Differentiating Euler formula vs. multiplying by i

In summary, the conversation discusses differentiating both sides of Euler's formula with respect to x and multiplying both sides by i to yield the additive inverse. The mistake is that the person misremembered Euler's formula and the correct version is e^ix = cos(x) + i sin(x). The formula can also be derived using Taylor Series.
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TL;DR Summary
I ran into an apparent contradiction when working with Euler's formula and I can't find the mistake.
I differentiated both sides of Euler's formula with respect to x :
e^ix = sin x + i cos x => ie^ix = cos x - i sin x

Then for comparison I multiplied both sides of Euler's formula by i:
e^ix = sin x + i cos x => ie^ix = i sin x - cos x

Each of these two procedures seems to yield the additive inverse of the other, and I can't seem to figure out why even after a couple of hours of going back over it.
 
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  • #2
ke7ijo said:
I differentiated both sides of Euler's formula with respect to x :
e^ix = sin x + i cos x
The mistake is that you have misremembered Euler's formula. The correct version is ##e^{ix} = \cos(x) + i\sin(x)##, which differs from what you wrote.

You can think of it this way. On the unit circle, with ##x## being the angle a ray makes with the horizontal axis, ##e^{ix}## represents the point on the unit circle. The coordinates of the point are ##(\cos(x), \sin(x))##. As a complex number, this point is ##\cos(x) + i\sin(x)##.
 
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  • #3
Thank you! That's it.
 
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Likes berkeman
  • #4
if you are mathematically curious, look at the different derivations of Euler's formula. My favorite is the one that uses Taylor Series.
 

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