Given z1=5(cosπ6+isinπ6)z_1=5\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right) and z2=3(cos3π4+isin3π4)z_2=3\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right), find the product z1・z2z_1・z_2.

z 1 ・ z 2 = 15 C i S ( 11 π 12 ) z_1・z_2=15CiS\left(\frac{11\pi}{12}\right)

Given z 1 = 5 ( cos π 6 + i sin π 6 ) z_1=5\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right) and z 2 = 3 ( cos 3 π 4 + i sin 3 π 4 ) z_2=3\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right) , find the product z 1 ・ z 2 z_1・z_2 .

this problem, we are given 2 complex numbers. We are given z 1, and we are given z 2, both in polar form. And we're asked to find the product of these two complex numbers. Now in order to find the product, what I'm first going to do is multiply the r values which are out in front. Now as I can see, we have this r value of 5 and that r value of 3. 5 times 3 is 15. Now recall that when dealing with this, we can take this these cosines and sines here, and we can abbreviate this all to be cis. So we'll have 15 cis and then we want to add the 2 angles together. So we have pi over 6 as angle 1, and we're going to add that to 3 pi over 4 radians as angle 2. So we're dealing with radians in this problem. Now from here, all I need to do is simplify everything that's within the parentheses. So we're going to have 15 on the outside. That's the r values being multiplied, and then this is going to be cis. And what I need to do is get common denominators for each of these fractions. Now to do it for this fraction, I can see that the lowest common denominator is gonna be 12 between 64. So you can take the top and bottom of this fraction and multiply it by 2, and I can take the top and bottom of this fraction, and I can multiply it by 3. So 2 times pi is 2 pi, and then that's over 2 times 6, which is 12. Then we'll have plus 3 times 3 which then have 9 pi on top, and that's divided by 3 times 4 which is 12. Notice we now have common denominators, so we can add the numerators. So adding the numerators, we're going to have 2 pi plus 9 pi, which is 11 pi. So we'll end up having 15, and that's going to be cis 11pi over 12, and that right there is going to be z 1 times z 2 and the solution to this problem. So that is how you can multiply these complex numbers in polar form. Hope you found this video helpful.

Given z 1 = 5 ( cos ⁡ π 6 + i sin ⁡ π 6 ) z_1=5\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right) and z 2 = 3 ( cos ⁡ 3 π 4 + i sin ⁡ 3 π 4 ) z_2=3\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right) , find the product z 1 ・ z 2 z_1・z_2 .

z 1 ・ z 2 = 15 C i S ( 11 π 12 ) z_1・z_2=15CiS\left(\frac{11\pi}{12}\right)

z 1 ・ z 2 = 15 C i S ( 11 π 2 12 ) z_1・z_2=15CiS\left(\frac{11\pi^2}{12}\right)

z 1 ・ z 2 = 15 C i S ( π 2 ) z_1・z_2=15CiS\left(\frac{\pi}{2}\right)

z 1 ・ z 2 = C i S ( π 2 2 ) z_1・z_2=CiS\left(\frac{\pi^2}{2}\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 $z_{1} = 5 (\cos \frac{π}{6} + i \sin \frac{π}{6})$ and $z_{2} = 3 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})$ , find the product $z sub 1 ・ z sub 2$ .

$z_{1} ・ z_{2} = 15 C i S (\frac{11 π^{2}}{12})$

$z_{1} ・ z_{2} = 15 C i S (\frac{π}{2})$

$z_{1} ・ z_{2} = C i S (\frac{π^{2}}{2})$

$z_{1} ・ z_{2} = 15 C i S (\frac{11 π}{12})$

Verified step by step guidance

Start by understanding that the given complex numbers are in polar form: $ z_1 = 5 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) $ and $ z_2 = 3 \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) $.

Recall the formula for multiplying two complex numbers in polar form: $ z_1 \cdot z_2 = r_1 r_2 \left( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right) $.

Apply the formula: Multiply the magnitudes $ r_1 = 5 $ and $ r_2 = 3 $ to get the magnitude of the product, which is $ 15 $.

Add the angles: $ \theta_1 = \frac{\pi}{6} $ and $ \theta_2 = \frac{3\pi}{4} $. The sum is $ \frac{\pi}{6} + \frac{3\pi}{4} = \frac{11\pi}{12} $.

Combine the results to express the product in polar form: $ z_1 \cdot z_2 = 15 \left( \cos \frac{11\pi}{12} + i \sin \frac{11\pi}{12} \right) $.

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