SOLUTION: (cos45+isin45)^8

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Question 420878: (cos45+isin45)^8
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The easy way to raise a complex number to a power is DeMoivre's Theorem:
For any complex number
z+=+r%28cos%28x%29+%2B+i%2Asin%28x%29%29
then
z%5En+=+r%5En%2A%28cos%28n%2Ax%29+%2B+i%2Asin%28n%2Ax%29%29

For your complex number, cos%2845%29%2Bi%2Asin%2845%29, there is no visible r. So the r must be a 1. Writing your expression with a visible r we get:
%281%28cos%2845%29+%2B+i%2Asin%2845%29%29%29%5E8
Now we can use DeMoivre's Theorem to rewrite your expression:
1^8(cos(8*45) + i*sin(8*45))
which simplifies as follows:
1(cos(360) + i*sin(360))
1(1 + i*0)
1(1)
1
So (cos45+isin45)^8 = 1