To express the complex number $-1 + i$ in polar form, we need to represent it in the form:

$r \left( \cos \theta + i \sin \theta \right)$

where $r$ is the modulus (magnitude) of the complex number, and $\theta$ is the argument (angle) of the complex number.

Step 1: Find the modulus $r$

The modulus of a complex number $a + bi$ is given by:

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

For $-1 + i$, where $a = -1$ and $b = 1$, we get:

$r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}$

Step 2: Find the argument $\theta$

The argument $\theta$ is the angle formed by the complex number in the complex plane, and it is calculated using the arctangent function. Specifically, $\theta = \tan^{-1} \left( \frac{b}{a} \right)$, but we must also consider the quadrant in which the complex number lies.

For $-1 + i$:

• $a = -1$
• $b = 1$

Since the complex number lies in the second quadrant (negative real part and positive imaginary part), we need to adjust the angle accordingly. The angle is:

$\theta = \pi - \tan^{-1} \left( \frac{1}{1} \right) = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$

Step 3: Write the polar form

Now, we can express $-1 + i$ in polar form as:

$-1 + i = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right)$

Final Polar Form:

$-1 + i = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right)$

Let me know if you'd like further details or if you have any questions!

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Here are 5 related questions for you:

1. How do you convert a complex number from polar form to rectangular form?
2. What is the significance of the modulus and argument in the polar form of complex numbers?
3. How do you find the argument of a complex number in the first quadrant?
4. What is the geometric interpretation of complex numbers in polar form?
5. How would you multiply two complex numbers in polar form?

Tip: When working with polar form, always check which quadrant the complex number lies in to determine the correct value for the argument!