Question

Euler's Equation Euler's equation defines e raised to an imaginary power in terms of sinusoidal functions as follows $$ e^{i \theta}=\cos \theta+i \sin \theta $$ Create a two-dimensional plot of this function as \(\theta\) varies from 0 to \(2 \pi\). Create a three-dimensional line plot using function plot 3 as \(\theta\) varies from 0 to \(2 \pi\) (the three dimensions are the real part of the expression, the imaginary part of the expression, and \(\theta\) ).

Short answer

The solution entails graphing Euler's Equation \(e^{i\theta}=\cos\theta+i\sin\theta\), first in 2-dimensional form and then as a 3-dimensional line plot. The real and imaginary parts of the equation are identified as \(\cos\theta\) and \(sin\theta\), respectively. For the 2D graph, both functions are plotted against \(\theta\) spanning from 0 to \(2\pi\) as both are periodic with the same period. The 3D line plot is slightly more complex, requiring coordinates where the real part, imaginary part and \(\theta\) are plotted in a 3D space. Again, \(\theta\) varies from 0 to \(2\pi\). As an interpretation, the 2D plot will show the real and imaginary components as oscillating sinusoidal functions. The 3D line plot will visualize these components in orthogonal planes, forming a helix shape that wraps around a central axis with each full turn corresponding to a \(2\pi\) interval. Tools like Desmos, Geogebra or a programming language with graphing libraries such as Matplotlib or Plotly can be used for these plots.

Step-by-step solution

Identify the real and imaginary parts of Euler's equation

The given expression is \(e^{i\theta} = \cos\theta + i\sin\theta\). The real part of the expression is \(\cos\theta\) and the imaginary part is \(\sin\theta\).

Plot the 2-dimensional graph of the real and imaginary parts of Euler's equation

To plot the real (\(\cos\theta\)) and imaginary (\(\sin\theta\)) parts of the Euler's equation in 2D, we plot them as two separate functions against the independent variable \(\theta\) on the horizontal axis. Both functions are periodic with the period \(2\pi\), so plot them for \(\theta\) ranging from 0 to \(2\pi\). You can use graphing software like Desmos or Geogebra, or a programming language like Python with Matplotlib library to create the plot.

Create the 3-dimensional line plot

To create a 3D line plot for the given Euler's equation, we need to plot the function in a coordinate system: (Real part, Imaginary part, \(\theta\)). The coordinates are: Real part: \(\cos\theta\) Imaginary part: \(\sin\theta\) \(\theta\): independent variable, varying from 0 to \(2\pi\) Use any 3D plotting software (such as Geogebra or Mayavi) or a programming language (like Python with Matplotlib or Plotly) to create the 3D line plot.

Interpret the results

The 2D plot of the Euler's equation will show two sinusoidal functions, the real (\(\cos\theta\)) and imaginary (\(\sin\theta\)) parts, oscillating between -1 and 1 for the entire range of 0 to \(2\pi\). The 3D line plot will show a helix-like shape, wrapping around a central axis, with each turn corresponding to a \(2\pi\) interval in the \(\theta\) axis. The helix will demonstrate that the real and imaginary parts of Euler's equation are sinusoidal functions that oscillate in orthogonal planes.

Key concepts

Complex Numbers

Complex numbers play a central role in Euler's Equation. A complex number is expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) being the imaginary unit (\( i^2 = -1 \)). Euler's formula \( e^{i \theta} = \cos \theta + i \sin \theta \) beautifully connects complex numbers with trigonometric functions, providing a powerful tool for analysis.

In Euler's formula, the magnitude of the complex number \( |e^{i\theta}| = 1 \), which implies that the point lies on the unit circle in the complex plane. The angle \( \theta \), called the argument, is the angle from the positive x-axis to the line connecting the origin to the point. This geometric interpretation plays an essential part in applications across various fields such as electrical engineering, quantum physics, and many more.

Complex numbers simplify the computation of rotations and oscillations. For example, multiplying a complex number by \( e^{i\theta} \) can rotate it by an angle \( \theta \). Understanding complex numbers is vital for decoding Euler’s Equation, as it reveals how real parts \( \cos \theta \) and imaginary parts \( \sin \theta \) work together to encode periodic behavior.

Trigonometric Functions

Euler's Equation directly relates to cosines and sines, which are the fundamental trigonometric functions defining the real and imaginary parts of the equation, respectively. These functions depict periodic oscillations.

Let's break them down:

• **Cosine Function**: \( \cos \theta \) oscillates between 1 and -1, modeling the horizontal component in the unit circle.
• **Sine Function**: \( \sin \theta \) oscillates similarly, representing the vertical component.

Both functions have a period of \( 2\pi \), meaning they repeat every \( 360^\circ \).

In the context of Euler's Equation, trigonometric functions express the rotational movement along the complex plane. This rotation is key to visualizing wave-like behaviors, a concept heavily used in physics and engineering fields. For example, phenomena such as alternating currents and signal waves rely on understanding these sinusoidal components.

Essentially, Euler's Equation bridges the gap between exponential functions and trigonometry, providing a unified framework to describe rotations and oscillations compactly.

3D Plotting

Visualizing Euler's Equation in three dimensions offers deeper insight into its properties. Here, we use a 3D plot to demonstrate the behavior of the real and imaginary parts of \( e^{i\theta} \) as \( \theta \) moves from 0 to \( 2\pi \).

In a 3D coordinate system:

• The **x-axis** represents the real part, \( \cos \theta \).
• The **y-axis** represents the imaginary part, \( \sin \theta \).
• The **z-axis** accounts for \( \theta \), showing its progression from 0 to 2\( \pi \).

As \( \theta \) advances, plotting these points traces a helical path, highlighting the intertwined nature of the real and imaginary components. This helical shape is crucial because it reveals how complex numbers oscillate and rotate in a spiraling trajectory over time.

Tools like Python's Matplotlib and Plotly, or software such as Geogebra, can create these visualizations, allowing us to perceive mathematical concepts in a 3D scope. Such plots are not only vital for educational purposes but also essential in fields like data visualization, where comprehending multidimensional interactions can lead to significant breakthroughs.