The exponential function has the property that it is its own derivative.
$$ \frac{d(e^x)}{dx} = e^x \qquad (1) $$
The is-its-own-derivative implies the following Taylor series:
$$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \qquad (2) $$
2-cycle
We also know two functions that are each other's derivative; a derivation 2-cycle.
$$ \sinh(x) = x +{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots \qquad (3) $$
$$ \cosh(x) = 1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots \qquad (4) $$
formula for the hyperbolic sine:
$$ \sinh(x) = \frac{e^x - e^{-x}}{2} \qquad (5) $$
formula for the hyperbolic cosine:
$$ \cosh(x) = \frac{e^x + e^{-x}}{2} \qquad (6) $$
We can count taking the derivative of the exponential function as a 1-cycle. For the $\sinh$ and $\cosh$ the split is achieved by using the cyclic property of successive multiplication of a factor of -1. With -1 you get a 2-cycle.
4-cycle
The next step is a sequence of two functions that are each others derivative, but after the second derivation the resulting function is the negative of the original, so the total cycle is a 4-cycle.
The Taylor series expansion of the trigonometric functions follows from the 4-cycle derivatives property:
$$ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \qquad (7) $$
$$ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \qquad (8) $$
At this point it is of course helpful to devise a compact notation for the above Taylor series. The $\sin$ and $\cos$ can be expressed in terms of the exponential function by using a factor that has the property of repeating in a 4-cycle, with sign alteration along the way. Interestingly, while we can connect this 4-cycle factor to other areas of mathematics, for the purpose of this demonstration making this connection is not a necessity.
We can pretend to invent a novel entity for this, we can call this factor the 'inventive number', abbreviated 'i'.
For this invented factor the following 4-cycle is defined:
$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$
The formulas for the sine and cosine then are:
$$ \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \qquad (9) $$
$$ \cos(x) = \frac{e^{ix} + e^{-ix}}{2} \qquad (10) $$
Euler's formula follows from (9) and (10):
$$ e^{ix} = \cos(x) + i\sin(x) \qquad (11) $$
The key points:
The connection between the exponential function and the trigonometric functions is the property of getting the same function back after taking the derivative. The difference is whether the original function comes back right away, or in a 2-cycle pattern, or in a 4-cycle pattern.
Euler's formula is a deep result of complex analysis that shows a surprising relationship between seemingly unrelated notions:
$$ e^{ix} = \cos(x) + i \sin(x) \, $$
where $e^{ix}$ denotes the exponential function and $ \cos(x)$, $\sin(x)$ are the usual trigonometric functions. If we evaluate this equation at $x= \pi$, we get
$$ e^{i\pi } = \cos(\pi) + i \sin(\pi) = -1 -i 0 = -1 \, .$$
This shows a deep relationship between the exponential function, the imaginary unit $i$ and $\pi$. (Pi is the ratio between circumference and diameter shared by all circles.)
Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence. Keith Devlin
[Euler's equation] is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth". Benjamin Peirce