SOLUTION: Can somebody fill me in a bit on Euler’s formula? e^(a + bi) = e^a*(cos b + i sin b) -- Euler's Formula. His equation gives rise to e^(πi) + 1 = 0. '0, 1,π , e,

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Question 1075983: Can somebody fill me in a bit on Euler’s formula?
e^(a + bi) = e^a*(cos b + i sin b) -- Euler's Formula.
His equation gives rise to e^(πi) + 1 = 0.
'0, 1,π , e, and i' Are all in one equation.
Could you explain how Euler’s formula could be applied to derive this equation? Thank you very much.

Found 3 solutions by Edwin McCravy, Fombitz, math_helper:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Just substitute a=0 and b=p, 
like I told you before.  Also notice that in your post
you needed to put ^ to indicate exponents, not just
write them next to the base, for that shows multiplication
only, not exponentiation.  Also you must enclose the
exponents in parentheses.

e%5E%28a+%2B+bi%29+=+e%5Ea%2A%28cos%28b%29%5E%22%22+%2B+isin%28b%29%29

e%5E%280+%2B+%28pi%29i%29+=+e%5E0%28cos%28pi%29%5E%22%22+%2B+isin%28pi%29%29

Now we use the facts that e%5E0=1, cos%28pi%29=-1, and sin%28pi%29=0:

e%5E%28pi%2Ai%29+=+1%28%28-1%29%5E%22%22+%2B+i%280%29%29

e%5E%28pi%2Ai%29+=+-1

Add 1 to both sides:

e%5E%28pi%2Ai%29%2B1=0

Edwin


Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
e%5E%28a%2Bbi%29=e%5Ea%2Ae%5E%28bi%29
OK so when a=0 and b=pi,
e%5E%280%2Bpi%2Ai%29=e%5E0%2A%28cos%28pi%29%2Bi%2Asin%28pi%29%29
e%5E%28pi%2Ai%29=1%28-1%2Bi%2A0%29
e%5E%28pi%2Ai%29=-1
e%5E%28pi%2Ai%29%2B1=0

Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!

Let a=0 so e%5Ea+=+e%5E0+=+1
Let b=pi so e%5E%28bi%29+=+e%5E%28%28pi%29i%29+=+cos%28pi%29+%2B+i%2Asin%28pi%29+=+-1+%2B+0%2Ai+=+-1+
Put the two together and the result is the famous Euler's Identity:
e%5E%28%28pi%29i%29+%2B+1+=+0+