Converting Complex Numbers to Polar Form: A Detailed Guide

Understanding how to convert complex numbers to polar form is a fundamental skill in various fields such as engineering, physics, and signal processing. One such example is the conversion of the complex number Z i - frac{1}{cos(pi/3) sin(pi/3)} to its polar form. This process involves understanding complex arithmetic, trigonometric identities, and the properties of complex numbers in polar format.

Step-by-Step Conversion

Let's start by rewriting the given complex number in a standard form to facilitate the conversion to polar form.

Step 1: Standard Form Conversion

The given number is:

[Z i - frac{1}{cos(pi/3) sin(pi/3)}]

Using the trigonometric identities, we know that (cos(pi/3) frac{1}{2}) and (sin(pi/3) frac{sqrt{3}}{2}). Therefore, we can rewrite the denominator as:

[cos(pi/3) sin(pi/3) frac{1}{2} cdot frac{sqrt{3}}{2} frac{sqrt{3}}{4}]

Thus, the expression becomes:

[Z i - frac{1}{frac{sqrt{3}}{4}} i - frac{4}{sqrt{3}}]

Expressing this in the standard form (Z a bi), we get:

[Z -frac{4}{sqrt{3}} i]

Step 2: Polar Form Conversion

To convert this to polar form, we need to find the magnitude (r) and the argument (alpha).

The magnitude (r) is given by:

[r sqrt{a^2 b^2} sqrt{left(-frac{4}{sqrt{3}}right)^2 1^2} sqrt{frac{16}{3} 1} sqrt{frac{16 3}{3}} sqrt{frac{19}{3}}]

The argument (alpha) is given by:

[alpha tan^{-1}left(frac{b}{a}right) tan^{-1}left(frac{1}{-frac{4}{sqrt{3}}}right) tan^{-1}left(-frac{sqrt{3}}{4}right)]

Using the rule that the argument of a complex number (a bi) in the fourth quadrant is calculated correctly by (alpha -tan^{-1}(frac{sqrt{3}}{4})), we can write:

[Z sqrt{frac{19}{3}} left(cosalpha isinalpharight)]

Using the fact that (alpha -tan^{-1}(frac{sqrt{3}}{4})), the polar form can be written as:

[Z sqrt{frac{19}{3}} e^{ialpha}]

Where (alpha -tan^{-1}(frac{sqrt{3}}{4})).

A Rule for Dividing Complex Numbers in Polar Form

When two complex numbers (Z r_1 text{cis} theta_1) and (W r_2 text{cis} theta_2) are given, their division can be simplified using the following rule:

Step 3: Simplification Using the Rule

For the complex number (Z i - frac{1}{cos(pi/3) sin(pi/3)}), we can rewrite it as:

[Z 1 - i]

The modulus and argument of (1 - i) are:

[r sqrt{2}, quad theta -frac{pi}{4}]

Therefore, the polar form is:

[1 - i sqrt{2} e^{-ipi/4}]

The given denominator (cos(pi/3) i sin(pi/3) e^{ipi/3}) simplifies to:

[cos(pi/3) i sin(pi/3) e^{ipi/3}]

The division of (1 - i) by (cos(pi/3) i sin(pi/3)) in polar form becomes:

[frac{1 - i}{cos(pi/3) i sin(pi/3)} frac{sqrt{2} e^{-ipi/4}}{e^{ipi/3}} sqrt{2} e^{-ipi/4 - ipi/3} sqrt{2} e^{-i(7pi/12)}]

This can be written in the standard polar form:

[sqrt{2} left(cos(-7pi/12) i sin(-7pi/12)right)]

Conclusion

Converting complex numbers to polar form is not only a theoretical exercise but also a practical tool in solving complex problems. The steps detailed above provide a clear and systematic way to perform such conversions. Understanding this process helps in simplifying complex number operations and solving problems in various mathematical and scientific domains.

The conversion of Z i - frac{1}{cos(pi/3) sin(pi/3)} to its polar form is a prime example of the utility of polar representation. By following the steps outlined, one can confidently convert any complex number from its standard form to its polar form, using fundamental trigonometric identities and properties.