I have to evaluate the exact value of tan(15).
To give some context, I have to find the area of a dodecagon, and I used the uber super cool formula:
ns^2/(4tan(180/n)) where n is the number of sides and s is the side length.
Thanks.
Use a half-angle formula.
I can never remember the formula for tan(a/2), so I do it this way:
sin(a/2) = (+/-)sqrt{1 - cos(a)}/sqrt{2}
cos(a/2) = (+/-)sqrt{1 + cos(a)}/sqrt{2}
(where we get the + or - from which quadrant the angle is in.)
So
tan(a/2) = sin(a/2)/cos(a/2) = (+/-)sqrt{1 - cos(a)}/sqrt{1 + cos(a)}
In this case, a = 30:
tan(15) = sqrt{1 - cos(30)}/sqrt{1 + cos(30)}
= sqrt{1 - sqrt{3}/2}/sqrt{1 + sqrt{3}/2}
= sqrt{2 - sqrt{3}}/sqrt{2 + sqrt{3}}
You will want to rationalize this, so multiply the numerator and denominator by sqrt{2 - sqrt{3}}. I'll just give you the answer:
tan(15) = sqrt{7 - 4sqrt{3}}
-Dan