Trig from the compound formula

Up a level : Euler's number e --- and a bit of pi
Previous page : Calculating pi - part 3

Let us go back to exploring the function

$\displaystyle {{e}^{ib}}=\underset{{n\to \infty }}{\mathop{{\lim }}}\,{{\left( {1+\frac{ib}{n}} \right)}^{n}}$

What if we plot the imaginary part and the complex part vs the angle? You can try this here:

The real and complex parts of the compound formula

Hopefully, you recognise the functions. As we saw before, the output of the above function will lie on a circle, and the angle in a counterclockwise direction from the x-axis will be proportional to b, and if we use radians, it will be b. That means that our real and imaginary parts must be the cos and sine functions. We have thus found another way to show that

${e^{i\theta }} = \cos \theta + i\sin \theta$

You can find a more traditional way of finding that here

Euler’s formula

Up a level : Euler's number e --- and a bit of pi
Previous page : Calculating pi - part 3

Last modified: May 9, 2026 @ 20:55