The trigonometric formulas for can be derived using the multipleangle formula
(1)

Letting and then gives
(2)

Factoring out one power of gives
(3)

Solving the quadratic equation for gives
(4)

But must be less than
(5)

so taking the minus sign and simplifying gives
(6)

Filling in the remainder of the trigonometry functions then gives
(7)
 
(8)
 
(9)
 
(10)
 
(11)
 
(12)

and
(13)
 
(14)
 
(15)
 
(16)
 
(17)
 
(18)
