# How do I see that right-angled triangles with a common angle 𝜃 (centered at the origin) have proportional side lengths?

I know that the angle $$\theta$$ of a right-angled triangle, centered at the origin, is defined as the radian measure of its intersection point with the unit circle, and that $$\cos(\theta)$$ and $$\sin(\theta)$$ are defined to be the x- and y-cordinates of that intersection point:

However, for larger right-angled triangles, we compute $$\cos(\theta)=A/C$$ and $$\sin(\theta)=B/C$$ by dividing the respective side length with the hypotenuse, respectively. But, for this to work, right-angled triangles with a common angle $$\theta$$ must have proportional side lengths.

How do I see that right-angled triangles with a common angle $$\theta$$ have proportional side lengths?

• They are similar triangles since they share the right angle as well. – Don Thousand Nov 5 '19 at 16:10

It simply is the theorem on intersecting lines (aka on confocal rays, or aka about similar triangles), which provides $$\frac{A}{\cos(\theta)}=\frac{C}1=\frac{B}{\sin(\theta)}$$ Solving the right or left equation then provides the searched for relations.