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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:

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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?

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    $\begingroup$ They are similar triangles since they share the right angle as well. $\endgroup$ – Don Thousand Nov 5 at 16:10
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Triangles that have the same angle measurements are similar. Since Both of these triangles are right triangles, and both of these triangles share a common angle, then these triangles have 2 angles that are the same. Since a triangle only has 3 angles, and if 2 angles are the same, then the third angle must be the same. So, right triangles that share a common angle (not including the right angle) are similar, meaning that they have proportional side lengths.

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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.

--- rk

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