I know that sine and cosine can be defined as certain ratios between the sides of a right-angled triangle. I use the laws very often, but this morning, I came up with the ascertainment that I don't know what justifies the definitions.
For instance, for sine to make sense as the ratio "opposite-over-hypotenuse", I need to be sure that, if I fix an angle and double the hypotenuse, the leg opposite the angle also doubles. What's the proof for this fact?
Suppose that we have right-angled triangles $ABC$ and $A_1B_1C_1$ with $\angle C=\angle C_1=90^\circ$ and $\angle A=\angle A_1$. The two triangles are similar.
So. we have $\displaystyle \frac{BC}{AB}=\frac{B_1C_1}{A_1B_1}$, $\displaystyle \frac{AC}{AB}=\frac{A_1C_1}{A_1B_1}$ and $\displaystyle \frac{BC}{AC}=\frac{B_1C_1}{A_1C_1}$
Say if I draw a $20^\circ$-$70^\circ$-$90^\circ$, measure and calculate the ratio of the side opposite to the $20^\circ$ to the hypotenuse. This ratio is not same no matter how big or small is my triangle. We can define it as $\sin20^\circ$.
If $\angle A=\theta$ and $\angle C=90^\circ$, we can define $\displaystyle \sin\theta=\frac{BC}{AB}$, $\displaystyle \cos\theta=\frac{AC}{AB}$ and $\displaystyle \tan\theta=\frac{BC}{AC}$. The size of the triangle does not matter.
We can actually construct a table for the trigonometric ratios by drawing and measuring right-angled triangles of different angles.