To write the complex number $z = -\sqrt{3} + 3i$ in trigonometric form, we need to determine its magnitude $|z|$ and its argument $\theta$.

1. Calculate the magnitude $|z|$:

The magnitude $|z|$ of a complex number $z = a + bi$ is given by:

$|z| = \sqrt{a^2 + b^2}$

Here, $a = -\sqrt{3}$ and $b = 3$. Therefore,

$|z| = \sqrt{(-\sqrt{3})^2 + (3)^2}$ $|z| = \sqrt{3 + 9}$ $|z| = \sqrt{12}$ $|z| = 2\sqrt{3}$

2. Calculate the argument $\theta$:

The argument $\theta$ is the angle made with the positive real axis, calculated using:

$\theta = \tan^{-1} \left(\frac{b}{a}\right)$

For $z = -\sqrt{3} + 3i$, we have:

$\theta = \tan^{-1} \left(\frac{3}{-\sqrt{3}}\right)$ $\theta = \tan^{-1} \left(-\sqrt{3}\right)$

Knowing that $\tan \left( \frac{2\pi}{3} \right) = -\sqrt{3}$ and the point $(- \sqrt{3}, 3)$ is in the second quadrant, the argument $\theta$ should be:

$\theta = \pi - \frac{\pi}{3}$ $\theta = \frac{2\pi}{3}$

Thus, the trigonometric form of the complex number $z = -\sqrt{3} + 3i$ is:

$z = 2\sqrt{3} \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)$

Would you like any further details or have any questions?

Further Questions:

1. How do you convert a complex number from rectangular to polar form?
2. What is the significance of the magnitude of a complex number?
3. Why is the argument of a complex number important?
4. How do you determine which quadrant a complex number lies in?
5. What are the common angles used in trigonometric forms of complex numbers?
6. How would the trigonometric form change if the complex number was in a different quadrant?
7. Can you express the argument in degrees instead of radians?
8. How can you verify the trigonometric form of a complex number?

Tip:

When converting complex numbers to trigonometric form, always make sure to check the quadrant to correctly determine the argument.