Why Euler’s Formula Works

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In many parts of engineering, you’ll run across something called Euler’s formula, which looks something like this.

(1)   \begin{equation*} e^{i\theta} = \cos(\theta) + i\sin(\theta) \end{equation*}

It’s something you can look at, memorize, and then use in the future.  But sometimes it’s nice to know that mathematically, the left side and the right side are in fact equivalent, even if it’s not obvious.  While reading Hecht’s Optics book, I came across a footnote that quickly shows that the two sides are equal.  Here is the gist of that footnote.

First, set the right hand side of the equation equal to a variable z.

(2)   \begin{equation*} z = \cos(\theta) + i\sin(\theta) \end{equation*}

Now take the derivative of z with respect to \theta.

(3)   \begin{equation*} \frac{dz}{d\theta} = -\sin(\theta) + i\cos(\theta) \end{equation*}

(4)   \begin{equation*} \frac{dz}{d\theta} = i\cos(\theta) - \sin(\theta) \end{equation*}

Now take our initial function z and multiply it by i.

(5)   \begin{equation*} iz = i[\cos(\theta) + i\sin(\theta)] = i\cos(\theta) + i^2\sin(\theta) \end{equation*}

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Since we know that i^2 = -1, we can write iz as the following, and note that it is the same as our derivative we just calculated.

(6)   \begin{equation*} iz = i\cos(\theta) -\sin(\theta) = \frac{dz}{d\theta} \end{equation*}

Now we can move the z variables to one side of the equation, and the rest to the other side of the equation.  Then we can take the integral.  The left hand side will be integrated with respect to z while the right hand side will be integrated with respect to \theta.

(7)   \begin{equation*} \int \frac{dz}{z} = \int i\, d\theta \end{equation*}

(8)   \begin{equation*} \ln(z) = i\theta \end{equation*}

Now we can take the exponential of both sides, and we can see the final equality proves that both sides of Euler’s formula are equivalent.

(9)   \begin{equation*} e^{\ln(z)} = e^{i\theta} \end{equation*}

(10)   \begin{equation*} z = e^{i\theta} = \cos(\theta) + i\sin(\theta) \end{equation*}

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