Let's solve the second and third questions from the image provided.

Question 2:

Problem: Find the principal argument of the complex number $e^{5+6i}$

Solution:

We start with the complex number in the form $e^{a+bi}$. Here, $a = 5$ and $b = 6$.

1. Recall Euler's formula: $e^{a+bi} = e^a (\cos(b) + i \sin(b))$

So, $e^{5+6i} = e^5 (\cos(6) + i \sin(6))$

2. The principal argument (θ) of a complex number $e^{bi}$ is $b \mod 2\pi$.

Therefore, $\text{Principal argument} = 6 \mod 2\pi$

Since $6$ is less than $2\pi \approx 6.2832$, the principal argument is:

$\theta = 6$

Question 3:

Problem: Find the rotation angle of the transformation $w = z^2$ at $z = 2i$

Solution:

1. First, express $z = 2i$ in polar form: $z = 2i = 2(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})$

2. The transformation $w = z^2$: $w = (2i)^2 = 4i^2 = 4(-1) = -4$

3. Find the argument of $w$: $w = -4 = 4(\cos \pi + i \sin \pi)$

So, the argument of $w$ is $\pi$.

4. The rotation angle for the transformation $w = z^2$ is twice the argument of $z$: $\text{Rotation angle} = 2 \times \frac{\pi}{2} = \pi$

Therefore, the rotation angle is $\pi$.

If you have any further questions or need additional details, please let me know.

Here are 5 questions you might consider next:

1. How do you find the magnitude of a complex number?
2. What is Euler's formula and how is it used in complex analysis?
3. How do you convert a complex number from rectangular form to polar form?
4. What is the principal argument of a complex number and how is it determined?
5. How do transformations affect the argument and magnitude of complex numbers?

Tip: When working with complex numbers, always try to convert them to polar form to make multiplication, division, and finding powers easier.