Let the right-angled triangle have angle $x$, opposite $B$, hypotenuse $H$, and adjacent $A$. (It is a bad choice of notation to use $O$ for opposite.)

(i) Let $A=0$ and $x=0$, then $\sin 0=0,$ $\cos 0=1$ and $\tan 0=0.$

(ii) Let $B=0$ and $x=\pi/2$, then $\sin{{\pi}\over{2}}=1,$ $\cos{{\pi}\over{2}}=0$ and $\tan x\rightarrow\infty$ as $x\rightarrow{{\pi}\over{2}}-$.

(iii) Let $B=A=1$ and $H=\sqrt{2}$ and $x={{\pi}\over{4}}$, then $\sin{{\pi}\over{4}}=\cos{{\pi}\over{4}}={{1}\over{\sqrt{2}}}$, and $\tan{{\pi}\over{4}}=1$.

(iv) Let $B=1,$ $H=2$ and $A=\sqrt{3}$, and $x={{\pi}\over{6}}$, then $\sin{{\pi}\over{6}}={{1}\over{2}},$ $\cos{{\pi}\over{6}}={{\sqrt{3}}\over{2}}$ and $\tan{{\pi}\over{6}}={{1}\over{\sqrt{3}}}$.

Instead of measuring triangles, computers calculate trig functions by using the Maclaurin series of 4.1 below.