Given z1=23(cos25°+isin25°)z_1=\frac23\left(\cos25\degree+i\sin25\degree\right)z1=32(cos25°+isin25°) and z2=52(cos15°+isin15°)z_2=\frac52\left(\cos15\degree+i\sin15\degree\right)z2=25(cos15°+isin15°), find the product z1・z2z_1・z_2z1・z2.

z 1 ・ z 2 = C i S ( 40 ° ) z_1・z_2=CiS\left(40\degree\right) z 1 ・ z 2 = C i S ( 40° )

Given z 1 = 2 3 ( cos 25 ° + i sin 25 ° ) z_1=\frac23\left(\cos25\degree+i\sin25\degree\right) z 1 = 3 2 ( cos 25° + i sin 25° ) and z 2 = 5 2 ( cos 15 ° + i sin 15 ° ) z_2=\frac52\left(\cos15\degree+i\sin15\degree\right) z 2 = 2 5 ( cos 15° + i sin 15° ) , find the product z 1 ・ z 2 z_1・z_2 z 1 ・ z 2 .

In this problem, we are given 2 complex numbers, z 1 and z 2. Both of these are in polar form, and we're asked to find their product. Now to find the product of z 1 and z 2, there are a couple steps that I need to take. The first step is going to be to take the r values out in front and multiply them. So you can see the r values are 2 thirds and 5 halves. So I'm going to have 2 thirds multiplied by 5 halves, that's multiplying the r values. Then what I need to do is I need to deal with these cosines and sines. Now recall that the cosine isine can be abbreviated to cis. So So we're going to have this multiplied by cis, and that's going to be the sum of the two angles. I can see that the first angle is 25 degrees. I can see that the second angle is 15 degrees. So this is what we end up getting. Now to solve the problem from here, well, all I really need to do is simplify everything that I have. I can simplify these 2 fractions by recognizing this 2 on top will cancel with that 2 on bottom. Meaning, all I'm gonna have is 5 thirds out in front. That's gonna be 5 thirds cis, and then we have 25 +15, which is 40 degrees. So this right here is the solution to z one times z 2, and that is how you can find the product of these two complex numbers. So hope you found this video helpful. Thanks for watching.

Given z 1 = 2 3 ( cos ⁡ 25 ° + i sin ⁡ 25 ° ) z_1=\frac23\left(\cos25\degree+i\sin25\degree\right) z 1 = 3 2 ( cos 25° + i sin 25° ) and z 2 = 5 2 ( cos ⁡ 15 ° + i sin ⁡ 15 ° ) z_2=\frac52\left(\cos15\degree+i\sin15\degree\right) z 2 = 2 5 ( cos 15° + i sin 15° ) , find the product z 1 ・ z 2 z_1・z_2 z 1 ・ z 2 .

z 1 ・ z 2 = C i S ( 40 ° ) z_1・z_2=CiS\left(40\degree\right) z 1 ・ z 2 = C i S ( 40° )

z 1 ・ z 2 = 5 3 C i S ( 375 ° ) z_1・z_2=\frac53CiS\left(375\degree\right) z 1 ・ z 2 = 3 5 C i S ( 375° )

z 1 ・ z 2 = 19 6 C i S ( 40 ° ) z_1・z_2=\frac{19}{6}CiS\left(40\degree\right) z 1 ・ z 2 = 6 19 C i S ( 40° )

z 1 ・ z 2 = 19 6 C i S ( 375 ° ) z_1・z_2=\frac{19}{6}CiS\left(375\degree\right) z 1 ・ z 2 = 6 19 C i S ( 375° )

17. Graphing Complex Numbers

Products and Quotients of Complex Numbers

Precalculus

17. Graphing Complex Numbers

Products and Quotients of Complex Numbers

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Multiple Choice

Given $z_1=\frac23\left(\cos25\degree+i\sin25\degree\right)$ and $z_2=\frac52\left(\cos15\degree+i\sin15\degree\right)$ , find the product $z_1・z_2$ .

$z_1・z_2=\frac53CiS\left(375\degree\right)$

$z_1・z_2=CiS\left(40\degree\right)$

$z_1・z_2=\frac{19}{6}CiS\left(40\degree\right)$

$z_1・z_2=\frac{19}{6}CiS\left(375\degree\right)$

Verified step by step guidance

Identify the given complex numbers in polar form: z1 = (2/3)(cos(25°) + i*sin(25°)) and z2 = (5/2)(cos(15°) + i*sin(15°)).

Recall the formula for multiplying two complex numbers in polar form: If z1 = r1*(cos(θ1) + i*sin(θ1)) and z2 = r2*(cos(θ2) + i*sin(θ2)), then z1 * z2 = (r1 * r2)(cos(θ1 + θ2) + i*sin(θ1 + θ2)).

Calculate the product of the magnitudes: r1 * r2 = (2/3) * (5/2).

Add the angles: θ1 + θ2 = 25° + 15°.

Express the product z1 * z2 in polar form using the calculated magnitude and angle: z1 * z2 = (r1 * r2)(cos(θ1 + θ2) + i*sin(θ1 + θ2)).

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