How do I evaluate tan(pi/3) without using a calculator?

tan (##pi/3##)==##sqrt3##
tan (##pi/3##)= tan(##(180^o)/3##) = tan(##60^o##)=##sqrt3##
{##NOTE##: I have substituted ##pi^c = 180^o## }
Now there’s a given way of remembering sine and cosine values of 5 crucial angles which are ##0^o## ##30^o## ##45^o## ##60^o## and ##90^o##.
The sine values:
sin(##0^o##) = ##0##
sin(##30^o##) = ##1/2##
sin(##45^o##) = ##1//(sqrt2)##
sin(##60^o##) = ##(sqrt3)//2##
sin(##90^o##) = ##1##
For the cosine values you need to refer to the same but in the reverse order.
##rArr## cos(##0^o##) = ##1##
cos(##30^o##) = ##(sqrt3)/2##
cos(##45^o##) = ##1/(sqrt2)##
cos(##60^o##) = ##1/2##
cos(##90^o##) = ##0##
Finally you can use this relation to find the tangents of any angle ##tan theta## = ##(sin theta)/(cos theta)##
FOR example ##tan 45^o = (sin 45^o)/(cos 45^o) ## = ##(1/(sqrt2))//(1/(sqrt2))## = ##1##
I hope it helps.