It's kinda silly, but trig as taught revolves mostly around two right triangles: the 30,60,90 and the 45,45,90. Learn those and you'll do well in trig.
The 30,60,90 right triangle is actually half an equilateral triangle. Let s be the side length of an equilateral triangle and h be the height. The height splits the equilateral triangle into two right triangles. The hypotenuse of each is the side s, the short side s/2 (since it's half the equilateral triangle's side), and the long side is the height h. By the Pythagorean Theorem
s^2 = (s/2)^2 + h^2
s^2 = s^2/4 + h^2
h^2 = \frac 3 4 s^2
h = \frac {sqrt{3}}{2} s
We can set our hypotenuse s=2 giving sides 1 and \sqrt{3} .
\pi/3 or 60^\circ is the angle opposite the long side. Let's calculate all its trig functions:
cos frac \pi 3 = cos 60^\circ = frac{ text{ adjacent } }{ text{ hypotenuse } } = 1/2
sin frac \pi 3 = sin 60^\circ = frac{ text{ opposite } }{ text{ hypotenuse } } = sqrt{3}/2
tan \frac pi 3 = tan 60^\circ = frac{ text{ opposite } }{ text{ adjacent } } = sqrt{3}
The remaining trig functions are the reciprocals:
sec frac \pi 3 = sec 60^\circ = frac{ text{ hypotenuse } }{ text{ adjacent } } = 2
csc frac \pi 3 = csc 60^\circ = frac{ text{ hypotenuse } }{ text{ opposite } } = 2/sqrt{3} = 2 / 3 sqrt{3}
cot \frac pi 3 = cot 60^\circ = frac{ text{ adjacent } }{ text{ opposite } } = 1/sqrt{3} = \sqrt{3}/3
