Show that $$ \operatorname{Re}\left[V e^{j \omega t}\right]=A \cos (\omega t+\varphi) $$ when \(V=A e^{j \varphi}\).

In electrical engineering, we often encounter complex numbers to simplify problem-solving. One essential concept is the complex exponential. It allows us to represent oscillatory functions, like sine and cosine functions, in a compact and efficient way. The complex exponential can be written as: \[ e^{j \theta} = \text{cos}(\theta) + j \text{sin}(\theta) \]This expression follows from Euler's Formula, which we'll discuss further.Understanding the complex exponential helps us analyze AC circuits and signals. Functions like voltage and current can be expressed using this method: \[ V(t) = A e^{j (\theta)} \]where \(A\) is the amplitude and \( \theta \) is the phase. By using complex exponentials, we can transform differential equations into simpler algebraic forms. Once we solve these forms, we convert the solutions back to their real or imaginary parts.

Euler's Formula is a crucial link between complex exponentials and trigonometric functions. It states: \[ e^{j \theta} = \text{cos}(\theta) + j \text{sin}(\theta) \]This shows how the exponential function with an imaginary exponent can be split into its real and imaginary components. Using Euler's Formula, we can easily handle oscillatory motions in both mathematical models and practical applications. For example, if you have a signal: \[ f(t) = A e^{j (\theta)} \] Euler's Formula allows us to rewrite it as: \[ f(t) = A (\text{cos}(\theta) + j \text{sin}(\theta)) \]This decomposition simplifies analysis by separating the real and imaginary parts of the functions. It's valuable in engineering disciplines for signal processing, control systems, and communications.

Extracting the real part of a complex number is essential in many engineering applications. Given a complex number in exponential form, such as: \[ A e^{j (\theta)} \]we use Euler's Formula to expand it: \[ A (\text{cos}(\theta) + j \text{sin}(\theta)) \]The real part, denoted as \( \text{Re}[...] \), is simply the cosine term multiplied by the amplitude: \[ \text{Re}[A e^{j (\theta)}] = A \text{cos}(\theta) \]Understanding this helps in converting complex signals back into their original real-valued forms.For example, in our exercise, we showed: \[ \text{Re}[V e^{j \theta}] = A \text{cos}(\theta + \theta_{0}) \]where \( V = A e^{j \theta_{0}} \). This transformation proves critical in analyzing and interpreting the behavior of AC signals in circuits, ensuring precise measurements and insights.