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Chapter 7: Problem 99

Explain how to find the quotient of two complex numbers in polar form.

Short Answer

Expert verified
To find the quotient of two complex numbers in polar form, one would divide the magnitudes of the two numbers and subtract their angles. If the result is needed in rectangular form, a conversion is made back from polar form.

Step by step solution

01

Convert to Polar Form

If the complex numbers are not already in polar form, convert them. The polar form of a complex number is \(r(\cos(\theta) +i\sin(\theta))\), where \(r\) is the absolute value or modulus of the complex number and \( \theta \) is the argument or phase.
02

Divide the Magnitudes

Once the complex numbers are in polar form, divide the magnitude of the first complex number by the magnitude of the second. In other words, if the first complex number has magnitude \(r_1\) and the second has magnitude \(r_2\), the quotient of their magnitudes is \(r_1/r_2\). This is the magnitude of the resulting complex number.
03

Subtract the Angles

Subtract the angle of the second complex number from the angle of the first to get the angle of the resulting complex number. If \( \theta_1 \) is the angle of the first complex number and \( \theta_2 \) is the angle of the second, the angle of the resulting complex number is \( \theta_1 - \theta_2 \).
04

Convert Back to Rectangular Form

If you need the resulting number in rectangular form, convert back using the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\), where \(x\) and \(y\) are the real and imaginary parts of the complex number, respectively.

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