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Explaination why sin(0)=0 requires to refer to the function definition.
sin(α) has been defined as:
Lets suppose that: alpha angle goes to zero (α -> 0) and adjacent side is constant (b = const).
It causes: opposite side goes to zero too (a -> 0) and hypotenuse side goes to adjacent side (h -> b). When angle is equal to 0 and a is equal to 0 then 0 divided by h is equal to 0. It means sin(0) is equal to 0 acording to above definition.
Simple visualisation is presented on below picture:
