There has been lurking in my mind a question for way too long, without being able to find a suitable, satisfying answer.

How are sin(), cos(), and tan() functions calculated on a calculator or computer?

I understand that mathematicians use the famous unit circle to define the trigonometric functions, but if

sin(?/4)=o/1

o= sin(?/4)*1

a=cos(?/4)*1

tan(?/4)=sin(?/4)

_______

cos(?/4)

So we are still left with the question, how is sine, cosine and tangent computed?

All the books and references that I read basically just suggest to use the calculator, in other words computed by magic or dogma.

Edit: I now remember that they use the Pythagorean theorem.

h²=o²+a²

?=?/4

o=a

1²=2o²

___

?1/2 = o

___

a= ?1/2

___

sin(?/4)=?1/2

_____

1

___

cos(?/4)=?1/2

_____

1

___

tan(?/4)=?1/2

_____

?1/2


But how do they compute the rest of angles?

Some hardware also uses interpolation (linear, quadratic, cubic, whatever) between lookup table entries.

Or you could go really old-school and try and figure out Ptolemy's Table of Chords. I only got to the first paragraph of the Special Angles section before I decided I have better things to do.