Given z1=12(cos30°+isin30°)z_1=12\left(\cos30\degree+i\sin30\degree\right) and z2=3(cos50°+isin50°)z_2=3\left(\cos50\degree+i\sin50\degree\right), find the quotient z1z2\frac{z_1}{z_2}.

z 1 z 2 = 4 C i S ( 340 ° ) \frac{z_1}{z_2}=4CiS\left(340\degree\right)

Given z 1 = 12 ( cos 30 ° + i sin 30 ° ) z_1=12\left(\cos30\degree+i\sin30\degree\right) and z 2 = 3 ( cos 50 ° + i sin 50 ° ) z_2=3\left(\cos50\degree+i\sin50\degree\right) , find the quotient z 1 z 2 \frac{z_1}{z_2} .

this problem, we have 2 complex numbers. We have z1 and z2, and we are asked to find the quotient between these complex numbers. Now notice we're given both the complex numbers in polar form, and the way that we can find the quotient is by following a certain pattern. Our first step is going to be to divide the r values, which is this number that we have out in front for each of these complex numbers. So we're going to have the first number 12 divided by 3. Now from here what I can do is take these cosines and sines and I can abbreviate them to be cis. This is a shortcut for dealing with the sine the cosines. Now what this is going to be is cis the first angle minus the second angle. Now the first angle I can see here is 30 degrees, and then that's gonna be minus the second angle, which I can see is 50 degrees. So all I need to do is simplify this to get my final answer. So we're going to have 12 divided by 3, which is 4, and that's going to be 4, sis 30 degrees minus 50 degrees which is negative 20 degrees. Now notice how how we ended up with a negative number, we typically don't want to leave this negative in the answer. I mean this is technically still correct, but if you think about this this complex number is going to look something, say, like this, where we have our point over here, and it's going to be negative 20 degrees. So if we have this negative 20 degrees, we can do a full rotation all the way around to get to this number and we do that by adding 360 degrees to this. So coming over here I can add 360 degrees to negative 20, which would give us 340 degrees. So what we're going to end up with is 4, and that's gonna be 4 cis 340 degrees. That's just by adding 360 degrees to this 20 degrees to get a full rotation and end up in the same place. So this right here is going to be our solution to the problem, and that's how you can solve these types of situations where you have to divide complex numbers in polar form.

Given z 1 = 12 ( cos ⁡ 30 ° + i sin ⁡ 30 ° ) z_1=12\left(\cos30\degree+i\sin30\degree\right) and z 2 = 3 ( cos ⁡ 50 ° + i sin ⁡ 50 ° ) z_2=3\left(\cos50\degree+i\sin50\degree\right) , find the quotient z 1 z 2 \frac{z_1}{z_2} .

z 1 z 2 = 4 C i S ( 340 ° ) \frac{z_1}{z_2}=4CiS\left(340\degree\right)

z 1 z 2 = 4 C i S ( 20 ° ) \frac{z_1}{z_2}=4CiS\left(20\degree\right)

z 1 z 2 = 9 C i S ( 340 ° ) \frac{z_1}{z_2}=9CiS\left(340\degree\right)

z 1 z 2 = 9 C i S ( 20 ° ) \frac{z_1}{z_2}=9CiS\left(20\degree\right)

11. Graphing Complex Numbers

Products and Quotients of Complex Numbers

Trigonometry

11. Graphing Complex Numbers

Products and Quotients of Complex Numbers

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Multiple Choice

Given $z_{1} = 12 (\cos 30 ° + i \sin 30 °)$ and $z_{2} = 3 (\cos 50 ° + i \sin 50 °)$ , find the quotient $the fraction with numerator z sub 1 and denominator z sub 2$ .

$\frac{z_{1}}{z_{2}} = 4 C i S (20 °)$

$\frac{z_{1}}{z_{2}} = 9 C i S (340 °)$

$\frac{z_{1}}{z_{2}} = 9 C i S (20 °)$

$\frac{z_{1}}{z_{2}} = 4 C i S (340 °)$

Verified step by step guidance

Identify the given complex numbers in polar form: $ z_1 = 12(\cos 30\degree + i\sin 30\degree) $ and $ z_2 = 3(\cos 50\degree + i\sin 50\degree) $.

Recall the formula for dividing two complex numbers in polar form: $ \frac{z_1}{z_2} = \frac{r_1}{r_2} \text{cis}(\theta_1 - \theta_2) $, where $ r_1 $ and $ r_2 $ are the magnitudes, and $ \theta_1 $ and $ \theta_2 $ are the angles.

Calculate the magnitude of the quotient: $ \frac{r_1}{r_2} = \frac{12}{3} = 4 $.

Determine the angle of the quotient: $ \theta_1 - \theta_2 = 30\degree - 50\degree = -20\degree $. Since angles in polar form are typically expressed as positive, convert $-20\degree$ to a positive angle by adding 360\degree, resulting in $ 340\degree $.

Express the quotient in polar form: $ \frac{z_1}{z_2} = 4 \text{cis}(340\degree) $.

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