Polar Number Multiplication and DivisionDate: 02/13/99 at 13:53:42 From: Matt Malensek Subject: Polar Number Multiplication and Division I understand that when two polar numbers are multiplied, the real parts multiply and the angles add. I also understand that when two polar numbers are divided, the real parts divide and the angles subtract. However, I do not understand how these rules work. Why is it that when I multiply or divide polar numbers, the angles add or subtract? I tried to convert the polar numbers into complex numbers and then FOIL (First Outer Inner Last) them out. Then I took the complex answer and converted it back to polar. This seemed to show me that two arctangents produce a third arctangent. For multiplication of polar numbers: arctan(imag1/real1) + arctan(imag2/real2) = yet another arctangent For division of polar numbers: arctan(imag1/real1) - arctan(imag2/real2) = yet another arctangent This did not seem to prove anything. Is there a proof that will show me the validity of these two polar rules? Do I have the right idea with the arctangent approach? I think I am missing something. Thank you for your interest and help. Date: 02/16/99 at 20:03:21 From: Doctor Schwa Subject: Re: Polar Number Multiplication and Division On the contrary, this proved exactly what you wanted, since arctan(imag1/real1) = angle1 arctan(imag2/real2) = angle2 yet another, arctan = angle3 then you have proved that, for multiplication, angle1 + angle2 = angle3 which is just as it should be. Another way to do the proof is to write the polar numbers like this r1 (cos theta1 + i sin theta1) * r2 (cos theta2 + i sin theta2) Then after foiling, you can use some trigonometric identities to show that the answer is r1*r2 (cos (theta1 + theta2) + i sin (theta1 + theta2)), which proves what you needed. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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