Find the exact value of sine cosine and tangent of the angle

Find the exact value of sine, cosine, and tangent of the angle 7 pi/12.

Solution

we have given angle 7pi/12

sin(7pi/12)=sin(pi/4 + pi/3)=sin(pi/4)cos(pi/3)+cos(pi/4)sin(pi/3) since sin(A+B)=sinAcosB+cosAsinB

sin(7pi/12)=1/sqrt(2) *(1/2) + 1/sqrt(2) *(sqrt(3)/2) =(1+sqrt(3))/(2*sqrt(2))

sin(7pi/12) =(1+sqrt(3))/(2*sqrt(2))

cos(7pi/12)=cos(pi/4+pi/3)

=cos(pi/4)cos(pi/3)-sin(pi/4)sin(pi/3) since cos(A+B)=cosAcosB-sinAsinB

=1/sqrt(2) *(1/2)- (1/sqrt(2))*sqrt(3)/2

=(1-sqrt(3))/(2*sqrt(2))

cos(7pi/12) =(1-sqrt(3))/(2*sqrt(2))

tan(7pi/12)=sin(7pi/12)/cos(pi/12)

=(1+sqrt(3))/(2*sqrt(2))/(1-sqrt(3))/(2*sqrt(2))

=(1+sqrt(3))/(1-sqrt(3))

tan(7pi/12) =(1+sqrt(3))/(1-sqrt(3))