In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. So, in this case our numbers are the two legs of a triangle. Given that this is a right triangle because one angle is equal to 90, the the relations are as follows:
[tex]opposite=(hypotenuse)(sin\alpha)[/tex]
[tex]adjacent=(hypotenuse)(cos \alpha)[/tex]
In the figure below α = 30, so
[tex]O=Hsin(30)[/tex]
[tex]A=Hcos(30)[/tex]
There are two ratios:
[tex]\frac{O}{A}[/tex] and
[tex] \frac{A}{O}[/tex]
Therefore:
[tex]\frac{O}{A}=\frac{\sin(30)}{cos(30)}=\frac{\sqrt{3}}{3}=\frac{1}{\sqrt{3}}[/tex]
[tex]\frac{A}{O}=\frac{\cos(30)}{sin(30)}=\sqrt{3}[/tex]
Finally, matching these results with the image above, the ratio:
[tex]1: \sqrt{3} = \frac{1}{ \sqrt{3} }[/tex]
is the only one that apply
