QuestionFind the product of $z_{1}=-1+i$ and $z_{2}=1+\sqrt{3} i$ and express it in polar form.

Studdy Solution

STEP 1

Assumptions1. The complex numbers are $z_{1}=-1+i$ and $z_{}=1+\sqrt{3} i$ . We need to find the product of these complex numbers in polar form

STEP 2

First, we need to convert the complex numbers into polar form. The polar form of a complex number is given by $r(\cos{\theta} + i\sin{\theta})$ , where $r$ is the modulus of the complex number and $\theta$ is the argument of the complex number.
The modulus of a complex number $z=a+bi$ is given by $\sqrt{a^2 + b^2}$ , and the argument is given by $\tan^{-1}(b/a)$ .

STEP 3

Let's find the modulus and argument of $z_{1}=-1+i$ .
The modulus of $z_{1}$ is given by $\sqrt{(-1)^2 +1^2}$ .
The argument of $z_{1}$ is given by $\tan^{-1}(1/-1)$ .

STEP 4

Calculate the modulus and argument of $z_{1}$ .
$r_{1} = \sqrt{(-1)^2 +1^2} = \sqrt{2}$ $\theta_{1} = \tan^{-1}(1/-1) = \frac{3\pi}{4}$ So, $z_{1}$ in polar form is $\sqrt{2}(\cos{\frac{3\pi}{4}} + i\sin{\frac{3\pi}{4}})$ .

STEP 5

Now, let's find the modulus and argument of $z_{2}=1+\sqrt{3} i$ .
The modulus of $z_{2}$ is given by $\sqrt{1^2 + (\sqrt{3})^2}$ .
The argument of $z_{2}$ is given by $\tan^{-1}(\sqrt{3}/1)$ .

STEP 6

Calculate the modulus and argument of $z_{2}$ .
$r_{2} = \sqrt{1^2 + (\sqrt{3})^2} =2$ $\theta_{2} = \tan^{-1}(\sqrt{3}/1) = \frac{\pi}{3}$ So, $z_{2}$ in polar form is $2(\cos{\frac{\pi}{3}} + i\sin{\frac{\pi}{3}})$ .

STEP 7

Now that we have the complex numbers in polar form, we can find their product. The product of two complex numbers in polar form is given by $r_{1}r_{2}(\cos{(\theta_{1}+\theta_{2})} + i\sin{(\theta_{1}+\theta_{2})})$ .

STEP 8

Plug in the values for $r_{1}$ , $r_{2}$ , $\theta_{1}$ , and $\theta_{2}$ to calculate the product.
$Product = \sqrt{2} \cdot2 (\cos{(\frac{3\pi}{4}+\frac{\pi}{3})} + i\sin{(\frac{3\pi}{4}+\frac{\pi}{3})})$

STEP 9

Calculate the product of the complex numbers.
$Product =2\sqrt{2} (\cos{\frac{11\pi}{12}} + i\sin{\frac{11\pi}{12}})$ So, the product of the complex numbers $z_{}=-+i$ and $z_{2}=+\sqrt{3} i$ in polar form is $2\sqrt{2} (\cos{\frac{11\pi}{12}} + i\sin{\frac{11\pi}{12}})$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms