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Converting Complex Numbers From Trigonometric Form to Rectangular - Problem 2 2,130 views

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.

I'm converting numbers from trigonometric form to rectangular form. Here I have a complex number 5 root 2, times the quantity cosine 7 pi over 4, plus i sine 7 pi over 4.

Just to refresh you memory, what does this mean? If I were to graph this number, the 5 root 2 is the distance of the number from the origin. The 7 pi over 4 is the argument of the number, which is its angle from the positive real axis, this axis. So 7 pi over 4 is all the way over here, leaving a remainder of pi over 4.

So this is going to be our point 7 pi over 4, and this length is 5 root 2. That's our complex number. Now let's convert it to rectangular form. I'll distribute the 5 root 2. I get 5 root 2 cosine 7 pi over 4, plus i times 5 root 2, sine 7 pi over 4. Now cosine 7 pi over 4 is root 2 over 2. So this is 5 root 2, times root 2 over 2 plus, now sine of 7 pi over 4 is negative root 2 over 2. So 5 root 2 times negative root 2 over 2. Now here root 2 times root 2 is 2, divided by 2 is 1. So this whole thing is just going to be 5.

Then over here, the same thing. This is really the same product only there's a minus sign. So I'll get minus 5i. Z equals 5 minus 5i. So those would be the coordinates of this point, 5 minus 5i. That means it's 5 units to the right, and 5 down. That's the rectangular form of z.