Find z1 z2 and (z1)/(z2). z1=3 √(2)(cos(5 π)/(4)+i sin(5 π)/(4)), z2=2(cos(5 π)/(3)+i sin(5 π)/(

Name: Problem 20, Chapter 10: Higher Level Mathematics
Uploaded: 2021-12-13T09:24:46-08:00
Duration: 4 min 40 s
Channel: Varsha Aggarwal
Description: Find z1 z2 and (z1)/(z2). z1=3 √(2)(cos(5 π)/(4)+i sin(5 π)/(4)), z2=2(cos(5 π)/(3)+i sin(5 π)/(3))

step 1 In this question we have to find z1 into z2 and z 1 by z 2. So we can write z 1 equal 3 root 2 i

Question

Find $z_{1} z_{2}$ and $\frac{z_{1}}{z_{2}}$. $$z_{1}=3 \sqrt{2}\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right), z_{2}=2\left(\cos \frac{5 \pi}{3}+i \sin \frac{5 \pi}{3}\right)$$

   Find $z_{1} z_{2}$ and $\frac{z_{1}}{z_{2}}$.
$$z_{1}=3 \sqrt{2}\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right), z_{2}=2\left(\cos \frac{5 \pi}{3}+i \sin \frac{5 \pi}{3}\right)$$

Higher Level Mathematics

Ibrahim Wazir, Tim… 2012 Edition

Chapter 10, Problem 20

Instant Answer

Solved by Expert Varsha Aggarwal

12/13/2021

Step 1

We can use the formula for the product of two complex numbers in polar form, which is $r_{1}r_{2}(\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2}))$. Here, $r_{1}$ and $r_{2}$ are the magnitudes of the complex numbers and $\theta_{1}$ and $\theta_{2}$ are Show more…

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Find $z_{1} z_{2}$ and $\frac{z_{1}}{z_{2}}$. $$z_{1}=3 \sqrt{2}\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right), z_{2}=2\left(\cos \frac{5 \pi}{3}+i \sin \frac{5 \pi}{3}\right)$$

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Key Concepts

Complex Numbers in Polar Form

Complex numbers can be represented in the form r(cos? + i sin?), where r is the magnitude and ? is the argument. This form is especially useful for performing multiplication and division, as it simplifies the operations by addressing the modulus and angle separately.

Multiplication of Complex Numbers in Polar Form

When multiplying two complex numbers expressed in polar form, the magnitudes multiply and the angles add. This process reflects a scaling of the distance from the origin and a rotation by the sum of the angles, which is fundamental in complex number arithmetic.

Division of Complex Numbers in Polar Form

Dividing complex numbers in polar form involves dividing the magnitudes and subtracting the angle of the divisor from the angle of the dividend. This method provides an intuitive understanding of the division process as a combination of scaling and rotation by the difference in angles.

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