Precalculus, Chapter 5, 5.4, Section 5.4, Problem 22

(5pi)/12=pi/4+pi/6

sin(u+v)=sin(u)cos(v)+cos(u)sin(v)
sin(pi/4+pi/6)=sin(pi/4)cos(pi/6)+cos(pi/4)sin(pi/6)
sin(pi/4+pi/6)=(sqrt2/2)(sqrt3/2)+(sqrt2/2)(1/2)=sqrt2/4(sqrt3+1)

cos(u+v)=cos(u)cos(v)-sin(u)sin(v)
cos(pi/4+pi/6)=cos(pi/4)cos(pi/6)-sin(pi/4)sin(pi/6)
cos(pi/4+pi/6)=(sqrt2/2)(sqrt3/2)-(sqrt2/2)(1/2)=sqrt2/4(sqrt3-1)

tan(u+v)=(tan(u)+tan(v))/(1-tan(u)tan(v))
tan(pi/4+pi/6)=(tan(pi/4)+tan(pi/6))/(1-tan(pi/4)tan(pi/6))=(1+(sqrt3/3))/(1-(1)(sqrt3/3))=((3+sqrt3)/3)/((3-sqrt3)/3)=(3+sqrt3)/(3-sqrt3)
After rationalizing the denominator the answer is 2+sqrt3.