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Dividing Complex Numbers - Problem 1
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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