How can one assume that the ratio altitude/hypotenuse is a function of angle. For a general right-angled triangle--->Let:
and angle opposite to altitude$=x$ .
Then by Pythagoras Theorem- $ a^2+b^2=c^2 \qquad(1)$
and for a fixed value of angle $x$ ratio of sides is constant,therefore- $a/c=S$, $b/c=C$ or $a=cS$ , $b=cC$ putting these values in equation(1): $$ (cS)^2+(cC)^2=c^2 \qquad S^2+C^2=1 \qquad(2) $$ Where $S$ and $C$ are some constants. My question is that how can I relate $S$ and $C$ with angle $x$.Do I simply write sin(x)=S,cos(x)=C,which according to me is just a declaration that S is a function of x.What is the real definition and how can it be derived?