Question
Using mathematically precise language, explain in detail how you would multiply the complex number lower z subscript 1 baseline equals lower r subscript 1 baseline left parenthesis cosine theta subscript 1 baseline plus sine theta subscript 1 baseline right parenthesis with the complex number lower z subscript 2 baseline equals lower r subscript 2 baseline left parenthesis cosine theta subscript 2 baseline plus sine theta subscript 2 baseline right parenthesis.
Answers
GPT-4o mini
To multiply two complex numbers given in polar form, we will follow the conventions of polar representation and the properties of complex number multiplication.
Let \( z_1 \) and \( z_2 \) be defined as follows:
\[
z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)
\]
\[
z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)
\]
Here, \( r_1 \) and \( r_2 \) are the magnitudes (or moduli) of the respective complex numbers, while \( \theta_1 \) and \( \theta_2 \) are the arguments (or angles) of those complex numbers.
To multiply the two complex numbers, we will use the fact that the product of two complex numbers in polar form can be calculated using the following properties:
1. The magnitudes of the product of two complex numbers is the product of their magnitudes. That is,
\[
|z_1 z_2| = |z_1| \cdot |z_2| = r_1 \cdot r_2.
\]
2. The argument of the product of two complex numbers is the sum of their arguments. That is,
\[
\arg(z_1 z_2) = \arg(z_1) + \arg(z_2) = \theta_1 + \theta_2.
\]
Now, applying these properties, we calculate the product \( z_1 z_2 \):
\[
z_1 z_2 = (r_1 (\cos \theta_1 + i \sin \theta_1))(r_2 (\cos \theta_2 + i \sin \theta_2)).
\]
Distributing the product gives:
\[
z_1 z_2 = r_1 r_2 \left( (\cos \theta_1 + i \sin \theta_1)(\cos \theta_2 + i \sin \theta_2) \right).
\]
Next, we apply the formula for the multiplication of two complex numbers expressed in rectangular form:
\[
(\cos \theta_1 + i \sin \theta_1)(\cos \theta_2 + i \sin \theta_2) = \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 + i (\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2).
\]
We can simplify this using the angle addition formulas for sine and cosine:
1. \(\cos(\theta_1 + \theta_2) = \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2\)
2. \(\sin(\theta_1 + \theta_2) = \sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2\)
Thus, we rewrite the expression for the product \( z_1 z_2 \) as:
\[
z_1 z_2 = r_1 r_2 \left( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right).
\]
Finally, we can express \( z_1 z_2 \) in polar form:
\[
z_1 z_2 = r_1 r_2 \left( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right).
\]
In summary, the product of two complex numbers in polar form, \( z_1 \) and \( z_2 \), results in a complex number whose magnitude is the product of the magnitudes of the two numbers, and whose argument is the sum of the arguments of the two numbers. Therefore, we have:
\[
z_1 z_2 = r_1 r_2 \operatorname{cis}(\theta_1 + \theta_2),
\]
where \( \operatorname{cis}(\theta) = \cos \theta + i \sin \theta \).
Let \( z_1 \) and \( z_2 \) be defined as follows:
\[
z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)
\]
\[
z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)
\]
Here, \( r_1 \) and \( r_2 \) are the magnitudes (or moduli) of the respective complex numbers, while \( \theta_1 \) and \( \theta_2 \) are the arguments (or angles) of those complex numbers.
To multiply the two complex numbers, we will use the fact that the product of two complex numbers in polar form can be calculated using the following properties:
1. The magnitudes of the product of two complex numbers is the product of their magnitudes. That is,
\[
|z_1 z_2| = |z_1| \cdot |z_2| = r_1 \cdot r_2.
\]
2. The argument of the product of two complex numbers is the sum of their arguments. That is,
\[
\arg(z_1 z_2) = \arg(z_1) + \arg(z_2) = \theta_1 + \theta_2.
\]
Now, applying these properties, we calculate the product \( z_1 z_2 \):
\[
z_1 z_2 = (r_1 (\cos \theta_1 + i \sin \theta_1))(r_2 (\cos \theta_2 + i \sin \theta_2)).
\]
Distributing the product gives:
\[
z_1 z_2 = r_1 r_2 \left( (\cos \theta_1 + i \sin \theta_1)(\cos \theta_2 + i \sin \theta_2) \right).
\]
Next, we apply the formula for the multiplication of two complex numbers expressed in rectangular form:
\[
(\cos \theta_1 + i \sin \theta_1)(\cos \theta_2 + i \sin \theta_2) = \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 + i (\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2).
\]
We can simplify this using the angle addition formulas for sine and cosine:
1. \(\cos(\theta_1 + \theta_2) = \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2\)
2. \(\sin(\theta_1 + \theta_2) = \sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2\)
Thus, we rewrite the expression for the product \( z_1 z_2 \) as:
\[
z_1 z_2 = r_1 r_2 \left( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right).
\]
Finally, we can express \( z_1 z_2 \) in polar form:
\[
z_1 z_2 = r_1 r_2 \left( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right).
\]
In summary, the product of two complex numbers in polar form, \( z_1 \) and \( z_2 \), results in a complex number whose magnitude is the product of the magnitudes of the two numbers, and whose argument is the sum of the arguments of the two numbers. Therefore, we have:
\[
z_1 z_2 = r_1 r_2 \operatorname{cis}(\theta_1 + \theta_2),
\]
where \( \operatorname{cis}(\theta) = \cos \theta + i \sin \theta \).