The complex exponential: $e^{i\omega t}$ is useful in calculating properties of many time-dependent equations. According to Euler, we can also write this function as: 1. $\cos(i \omega t) + \sin (i \omega t)$ 2. $\sin (\omega t) + i \cos(\omega t)$ 3. $\cos(\omega t) + i \sin (\omega t)$ 4. MORE than one of these is correct 5. None of these is correct! Note: * Correct Answer: C * Just a reminder of the euler equation for the future
## Announcements * Quiz 3 (next Friday 2/22) - RLC circuits * Solve a circuit problem using the phasor method * Discuss limits on the response and how it might act as a filter
What is $\vert 2+i \vert$? 1. $1$ 2. $\sqrt{3}$ 3. $5$ 4. $\sqrt{5}$ 5. Something else! Note: * Correct Answer: D * Use pythagoras in the complex plane
What is $(1+i)^2/(1-i)$? 1. $e^{i\pi/4}$ 2. $\sqrt{2}e^{i\pi/4}$ 3. $e^{i3\pi/4}$ 4. $\sqrt{2}e^{i3\pi/4}$ 5. Something else! Note: * Correct Answer: D * Best to convert to euler with phase and just use the rules for adding and subtracting of exponents
Which point below best represents $4e^{i3\pi/4}$ on the complex plane? <img src="./images/complex_numbers_graph.png" align="center" style="width: 600px";/> Note: * Correct Answer: D
What is $Re\left[\frac{e^{i\omega t}}{1+i}\right]$? 1. $\frac{1}{\sqrt{2}}\cos(\omega t + \pi/4)$ 2. $\frac{1}{\sqrt{2}}\cos(\omega t - \pi/4)$ 3. $\frac{1}2\cos(\omega t + \pi/4)$ 4. $\frac{1}2\cos(\omega t - \pi/4)$ 5. Something else Note: * Correct Answer: B