To solve the problem, we need to find the argument of the complex number $z = \sqrt{3} + 3i$ and express the answer in degrees.

Step-by-Step Solution:

1. Recall the formula for the argument of a complex number: $\arg(z) = \arctan\left(\frac{\text{Im}(z)}{\text{Re}(z)}\right),$ where $\text{Im}(z)$ and $\text{Re}(z)$ are the imaginary and real parts of $z$, respectively.

2. Identify the real and imaginary parts: $\text{Re}(z) = \sqrt{3}, \quad \text{Im}(z) = 3.$

3. Substitute into the formula: $\arg(z) = \arctan\left(\frac{3}{\sqrt{3}}\right).$

4. Simplify the fraction: $\frac{3}{\sqrt{3}} = \sqrt{3}.$ So: $\arg(z) = \arctan(\sqrt{3}).$

5. Find the angle: From trigonometry, $\arctan(\sqrt{3}) = 60^\circ$ (in degrees).

Final Answer:

The argument of $z = \sqrt{3} + 3i$ is: $\boxed{60^\circ}.$

Would you like me to elaborate further or solve related problems?

Related Questions:

1. What is the magnitude (modulus) of $z = \sqrt{3} + 3i$?
2. How do you convert an argument in radians to degrees?
3. Can we find the argument for a general complex number $a + bi$?
4. How does the quadrant of a complex number affect its argument?
5. What is the polar form representation of $z = \sqrt{3} + 3i$?

Tip:

Always double-check which quadrant the complex number is in to determine the correct angle, especially when the argument is negative or over $180^\circ$.