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In right triangle trigonometry, the sine of an angle $A$ is defined as the ratio of two lengths, the opposite leg $a$ and the hypotenuse $c$, that's to say, $\sin A= \frac{a}{c}$?
My question is: supposing that the hypotenuse is not equal to $1$, how to see this ratio geometrically? I'm looking for a geometric interpretation of the ratio of two lengths, and how this construction can be done, if these lengths are the sides of a right triangle.
You can "see" the ratio geometrically by using similarity of triangles. Suppose everything is as in your example. (I will let a symbol denote a segment or its length below, as it can be discerned by context, although that is technically not mathematically correct.)
If $c > 1$, than tracing a line parallel to the segment that is NOT $a$ or $c$, you will create a triangle that is similar to the original one. Let $a'$ and $c'$ be the legs that correspond to $a$ and $c$ within the triangle similarity, respectively.
Now let the length of $c'$ be $1$. (Note we had not specified that thus far.) Because all corresponding ratios in similar triangles must be the same, than $a' = a'/c' = a / c = SinA$.
If $1 > c$, than analogous reasoning can be derived by forming a similar triangle by extending the legs of the triangle and tracing the parallel line.
When you first divided the opposite by the hypotenuse you needlessly took care to keep the hypotenuse equal to 1. However it is not necessary to force hypotenuse to be equal to 1 from the beginning itself.
For a 30 degrees angle the opposite side can be 100 and the hypotenuse can be 200, no problem, the whole triangle can be zoomed maintaining similarity and ratios of corresponding sides. Construction involves drawing lines parallel to the sides like it is done here. They are not only similar but similarly placed.
