The image you provided contains several exercises involving complex numbers, geometric arguments, and trigonometric forms. Let's break down a couple of the exercises to help you out.

Exercise 12:

This problem involves a geometric figure where $OAB$ is an equilateral triangle. The task is to determine the modulus and the argument of the affixes $z_A$, $z_B$, $z_C$, and $z_D$ of the points $A$, $B$, $C$, and $D$.

Approach:

• Modulus: The modulus of a complex number $z = x + yi$ is given by $|z| = \sqrt{x^2 + y^2}$.
• Argument: The argument is the angle $\theta$ that the line joining the origin to the point makes with the positive real axis, given by $\arg(z) = \tan^{-1}\left(\frac{y}{x}\right)$.

By examining the graph:

• $z_A$ and $z_B$ can be calculated directly from their coordinates on the complex plane.
• For $z_C$ and $z_D$, these could be calculated by considering the rotational symmetry of the equilateral triangle.

Exercise 13:

Here, you need to determine the modulus and argument of points on a circle. The points are clearly positioned on the unit circle, which simplifies the calculations:

• Since they lie on the unit circle, the modulus of each point is 1.
• The argument can be calculated directly based on their positions (for example, $z_A$ lies at an angle corresponding to the points on the unit circle).

Exercise 14:

This problem asks you to convert complex numbers into their trigonometric form. The trigonometric form of a complex number is $z = r(\cos\theta + i\sin\theta)$, where $r$ is the modulus and $\theta$ is the argument.

To convert the given numbers:

• First, calculate the modulus $r$ using $r = \sqrt{x^2 + y^2}$.
• Then, determine the argument $\theta$ using $\theta = \tan^{-1}\left(\frac{y}{x}\right)$.
• Express $z$ in the form $r(\cos\theta + i\sin\theta)$.

Exercise 15:

You are given two complex numbers $z_1 = 1+i$ and $z_2 = \sqrt{3} - i$ and asked to:

1. Find their modulus and argument.
2. Express the product $z_1z_2$ in trigonometric form.
3. Derive the exact values of $\cos\frac{\pi}{12}$ and $\sin\frac{\pi}{12}$.

For this, compute:

• Modulus $|z_1|$ and $|z_2|$ and arguments $\arg(z_1)$ and $\arg(z_2)$.
• Then, multiply $z_1$ and $z_2$ in their trigonometric forms and simplify.

Would you like to explore any of these problems in detail or have any specific questions? Here are some relative questions to help guide your understanding:

1. How do you calculate the argument of a complex number geometrically?
2. What is the relationship between a complex number’s trigonometric and algebraic forms?
3. How do rotations in the complex plane affect the modulus and argument?
4. Why is the modulus of any point on the unit circle always 1?
5. How do you derive the exact values for trigonometric functions of specific angles using complex numbers?

Tip: When working with complex numbers, always double-check the quadrant in which your point lies to correctly determine the argument.