Tuesday, 2 April 2013

calculus - 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:



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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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