TITLE: Why does $\sin(0)$ exist?
QUESTION [15 upvotes]: I can't understand why should $\sin(0)$ exist, because if an angle is $0^{\circ}$, then the triangle doesn't exist i.e. there is no perpendicular or hypotenuse. However, if we take $\lim_{x \to 0} \sin(x)$, then I can understand
$$\lim_{x \to 0} \sin(x) = 0$$
since perpendicular $\approx$ 0.
So although $\lim_{x \to 0} \sin(x)$ is $0$, I can't understand how $\sin(0)=0$, and if $\sin(0)$ is not defined, then why is graph of $\sin(x)$ continuous ?

REPLY [4 votes]: I don't see why you'd have to take $\lim_{x \to 0} \sin(x)$. I'd like to argue that every line segment is a (degenerate) triangle. So with the definition of $\text{sin} = \frac{\text{perpendicular}}{\text{hypotenuse}}$, we get $\sin(0) = \frac{0}{\text{hypotenuse}} = 0.$
Similar reasoning can be applied to $\sin{90^{\circ}}$, $\cos{0^{\circ}}$, $\cos{90^{\circ}}$ and so on.