Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ y=\cos x $$

Step-by-Step Solution

Answer

The graph of \\(y = \\cos x\\) in three-dimensions is a wave-like surface extending indefinitely along the \\(z\\)-axis.

Step 1: Understand the Equation

The given equation is a function of two variables where the output \(y\) depends on \(x\). In three-dimensional space, we assume \(z\) to be the third variable that remains independent, giving the graph its depth.

Step 2: Plot the Base Curve in the xy-plane

We start by considering the curve \(y = \cos x\) in the two-dimensional \(xy\)-plane. This is the standard cosine curve which oscillates between \(-1\) and \(1\) along the \(y\)-axis, while the \(x\)-axis represents the angle in radians.

Step 3: Extend into Three Dimensions

In three-dimensional space, we extend the curve along the \(z\) axis. Since the equation \(y = \cos x\) doesn't involve \(z\), \(z\) can take any real value. This means the cosine wave stretches infinitely along the \(z\)-axis, forming a wave-like surface.

Step 4: Sketch the Graph

Imagine the \(xy\) cosine curve as a surface that moves parallel to the \(z\)-axis. The graph will appear as stacked layers of cosine curves, each plane parallel to the \(xy\)-plane, but extending indefinitely up and down along \(z\).

Key Concepts

Cosine Function

The cosine function is one of the basic trigonometric functions, commonly abbreviated as "cos." It transforms an angle, often measured in radians, into a value. The output of the cosine function varies between -1 and 1. In a regular two-dimensional scenario, its graph is a wavy line, repeating its peaks and troughs over a set interval of \[2\pi\]. This characteristic periodic nature is handy for modeling cyclic patterns, like sound waves or tides.
The formula for the cosine function is \(y = \cos x\), where \(x\) represents the angle. Knowing that it repeats its pattern, understanding the cosine helps in grasping the basics of wave behavior.
When dealing with three-dimensional space, this familiar oscillation serves as a base for more complex surfaces.

Three-Dimensional Space

Three-dimensional space, often abbreviated as 3D, extends beyond the traditional \(xy\)-plane by introducing a third axis, typically denoted as \(z\). This extra depth gives objects a volume and allows us to visualize complex, real-world forms.
In mathematical terms, three-dimensional space introduces another layer of variability. When graphing functions in 3D, we have - Width (\(x\)-axis),- Height (\(y\)-axis), and - Depth (\(z\)-axis).
This depth is not just visual but provides crucial insight into relationships between more than two variables.
To represent functions like \(y = \cos x\) in 3D, we view them as extending surfaces. The introduction of the \(z\)-axis allows such graphs to capture changes across different dimensions, reflecting scenarios such as sound frequency over time or temperature across a plane.

Mathematical Graphing

Mathematical graphing is the visual representation of functions or data models via coordinates on different kinds of planes. This visualization is especially vital in understanding the properties and interaction of mathematical equations.
In a two-dimensional setup, graphs typically involve \(x\) and \(y\) planes. However, in three-dimensional graphing, we bring in the \(z\)-axis, making the graphical interpretation richer and more detailed.
To graph \(y = \cos x\) in 3D effectively:

• First, recognize the familiar sine wave in the \(xy\)-plane.
• Next, extend this wave parallel along the \(z\)-axis.

This approach forms a surface that illustrates how the cosine function behaves continuously across a new dimension. Such graphing provides insights into phenomena not just in mathematics but in fields like physics and engineering, where understanding dynamic systems in three dimensions is crucial.

Surface in Mathematics

In mathematical terms, a surface is a two-dimensional shape that exists within three dimensions, offering a visualization of functions and equations. Surfaces allow us to see the behavior of complex functions as they expand across space.
When considering the cosine function \(y = \cos x\) in three-dimensional space, the concept of a surface comes into play. Here, the classic wavy line, now free from \(z\), becomes a continuous undulating formation that extends infinitely along the third dimension.
Such surfaces help in modeling situations where data changes in more than one dimension, such as in meteorology, where temperatures can be plotted not just against time, but also altitude.
Additionally, mathematical surfaces make abstract ideas tangible, enabling us to harness and predict the patterns they form. Understanding the concept of a surface in mathematics equips us with tools to analyze functions' behaviors under broader conditions, enriching our insights into dynamic systems both theoretical and practical.