Polar Representation of Complex NumbersDate: 11/04/2002 at 23:48:20 From: Christie Shroyer Subject: Pre-Calculus - Complex numbers A) The polar representation of a complex number z=a+ib is z=(cos(x) + sin(x)). Express the quantities r and x in terms of a and b. B) Represent the quantities r and x geometrically. C) Give two examples of complex numbers and find their corresponding polar representation. D) What is/are the value(s) of x for a number that is purely real. E) What is/are the value(s) of x for a number that has real part equal to zero. Date: 11/05/2002 at 12:44:21 From: Doctor Peterson Subject: Re: Pre-Calculus - Complex numbers Hi, Christie. You copied something wrong in the first question; it should be z = r[cos(x) + i sin(x)] What you are calling x, more often denoted by the Greek letter theta, is the angle from the positive x axis to the number z; r is the distance from the origin to z: | | z b+------+ | /| | / | | r/ | | / | | / | |/ x | -------------------+------+------------- | a | | | | | | | | You can use trigonometry to show that, as indicated in the problem, the real part a will be r cos(x) and the imaginary part b will be r sin(x). Similarly, you can find r and x from the picture, using the Pythagorean theorem and the definition of a trig function. You can pick any numbers you want for part (c); try 1+i and 1-i for starters, and then try one where x is 60 degrees. A purely real number, with b=0, will lie on the x-axis (positive or negative). What will the angle x be in my picture? (There are two answers.) A purely imaginary number, with a=0, will lie on the y-axis. What will the angle be? If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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