The equation tan(x)+sqrt3=0tan(x)+√3=0 can be rewritten as
tan(x)=-sqrt3tan(x)=−√3
Knowing that tan(x) = sin(x)/cos(x)tan(x)=sin(x)cos(x)
and knowing some specific values of coscos and sinsin functions:
cos(0)=1cos(0)=1 ; sin(0)=0sin(0)=0
cos(pi/6)=sqrt3/2cos(π6)=√32 ; sin(pi/6)=1/2sin(π6)=12
cos(pi/4)=sqrt2/2cos(π4)=√22 ; sin(pi/4)=sqrt2/2sin(π4)=√22
cos(pi/3)=1/2cos(π3)=12 ; sin(pi/3)=sqrt3/2sin(π3)=√32
cos(pi/2)=0cos(π2)=0 ; sin(pi/2)=1sin(π2)=1
as well as the following coscos and sinsin properties:
cos(-x)=cos(x)cos(−x)=cos(x) ; sin(-x)=-sin(x)sin(−x)=−sin(x)
cos(x+pi)=-cos(x)cos(x+π)=−cos(x) ; sin(x+pi)=-sin(x)sin(x+π)=−sin(x)
We find two solutions:
1) tan(-pi/3) = sin(-pi/3)/cos(-pi/3) = (-sin(pi/3))/cos(pi/3) = - (sqrt3/2)/(1/2) = -sqrt3tan(−π3)=sin(−π3)cos(−π3)=−sin(π3)cos(π3)=−√3212=−√3
2) tan(pi-pi/3) = sin(pi-pi/3)/cos(pi-pi/3) = (-sin(-pi/3))/(-cos(-pi/3)) = sin(pi/3)/(-cos(pi/3)) = - (sqrt3/2)/(1/2) = -sqrt3tan(π−π3)=sin(π−π3)cos(π−π3)=−sin(−π3)−cos(−π3)=sin(π3)−cos(π3)=−√3212=−√3
