Defining only in terms of a unit circle

We ofcourse know that given a unit circle, certain values of trigonometric functions apply ($\displaystyle \displaystyle sin\frac {\pi}{4}=\frac {\sqrt2}{2}$, for example).

However, if we were to take a circle with a bigger radius, that would automatically lead the values of such functions to change.

Whenever i used sin(pi/4) in a problem i always took its value as the one mentioned in terms of a unit circle, when in fact it varies according to the lengths of the triangle.

So my question would be: How can we take $\displaystyle \displaystyle sin\frac {\pi}{4}$ as having a standard value of $\displaystyle \frac {\sqrt2}{2}$ when in fact it varies. What am i missing here?