Cosine

The cosine function, $\cos \colon \mathbb{R} \rightarrow \mathbb{R}$ is the unique solution of the second order initial value problem $$y=-y;y(0)=1,y'(0)=0.$$

• Cosine.png

  Graph of $\cos$ on $\mathbb{R}$.

• Complex cos.jpg

  Domain coloring of analytic continuation of $\cos$ to $\mathbb{C}$.

Properties

Proposition: $$\cos(x) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k x^{2k}}{(2k)!}$$

Proof: █

Proposition: $$\cos(x) = \prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$$

Proof: █