The complex exponential function, \(x=e^{i \omega t}\), can be used to describe harmonic motion (the function can be defined in MATLAB \({ }^{\circledR}\) using \(\mathrm{x}=\exp\) \(\left(1 i *\right.\) omega \(\left.\left.{ }^{*} t\right) ;\right)\). Complete the following to explore this function.

(a) Plot the real part of the function for \(\omega=\pi \mathrm{rad} / \mathrm{s}\) over a time interval of \(t=0\) to \(10 \mathrm{~s}\) using time steps of \(0.05 \mathrm{~s}\). Use the command plot ( \(t\), real (x)) to complete this task.

(b) Plot the imaginary part of the function. Use the command plot ( \(t\), imag (x)).

(c) Describe your results from parts (a) and (b) in terms of sine and cosine functions.

(d) Sketch the Argand diagram for \(x\) at \(t=0.25 \mathrm{~s}\) and show its projections on the real and imaginary axes. What is the numerical value of these projections? How do these results relate to parts (a) and (b)?