Given z1=15(cosπ2+isinπ2)z_1=\frac15\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right) and z2=5(cosπ5+isinπ5)z_2=5\left(\cos\frac{\pi}{5}+i\sin\frac{\pi}{5}\right), find the quotient z1z2\frac{z_1}{z_2}.

z 1 z 2 = 1 25 C i S ( 3 π 10 ) \frac{z_1}{z_2}=\frac{1}{25}CiS\left(\frac{3\pi}{10}\right)

Given z 1 = 1 5 ( cos π 2 + i sin π 2 ) z_1=\frac15\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right) and z 2 = 5 ( cos π 5 + i sin π 5 ) z_2=5\left(\cos\frac{\pi}{5}+i\sin\frac{\pi}{5}\right) , find the quotient z 1 z 2 \frac{z_1}{z_2} .

in this problem, we are given these 2 complex numbers, z 1 and z 2, and we're asked to find the quotient z 1 divided by z 2. Now we've talked about how to do quotients, and what you first wanna do, if you wish to find what the solution is, is you wanna start by dividing the r values, and the r values are these numbers out in front here. So if we go to this fraction, we're going to have r one, which is 1 over 5, divided by r two, which is 5. So that's what's gonna go in front of this. Now what I can do is take these cosines and these I sines here, and I can abbreviate this to be cis. So we're going to have cis, the first angle minus the second angle. Now I can see the first angle here is pi over 2 radians, and I can see the second angle is pi over 5 radians. So we're gonna subtract these two angles, because we're doing division here, so you want to subtract the angles. Now what I need to do from here is simplify, and to simplify this fraction here, well, 5 is the same thing as 5 over 1. And so I can actually flip this fraction bring it to the top, meaning we'll have 1 5th times 1 5th, which is the same thing as 125th. And then we're going to have that, and it's gonna be cis, and we need to get common denominators here. Now what the common denominator between 25 is, is 10. So I can multiply the top and bottom of this fraction by 5 to get 5 pi over 10. And and then I can multiply the top and bottom of this fraction by 2 to get minus 2 pi over 10. Notice we have the same denominator, so I just need to subtract the numerators. So we're going to end up having is 1 25th as the r value out in front the fraction of the 2 r values divided, then we're gonna have cis, and we're gonna have 5 pi minus 2 pi, which is gonna be 3 pi over 10. And this right here would be the solution to the problem, and that's how you can find z 1 divided by z 2. So hope you found this video helpful. Thanks for watching.

Given z 1 = 1 5 ( cos ⁡ π 2 + i sin ⁡ π 2 ) z_1=\frac15\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right) and z 2 = 5 ( cos ⁡ π 5 + i sin ⁡ π 5 ) z_2=5\left(\cos\frac{\pi}{5}+i\sin\frac{\pi}{5}\right) , find the quotient z 1 z 2 \frac{z_1}{z_2} .

z 1 z 2 = 1 25 C i S ( 3 π 10 ) \frac{z_1}{z_2}=\frac{1}{25}CiS\left(\frac{3\pi}{10}\right)

z 1 z 2 = 1 25 C i S ( 5 π 2 ) \frac{z_1}{z_2}=\frac{1}{25}CiS\left(\frac{5\pi}{2}\right)

z 1 z 2 = − 24 5 C i S ( 3 π 10 ) \frac{z_1}{z_2}=-\frac{24}{5}CiS\left(\frac{3\pi}{10}\right)

z 1 z 2 = − 24 5 C i S ( 5 2 ) \frac{z_1}{z_2}=-\frac{24}{5}CiS\left(\frac52\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 $z_{1} = \frac{1}{5} (\cos \frac{π}{2} + i \sin \frac{π}{2})$ and $z_{2} = 5 (\cos \frac{π}{5} + i \sin \frac{π}{5})$ , find the quotient $the fraction with numerator z sub 1 and denominator z sub 2$ .

$\frac{z_{1}}{z_{2}} = \frac{1}{25} C i S (\frac{3 π}{10})$

$\frac{z_{1}}{z_{2}} = \frac{1}{25} C i S (\frac{5 π}{2})$

$\frac{z_{1}}{z_{2}} = - \frac{24}{5} C i S (\frac{3 π}{10})$

$\frac{z_{1}}{z_{2}} = - \frac{24}{5} C i S (\frac{5}{2})$

Verified step by step guidance

Identify the given complex numbers in polar form: $ z_1 = \frac{1}{5} \left( \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} \right) $ and $ z_2 = 5 \left( \cos\frac{\pi}{5} + i\sin\frac{\pi}{5} \right) $.

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 magnitudes: $ r_1 = \frac{1}{5} $ and $ r_2 = 5 $. Therefore, $ \frac{r_1}{r_2} = \frac{1}{5} \times \frac{1}{5} = \frac{1}{25} $.

Determine the angles: $ \theta_1 = \frac{\pi}{2} $ and $ \theta_2 = \frac{\pi}{5} $. Subtract the angles: $ \theta_1 - \theta_2 = \frac{\pi}{2} - \frac{\pi}{5} = \frac{5\pi}{10} - \frac{2\pi}{10} = \frac{3\pi}{10} $.

Combine the results to express the quotient in polar form: $ \frac{z_1}{z_2} = \frac{1}{25} \text{cis}\left(\frac{3\pi}{10}\right) $.

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