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Multiplying 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 talking about multiplying complex numbers in trigonometric form. I have two products written here. Let's simplify these products.
First we have 2 cosine 24 degrees plus i sine 24 degrees, times 5 cosine 36 degrees plus i sine 36 degrees. Now when you multiply complex numbers in trig form, you multiply their moduli; 2 and 5. So we get 10, and then you add their arguments. So I'll get cosine of 24 plus 36, which is 60 degrees. Sine of 24 plus 36, 60 degrees.
Now cosine of 60 is 1/2, so this is 10 times 1/2 plus i times the sine of 60, which is root 3 over 2. So this will simplify very nicely. We get 10 times 1/2, 5, plus 10 times root 3 over 2, 5 root 3. I'll put i times 5 root 3. That's the simplified form.
Now 9 times cosine 3 pi over 5 plus i sine 3 pi over 5, times 2 cosine 3 pi over 20, plus i sine 3 pi over 20. Same rules here, we're multiplying. So we first multiply the moduli, and we get 18, 9 times 2. Then we have cosine of 3 pi over 5 plus 3 pi over 20. So I have to add those 2, plus i sine 3 pi over 5 plus 3 pi over 20. I have to think about that.
In order to add these, I'm going to want to get a common denominator. The common denominator I want is 20. So I'm going to multiply the top and bottom of this fraction by 4, and get 12 pi over 20 plus 3 pi over 20. That gives me 15 pi over 20, which reduces to 3 pi over 4. So this is all going to be 18 cosine 3 pi over 4 plus i sine 3 pi over 4.
What's the cosine of 3 pi over 4? It's in the second quadrant. It's going to be negative root 2 over 2. Negative root 2 over 2 plus i times, and the sine of 3 pi over 4 is positive root 2 over 2. So 18 times negative root 2 over. 2 and 18 will cancel leaving a 9. I get -9 root 2. 18 times root 2 over 2 again the 18, and 2 cancel leaving a 9. So I get plus i times 9 root 2. That's my simplified answer in rectangular form.
So just remember when you're multiplying complex numbers in trig form, multiply the moduli, and add the arguments.
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