Sketch the imaginary part of \(A e^{j \varphi} e^{j \omega t}\) to show that this is \(A \sin (\omega t+\varphi)\). What do we mean when we say that the real and imaginary parts of \(A e^{j \varphi} e^{j \omega t}\) are " \(90^{\circ}\) out of phase"?

In complex numbers, the imaginary part refers to the coefficient of the imaginary unit, denoted by 'j' (or 'i' in mathematics). To extract the imaginary part from the given expression, we first need to rewrite it in a form where it is easy to separate real and imaginary components.
Combining the exponents as per the rules, the expression becomes: \[ A e^{j (\text{\textomega t + \textphi})} \] Using Euler's formula, we can expand this to: \[ A (\text{cos} (\text{\textomega t + \textphi}) + j \text{sin} (\text{\textomega t + \textphi})) \] The imaginary part is represented by the term that includes 'j', which is \[ A \text{sin} (\text{\textomega t + \textphi}) \]
This sine term is what we sketch in the exercise to show the imaginary part. By focusing on the imaginary part, we see how it fits into the overall picture of the complex exponential function. Understanding each component is crucial for deeper insights into complex analysis.

Euler's formula is a key tool in dealing with complex exponentials. It states that \[ e^{j \theta} = \text{cos} \theta + j \text{sin} \theta \]
This formula helps us move between exponential and trigonometric forms of complex numbers. In the given exercise, we leverage Euler's formula to expand \[ A e^{j (\text{\textomega t + \textphi})} \] into its real and imaginary components:
\[ A (\text{cos} (\text{\textomega t + \textphi}) + j \text{sin} (\text{\textomega t + \textphi})) \]
By doing so, we quickly identify the real part as \[ A \text{cos} (\text{\textomega t + \textphi}) \] and the imaginary part as \[ A \text{sin} (\text{\textomega t + \textphi}) \]
Euler's formula simplifies the complexity of dealing with exponentials involving imaginary units, making it easier to study phase relationships and other properties.

Phase difference refers to the amount by which one wave (or function) is shifted relative to another. In the context of sine and cosine, these functions are out of phase by 90 degrees (or π/2 radians).
Consider the expanded expression: \[ A (\text{cos} (\text{\textomega t + \textphi}) + j \text{sin} (\text{\textomega t + \textphi})) \]
The real part is \[ A \text{cos} (\text{\textomega t + \textphi}) \] and the imaginary part is \[ A \text{sin} (\text{\textomega t + \textphi}) \]
Sine and cosine functions are 90 degrees out of phase because if you graph \[ \text{cos} (\text{\textomega t}) \] and \[ \text{sin} (\text{\textomega t}) \], sine is simply the cosine shifted by a quarter of a period, which corresponds to 90 degrees. Understanding this phase difference is essential in various applications, such as signal processing and AC circuit analysis. It shows how even slight shifts can change the behavior and interplay of different components in a system.