Expert: Sherry Wallin - 4/12/2010

Question
I am trying to understand why e^ipi=cos(pi)+isin(pi)...I get that cos pi=-1, and that isin pi=0 (obviously as the sin or y-value of 180 degrees=zero- and that this has something to do with the complex number plane)- but I can't seem to understand the why of this specific relationship. Your help will be much appreciated. Regards, Dave

Answer
Dave~
    You are right in everything you say. Euler's identity is e^ipi + 1 = 0 and it is derived from e^ipi = cos pi + isin pi = -1 + 0 = -1 -> e^ipi+1=0.

Your comment about the isin pi = 0 because of the complex number plane is not exactly correct because by definition e^ix = cos x + isin x for any REAL NUMBER x measured in radians. So really the complex plane is not involved when x = pi because you are on the real axis. Remember the y-axis is your imaginary axis. Granted there is an 'i' in isin pi but as you stated sin pi = 0 and i*0 = 0.

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