Exponential, Cosine and Sine Functions
The intuition on the meaning of $e^x$ with $x \in \mathbb{Z}^{+}$ is multiplying $e$ by itself $x$ times and $e^{-x} = 1/e^x$ being the inverse of that. However, when $x$ is irrational, or even complex, this intuition doesn’t work very well and hence we need a better definition which caters that.
Similarily, the trigonometric functions $\cos(x)$ and $\sin(x)$ can also be extended beyond $x$ being some sort of angles, e.g. $x \in \mathbb{C}$. The definition of them by power series provides a great extension to evaluate them on different kinds of numbers, with the basic properties preserved.
Exponential Function
Definition. The exponential function, $\exp(x)$, is defined by the power series
\[\exp(x) = \sum_{n=0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + ...\]
Lemma.
\[\sum_{n=0}^\infty \sum_{m=0}^\infty a_{mn} = \sum_{n=0}^\infty \sum_{r=0}^n a_{n-r, r}\]Proof.
\[\begin{align*} \sum_{n=0}^\infty \sum_{m=0}^\infty a_{mn} &= a_{00} + a_{01} + a_{02} + \cdots \\ &+ a_{10} + a_{11} + a_{12} + \cdots \\ &+ a_{20} + a_{21} + a_{22} + \cdots \\ &= a_{00} + (a_{10} + a_{01}) + (a_{20} + a_{11} + a_{02}) + \cdots \\ &= \sum_{n=0}^\infty \sum_{r=0}^n a_{n-r, r} \end{align*}\]
Theorem. $\exp(x)\exp(y) = \exp(x+y)$.
Proof.
For $x, y \in \mathbb{R}$,
\[\begin{align*} \exp(x)\exp(y) &= \left( \sum {x^n \over n!} \right) \left( \sum {y^n \over n!} \right) \\ &= \sum_{n=0}^{\infty}\sum_{m=0}^{\infty} {x^n \over n!}{y^m \over m!} \\ &= \sum_{n=0}^{\infty}\sum_{r=0}^{n} {x^{n-r} \over (n-r)!}{y^{r} \over (r)!} \\ &= \sum_{n=0}^{\infty} {1 \over n!} \sum_{r=0}^{n} {n! \over r!(n-r)!}x^{n-r}y^{r} \\ &= \sum_{n=0}^{\infty} {(x+y)^n \over n!} \\ &= \exp(x+y) \end{align*}\]
The power series definition is consistent with the intuition described above.
Theorem. For $x \in \mathbb{Q}$, $\exp(x) = e^x$.
Proof.
\[\exp(-1) = 1/e = e^{-1}\]
$\exp(a)\exp(b) = \exp(a + b)$ as proved above.
Substituding $x = 0$ into the definition, we have $\exp(0) = 1$.
Let $\exp(1) = e$. As $\exp(0) = \exp(1 - 1) = \exp(1)\exp(-1) = (e)\exp(-1) = 1$, we have
By induction, we have $\exp(x) = e^x$ for $x \in \mathbb{Q}$.
We can then define $e^x$ beyond rational numbers, i.e.
Definition. For $x \in \mathbb{C}$, $e^x \equiv \exp(x)$.
Trigonometric Functions
Similarily, we have
Definition. The cosine and sine functions are defined by the power series
\[\cos(x) = \sum_{n=0}^{\infty} (-1)^n {x^{2n} \over (2n)!}\] \[\sin(x) = \sum_{n=0}^{\infty} (-1)^n {x^{2n+1} \over (2n+1)!}\]
which allows us to extend the definition to having $x \in \mathbb{C}$.
Relationship between them
When $x \in \mathbb{R}$, we don’t really see a relationship between them as $\exp(x)$ appears to be ever-increasing function while $\cos(x)$ and $\sin(x)$ are periodic functions oscillating between $1$ and $-1$.
However, with the extension of defintion to complex number, we see an astonishing connection between them:
Theorem. $\exp(ix) = \cos(x) + i\sin(x)$.
Proof.
\[\begin{align*} \exp(ix) &= \sum_{n=0}^{\infty} {(ix)^n \over n!} \\ &= 1 + ix - {x^2 \over 2!} - i{x^3 \over 3!} + ... \\ &= \left( 1 - {x^2 \over 2!} + {x^4 \over 4!} + ...\right) + i\left( x - { x^3 \over 3!} + {x^5 \over 5!} + ... \right) \\ &= \sum_{n=0}^{\infty} (-1)^n {x^{2n} \over (2n)!} + i \sum_{n=0}^{\infty} (-1)^n {x^{2n+1} \over (2n+1)!} \\ &= \cos(x) + i\sin(x) \end{align*}\]
It produces what is probably the most striking formula in mathematics, namely
Theorem. [Euler Identity]
\[e^{i\pi} = -1\]
Remarks
Most of the proof above aims to illustrate the ideas behind and should not be considered rigorous. As the power series are infinite sum, a rigorous proof has to consider the convergence of the series in different situations.