## 17Calculus Precalculus - Euler's Formula

##### 17Calculus

Euler's Formula is where exponentials, complex numbers and trig all combine into one elegant formula, $$e^{i\theta} = \cos(\theta)+i\sin(\theta)$$. This may not seem very useful but in electronics, for example, we convert trig functions to complex exponental form, using this equation, to make calculations easier. But how do we do that?

The idea is replace $$\theta$$ with $$-\theta$$ to get $$e^{-i\theta} = \cos(-\theta)+i\sin(-\theta) = \cos(\theta)-i\sin(\theta)$$. Then we add this to $$e^{i\theta}$$ and solve for cosine. Similarly, we subtract them and solve for sine. This process yields these equations.

Rather than just taking Euler's Formula for granted, let's look closely at what it is telling us. Here is a great video giving an intuitive explanation of the equation and what it means. This guy is great. He takes complicated concepts and makes them easier to see and understand.

### BetterExplained - Understanding Euler's Formula [9min-53secs]

video by BetterExplained

Now all this may seem a bit overwhelming but let's watch a great video explaining more intuitively why $$e^{i\pi}=-1$$ holds.

### 3Blue1Brown - Understanding e to the pi i [6min-13secs]

video by 3Blue1Brown

Proofs of Euler's Formula

These proofs require calculus.

### Derek Owens - Calculus 6.11 - Euler's Identity (using calculus) [12min-25secs]

video by Derek Owens

### Dr Chris Tisdell - Euler's formula: A cool proof (using differential equations) [11min-55secs]

video by Dr Chris Tisdell

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