Trigonometric ratios of standard angles if found out by following simple method.
Trigonometric ratio for 30° and 60°
As shown in the figure, triangle ABC is equilateral triangle with each side of length 2X.
AD is the bisector of angle BAC.
As the triangle is equilateral, the length of BD and DC will be X.
Triangle ADB is right-angle triangle, so from the Pythagorean theorem we can say that,
AD2 + BD2 = AB2
AD2 + X2 = 4X2
AD2 = 3X2
AD = √3X
So for the right-angle triangle ADB, we have values of all the sides,
AB = 2X, BD = X and AD = √3X
Simillarly for 60°
Trigonometric ratio for 45°
As shown in figure, triangle XYZ is right-angle triangle and two sides of the triangle are equal. So the value of the remaining angles are also same which is 45°.
By Pythagorean theorem, we can say that
XY2 + YZ2 = XZ2
a2 + a2 = XZ2
XZ2 = 2a2
XZ = √2a
So for the triangle XYZ, we have values of all the sides.
XY = a, YZ = a and XZ = √2a
The table for trigonometric ratio of standard angle is given below.