To solve the problem, we calculate the argument of the complex number $z = \sqrt{3} + 3i$. The argument of a complex number $z = a + bi$ is given by:

$\theta = \arctan\left(\frac{b}{a}\right)$

where:

• $a = \text{Re}(z) = \sqrt{3}$,
• $b = \text{Im}(z) = 3$.

Step 1: Plug in the values

$\theta = \arctan\left(\frac{3}{\sqrt{3}}\right)$

Step 2: Simplify

$\frac{3}{\sqrt{3}} = \sqrt{3}$

$\theta = \arctan(\sqrt{3})$

Step 3: Find the angle

From trigonometric tables, $\arctan(\sqrt{3}) = 60^\circ$.

Thus, the argument of $z$ is: $\theta = 60^\circ$

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Related Questions:

1. What is the modulus of the complex number $z = \sqrt{3} + 3i$?
2. How do you convert this complex number to polar form?
3. What is the argument of $z$ in radians?
4. How does the quadrant of $z$ affect the calculation of its argument?
5. How can the argument be calculated geometrically?

Tip:

To find the argument, always consider the quadrant of the complex number to adjust the angle if necessary. Here, both parts are positive, so $z$ lies in the first quadrant.