Can one give example or picture to how the trigonometric functions (for example, sinus) are presented in Riemann Sphere?
Exactly the same as in the complex plane since there is no value where $\sin x$ is undefined. Trigonometric functions like $\sec x$, $\csc x$, $\tan x$, and $\cot x$ look exactly the same, as well, but they become continuous in the Riemann Sphere.