There are two triangles which you must know like the back of your hands.
1. Isosceles Right Angled Triangle
2. Equilateral Triangle
ISOSCELES RIGHT ANGLED TRIANGLE
The trigonometric ratios of Sin Cos Tan 45 comes from here.
The trigonometric ratios of angles 30 and 60 degrees come from this Equilateral triangle. It is very easy to see that
a Cos 60 = a Sin 30 and
So, now we know the source of the three vital angles 45, 30 & 60 degrees and can see how the geometry leads to the trigonometry.
Next we will place these triangles in Coordinate Geometry and look for insights.
A very important point is that half of a SQUARE is precisely this isosceles right angled triangle. Study the diagram below.