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Chapter 6: Problem 37

Find the product of the complex numbers. Leave answers in polar form. $$\begin{array}{l} z_{1}=6\left(\cos 20^{\circ}+i \sin 20^{\circ}\right) \\ z_{2}=5\left(\cos 50^{\circ}+i \sin 50^{\circ}\right) \end{array}$$

Short Answer

Expert verified
The product of the given complex numbers \(z_{1}\) and \(z_{2}\) is: \[ z_{1} \cdot z_{2} = 30(\cos 70^{\circ} + i\sin 70^{\circ}) \]

Step by step solution

01

Write out the complex numbers

The provided complex numbers are: \[z_{1}=6\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)\] and \[z_{2}=5\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)\] These complex numbers are represented in module-argument(Polar) form \(r(\cos \theta + i\sin \theta)\). Where \(r\) is the modulus and \(\theta\) is argument.
02

Multiply the moduli and add the arguments using the product rule of complex numbers in Polar form

The product rule of complex numbers in polar form states: For any complex numbers \(z_{1}\) and \(z_{2}\), \(z_{1} * z_{2} = r_{1}r_{2}\left(\cos(\theta_{1} + \theta_{2}) + i\sin(\theta_{1} + \theta_{2})\right)\), where \(r_{1}\) and \(r_{2}\) are the moduli and \(\theta_{1}\) and \(\theta_{2}\) are the arguments of \(z_{1}\) and \(z_{2}\) respectively. Here, the moduli are \(r_{1} = 6\) and \(r_{2} = 5\) and the arguments \(\theta_{1} = 20^{\circ}\) and \(\theta_{2} = 50^{\circ}\). Thus multiplying the moduli gives us \(6*5=30\) and adding the arguments we have \(20^{\circ} + 50^{\circ} = 70^{\circ}\).
03

Write the result

The solution will be the modulus, which is the product of the individual moduli, and the argument, which is the sum of the individual arguments. So our result is: \[ z_{1} \cdot z_{2} = 30(\cos 70^{\circ} + i\sin 70^{\circ}) \]

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Most popular questions from this chapter

Find the angle, in degrees, between \(\mathbf{v}\) and \(\mathbf{w}.\) $$\mathbf{v}=3 \cos \frac{5 \pi}{3} \mathbf{i}+3 \sin \frac{5 \pi}{3} \mathbf{j}, \quad \mathbf{w}=2 \cos \pi \mathbf{i}+2 \sin \pi \mathbf{j}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j}$$

Use the vectors $$\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad \text { and } \quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j},$$ to prove the given property. $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=2 \mathbf{i}-2 \mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j}$$

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$r=\sin ^{5} \theta+8 \sin \theta \cos ^{3} \theta$$
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