There is a question with the same title on the math StackExchange here. However, I am asking a different question: under the definition of them as the right triangle ratios $$\sin(\theta)=\frac{\textrm{opposite}}{\textrm{hypotenuse}}$$ and $$\cos(\theta)=\frac{\textrm{adjacent}}{\textrm{hypotenuse}},$$ how do we (rigorously) know that they stay constant for a given $\theta$ for any right triangle?
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1corresponding sides of similar triangles are proportional – J. W. Tanner Nov 07 '19 at 02:00
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Yeah... how do we know this ? – Lavie Nov 07 '19 at 02:01
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@A.Lavie: Thales theorem – Vasili Nov 07 '19 at 02:06
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1https://proofwiki.org/wiki/Equiangular_Triangles_are_Similar – msm Nov 07 '19 at 02:07