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Dividing Complex Numbers - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

We're dividing complex numbers in trigonometric from. Here I have a problem that asks me to divide 12 times cosine of 50 degrees plus i sine 50, divided by 2 cosine 5 degrees plus i sine 5 degrees.

Now remember, when you divide complex numbers in trig form, you divide the moduli, and you subtract the arguments. So dividing the moduli 12 divided by 2, I get 6. Then you subtract the arguments; 50 minus 5, so I get cosine of 45 degrees plus i sine 45 degrees.

Now the problem asks for me to write the final answer in rectangular form. So I have to multiply this out. I get 6 times, cosine of 45 is root 2 over 2, and the sine of 45 is the same, root 2 over 2. So I'm going to get 6 times root 2 over 2, 3 root 2, plus i times 3 root 2. That's my rectangular form for the quotient of these two numbers.

Let's do another example. 20 cosine 5 pi over 3 plus i sine 5 pi over 3, divided by, 5 cosine 2 pi over 3 plus i sine 2 pi over 3. Again, first we divide the moduli. 20 divided by 5, and we get 4. Then we subtract the arguments; 5 pi over 3 minus 2 pi over 3, is 3 pi over 3, or just pi. Cosine pi plus i sine pi. Cosine pi is -1, so this is 4 times -1 plus i sine times (sine of pi is 0), i times 0. This is just going to be -4. So this quotient simplifies to -4 very nice in rectangular form.

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