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 \(\theta\) ).

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.