Sine rule:
$$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{C}{\sin(C)}=2R$$
But I want to know what is
$$\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}=?$$
On wikipedia it says it is equal to $\dfrac{2\Delta}{abc}$ shouldn't it be simply $\frac 1{2R}$ ?
There is no contradiction: $$\dfrac{2\Delta}{abc} = \dfrac1{2R}$$ i.e. the area of the triangle is $$\Delta = \dfrac{abc}{4R}$$ where $R$ is the radius of the circumcircle.
For example, with a $3,4,5$ right angled triangle, the area is $6$ and the circumcircle radius is $\frac52$ and you have $$\dfrac1{2R}=\dfrac1{2 \times \frac52}=\dfrac15=\dfrac{2 \times 6}{3\times 4 \times 5} = \dfrac{2\Delta}{abc}$$