The problem in the image asks to express the complex number $1 + i\sqrt{3}$ in polar form.

Steps to Convert to Polar Form:

1. Identify the complex number: $z = 1 + i\sqrt{3}$, where the real part $x = 1$ and the imaginary part $y = \sqrt{3}$.

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

3. Find the argument (θ): $\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$

4. Express in polar form: The polar form of a complex number is given by: $z = r(\cos \theta + i\sin \theta)$ Substituting $r = 2$ and $\theta = \frac{\pi}{3}$, the polar form becomes: $z = 2\left(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\right)$

Thus, the complex number $1 + i\sqrt{3}$ in polar form is: $z = 2\left(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\right)$

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions you might explore:

1. How would you convert a complex number in polar form back to rectangular form?
2. What are the advantages of expressing a complex number in polar form?
3. How does the polar form help in multiplying and dividing complex numbers?
4. Can the argument $\theta$ be negative? If so, what does it represent geometrically?
5. How does De Moivre’s Theorem apply to powers of complex numbers in polar form?

Tip: Always check whether the argument $\theta$ lies in the correct quadrant based on the signs of the real and imaginary parts.