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Chapter 6: Problem 10

Write $$ \left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right)\left(\cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\right) $$ in polar form.

Short Answer

Expert verified
The product of the given complex numbers in polar form is: \[ e^{i\left(\frac{\pi}{7} + \frac{\pi}{9}\right)} \]

Step by step solution

01

Identify the two complex numbers in polar form

We have the two complex numbers in polar form given as: \(A = \cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\) and \(B = \cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\)
02

Identify r and 胃 for each complex number

For complex number A: \[r_1 = 1\] and \[\theta_1 = \frac{\pi}{7}\] For complex number B: \[r_2 = 1\] and \[\theta_2 = \frac{\pi}{9}\]
03

Multiply the complex numbers in polar form

Using the formula for the multiplication of complex numbers in polar form: \(re^{i\theta} \cdot re^{i\phi} = r_1r_2e^{i(\theta + \phi)}\) We have: \(r_1r_2 = 1 \cdot 1 = 1\) And for the arguments: \(\theta_1 + \theta_2 = \frac{\pi}{7} + \frac{\pi}{9}\)
04

Write the result

With the product of the magnitudes and the sum of the arguments, the result in polar form is: \[ \left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right)\left(\cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\right) = e^{i\left(\frac{\pi}{7} + \frac{\pi}{9}\right)} \]

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Most popular questions from this chapter

Show that if \(\mathbf{u}\) and \(\mathbf{v}\) are vectors and \(t\) is a real num- ber, then $$ (t \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(t \mathbf{v})=t(\mathbf{u} \cdot \mathbf{v}) $$

Describe the subset of the complex plane consisting of the complex numbers \(z\) such that \(z^{3}\) is a real number.

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (5+7 i)+(4+6 i) $$

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (4-3 i)(2-6 i) $$

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