Sine

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

• Sine.png

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

• Complex sin.jpg

  Domain coloring of analytic continuation of $\sin$.

Properties

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

Proof: proof goes here █

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

Proof: proof goes here █

References

The sine product formula and the gamma function