What does "radius" mean when talking about reference triangles?

Question

I'm watching this trig tutorial and at several points the guy refers to the hypotenuse of the triangle as the "radius" and explicitly writes $2 = r$. To be clear, it's a $30^\circ - 90^\circ - 60^\circ$ triangle, with $2$ being the hypotenuse, $\sqrt3$ being the adjacent side and $1$ being the opposite side. So $\cos\theta = \frac{\sqrt3}{2}$, etc. I know that $\cos\theta = \frac{x}{r}$ where "r" is the radius in the Pythagorean formula $r^2 = x^2 + y^2$. But I thought the radius of the unit circle is $1$. I understand that a reference triangle is not actually part of the unit circle, but how does this language of calling the hypotenuse of the triangle "radius" relate to everything else?

Answer 1

The hypotenuse of a reference triangle that lies on the unit circle is the radius of the unit circle and has length equal to $1$. Any similar triangle being used as a reference also has a hypotenuse that is the radius of some circle centered at the origin.

In short, when working with the unit circle or referring to it, it can be useful to refer to the hypotenuse of a reference triangle as some radius.

Answer 2

The actual "radius" of the triangle, that is the length of the hypotenuse doesn't matter as long as the ratios result in the same value. For example if we set the hypotenuse to $1$, the adjacent side to $\frac{\sqrt3}{2}$ and the opposite side to $\frac{1}{2}$, the length of the hypotenuse is the same as the radius of the unit circle and the ratios are the same.

$$\cos(45^\circ) = \frac{\frac{\sqrt3}{2}}{1} = \frac{\sqrt3}{2}$$