Write the following complex number in trigonometric form. Write the argument in radians, and if possible, use the exact form. Otherwise, use a decimal approximation rounded to the hundredths place.

−3⎯⎯√+3i

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.

Learn how to convert the complex number -√3 + 3i to trigonometric form with a step-by-step solution. This problem covers key concepts in complex numbers, including calculating magnitudes and arguments. Suitable for high school students studying complex numbers in Grades 10-12.

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