axis. The function repeats every \( 2\pi \) and has its maximum at even multiples of \( \pi \), minimum at odd multiples of \( \pi \), and zeros at \( \left(2n + 1\right)\frac{\pi}{2} \).
03

Sketch the Graph

Begin by marking key points on the graph: the peaks of the cosine wave at \( x = 0, \pm 2\pi, \pm 4\pi, \ldots \) with the value 1, the troughs at \( x = \pm \pi, \pm 3\pi, \ldots \) with the value -1, and the zeros at \( x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots \). Connect these points with a smooth curve displaying the characteristic wave-like pattern of the cosine function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
When graphing functions, especially trigonometric ones, it's essential to understand their basic shape and behavior. The function provided in the exercise simplifies to the cosine function. When you graph a function like this, several elements need consideration:
  • Period: The cosine function repeats every \(2\pi\). This is its period. You can recognize a cycle of the graph by finding where it starts repeating its values.
  • Amplitude: This refers to the height of the wave from its centerline to its peak. For the standard cosine function, it's 1.
  • Axis intersections: For \(y = \cos(x)\), the function crosses the x-axis at points where \(x = (2n + 1)\frac{\pi}{2}\), where \(n\) is an integer.
  • Vertical and horizontal transformations: There are none here, since this is a basic cosine function.
To sketch it, plot key points like peaks, troughs, and x-axis intersections. Then connect these points with smooth, rounded curves to illustrate the repeating wave structure.
Cosine Function
The cosine function is one of the fundamental trigonometric functions, noted as \(\cos(x)\). It is periodic, meaning it repeats its values in a regular cycle. Here are some key characteristics:
  • Range and Domain: The domain of \(\cos(x)\) includes all real numbers, while its range is from -1 to 1.
  • Symmetry: Cosine is an even function. This means that \(\cos(-x) = \cos(x)\), resulting in symmetry about the y-axis.
  • Zero points: The function crosses zero at points \(x = (2n+1)\frac{\pi}{2}\).
  • Maxima and Minima: Cosine achieves its maximum value of 1 at even multiples of \(\pi\) and its minimum of -1 at odd multiples of \(\pi\).
Understanding these features helps with graph recognition and manipulation by identifying patterns and transformations.
Double Angle Identity
The double angle identity is a trigonometric tool that can significantly simplify expressions, especially when dealing with functions like tangent or sine. One essential identity is \(\cos(2\theta) = 1 - 2\sin^2(\theta) = 2\cos^2(\theta) - 1\). However, the exercise at hand requires using the tangent-based identity for cosine:
  • \(\cos(x) = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}\)
This identity allows transforming expressions involving tangent into a form relatable to cosine, allowing us to graph it using familiar properties of the cosine function. This transformation is important when simplifying tasks in calculus, such as integration, where cosine exhibits steadier patterns.

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Most popular questions from this chapter

Touch-tone keypads have the following simultaneous low and high frequencies. $$\begin{array}{|l|c|c|c|} \hline \text { Freauency } & 1209 \mathrm{Hz} & 1336 \mathrm{Hz} & 1477 \mathrm{Hz} \\ \hline 697 \mathrm{Ilz} & 1 & 2 & 3 \\ \hline 770 \mathrm{Hz} & 4 & 5 & 6 \\ \hline 852 \mathrm{Hz} & 7 & 8 & 9 \\ \hline 941 \mathrm{Hz} & ^{*} & 0 & \\# \\ \hline \end{array}$$ The signal given when a key is pressed is \(\sin \left(2 \pi f_{1} t\right)+\sin \left(2 \pi f_{2} t\right)\) where \(f_{1}\) is the low frequency and \(f_{2}\) is the high frequency. What is the mathematical function that models the sound of dialing \(3 ?\)

Graphing calculators can be used to find approximate solutions to trigonometric equations. For the equation \(f(x)=g(x),\) let \(Y_{1}=f(x)\) and \(Y_{2}=g(x) .\) The \(x\) -values that correspond to points of intersections represent solutions. With a graphing utility, find all of the solutions to the equation \(\cos \theta=e^{\theta}\) for \(\theta \geq 0\).

Determine whether each statement is true or false. If a trigonometric equation has all real numbers as its solution, then it is an identity.

Solve each trigonometric equation on \(0^{\circ} \leq \theta<360^{\circ} .\) Express solutions in degrees and round to two decimal places. $$\sec ^{2} x+\tan x-2=0$$

In calculus, the method of separation of variables is used to solve certain differential equations. Given an equation with two variables, the method consists of writing the equation in such a way that each side of the equation contains only one type of variable. Use the product-to-sum and sum-to-product identities to separate the variables \(x\) and \(y\) in each equation. $$2+\cos (x+y)=\cos (x-y)$$
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