The problem requires representing complex numbers in their trigonometric form. Let me break it down step by step for each case.

The general trigonometric form of a complex number $z = x + yi$ is: $z = r (\cos \theta + i \sin \theta),$ where:

• $r = \sqrt{x^2 + y^2}$ is the modulus of the complex number,
• $\theta = \arg(z)$ is the argument (angle) of the complex number.

a) $z = -2$

Here, $z$ is purely real and negative. Its components are:

• Real part $x = -2$,
• Imaginary part $y = 0$.

1. Find the modulus $r$: $r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2.$

2. Find the argument $\theta$: Since $z = -2$ lies on the negative real axis, the argument is: $\theta = \pi \, \text{(radians)}.$

3. Write the trigonometric form: $z = 2 (\cos \pi + i \sin \pi).$

b) $z = \sqrt{3} - i$

Here, the components of $z$ are:

• Real part $x = \sqrt{3}$,
• Imaginary part $y = -1$.

1. Find the modulus $r$: $r = \sqrt{x^2 + y^2} = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2.$

2. Find the argument $\theta$: The complex number is in the fourth quadrant (real positive, imaginary negative). The angle $\theta$ can be determined as: \theta = -\frac{\pi}{6} \, \text{(radians)} \quad \text{(or equivalently \frac{11\pi}{6})}.

3. Write the trigonometric form: $z = 2 \left( \cos\left(-\frac{\pi}{6}\right) + i \sin\left(-\frac{\pi}{6}\right) \right).$

Final Answer:

a) $z = 2 (\cos \pi + i \sin \pi)$
b) $z = 2 \left( \cos\left(-\frac{\pi}{6}\right) + i \sin\left(-\frac{\pi}{6}\right) \right)$

Would you like a more detailed explanation for any of the steps?

Follow-Up Questions:

1. How do you convert rectangular form to polar form for complex numbers?
2. What is the significance of the modulus and argument of a complex number?
3. Can you generalize the trigonometric form for real and imaginary axes?
4. How do you find the argument for complex numbers in different quadrants?
5. How is Euler's formula related to the trigonometric form of complex numbers?

Tip: The modulus $r$ and argument $\theta$ are critical when working with complex numbers in polar form, especially for multiplication and division operations.