The answer might be quite obvious here but it's something I've been thinking over for a while as I've been going through study of trigonometry. Why, as an angle increases, does the side length corresponding to this angle not increase proportionally?
I'm aware that there is the sine rule used to describe this ratio, a/sin(A) = b/sin(B) = c/sin(C), just wondering why this is?
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Given a triangle $\triangle$ either through the coordinates of its vertices or through the lengths of its sides the trigonometric functions $\cos$, $\sin$, $\tan$ etc. of the angles are algebraic, sometimes even rational functions of the data, but the angles themselves are transcendental functions of the data. Only in special cases, e.g., if the side lengths are $3$, $4$, $5$, some angles are rational multiples of $\pi$. This deep mystery about angles is glossed over in the high school treatment of elementary geometry.
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Interesting question!
When you increase an angle by moving the side(s), I believe you keep their length(s) unchanged. Otherwise, it is independent of the angle.
Now recall the Cosine Theorem: $$c^2=a^2+b^2-2ab\cos{C}.$$ When the angle $C$ is close to $0$, then $\cos{C}$ will be close $1$ and almost maximum, hence maximum value is subtracted which makes $c^2$ minimum.
When the angle $C$ starts to increase from almost $0$, the value of $\cos{C}$ starts to decrease, hence smaller value is subtracted which makes $c^2$ greater.
When the angle $C$ goes over $90^o$, the value of $\cos{C}$ becomes negative, which means the cosine term is added and it keeps making $c^2$ greater.
In conclusion, as an angle changes, the opposite side also changes by the Cosine Theorem.
I suggest you look at the circumcircle of the triangle. The center of which is the circumcenter and the angle that each side subtends from the center is twice the angle it subtends from the corresponding opposite angle of the triangle. Thus there is a direct relationship between corresponding sides and angles. Any of the triangle sides is a chord of the circumcircle and so its length is twice the sine of half the central angle, and we already know that central angle is twice the corresponding opposite angle of the triangle. This result is where the law of sines comes from.