The question in the image asks to express $1 + i$ in polar form.

To express a complex number $z = a + bi$ in polar form, we use the formula:

$z = r(\cos \theta + i \sin \theta)$ where:

• $r$ is the magnitude of the complex number,
• $\theta$ is the argument (or angle) of the complex number.

For the complex number $1 + i$:

• The real part $a = 1$,
• The imaginary part $b = 1$.

Step 1: Find the magnitude $r$

The magnitude $r$ is given by:

$r = \sqrt{a^2 + b^2}$

Substitute $a = 1$ and $b = 1$:

$r = \sqrt{1^2 + 1^2} = \sqrt{2}$

Step 2: Find the argument $\theta$

The argument $\theta$ is the angle that the complex number makes with the positive real axis. It can be found using the formula:

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

For $a = 1$ and $b = 1$:

$\theta = \tan^{-1} \left( \frac{1}{1} \right) = \frac{\pi}{4}$

Final Answer:

So, the polar form of $1 + i$ is:

$z = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$

Let me know if you would like more details or if you have any questions!

Related Questions:

1. How do you convert from polar to rectangular form?
2. Can polar form be used for any complex number?
3. What is the general formula for converting to polar form?
4. How do you find the argument of a complex number in different quadrants?
5. What are the applications of polar form in mathematics?

Tip:

The magnitude $r$ always represents the distance of the complex number from the origin in the complex plane!