In trigonometry, sometimes we require students to learn the special values of sine, cosine and tangent at 30, 45, 60 degrees, because these are nice values. But I wonder if these facts are explained or do teachers just say "sin(30) = 1/2 , memorise this!"
Here's my explanation.(I am not sure if they teach this in schools, but I'll look up the textbook the next time I visit popular bookstore.)
Show that sin(30) = Opp/ Hyp = 1/2.
1) Draw a new line dividing the right-angle into 30 and 60 degrees respectively.
Now triangle ABD and BCD are both isosceles. In fact, BCD is an equilateral triangle.
2) Hence, CD= BD = BC
3) Since BD lies on the isosceles triangle ABD, BD= AD.
4) Conclude sin(30) = opp/hyp = BC / AC = 1/2
5) For good measure, using pythagoras theorem, we can now show that