Question 149789
{{{tan(-3pi)}}}



{{{-tan(3pi)}}} Use the identity {{{tan(-u)=-tan(u)}}} to rewrite the expression. 



{{{-sin(3pi)/cos(3pi)}}} Rewrite tangent in terms of sine and cosine.



{{{-sin(2pi+pi)/cos(2pi+pi)}}} Rewrite {{{3pi}}} as {{{2pi+pi}}}



{{{-(sin(2pi)cos(pi)+cos(2pi)sin(pi))/(cos(2pi)cos(pi)-sin(2pi)sin(pi))}}} Rewrite the expression using the identities {{{sin(A+B)=sin(A)cos(B)+cos(A)sin(B)}}} and {{{cos(A+B)=cos(A)cos(B)-sin(A)sin(B)}}} 



Now, let's reference the unit circle


<img src="http://www1.fccj.edu/lchandouts/trigresources/Unit_circle_angles.png"></img>



From the unit circle, we can see that at the angle {{{pi}}}, there is a point on the unit circle *[Tex \LARGE \left(-1,0\right)]. So this tells us that {{{cos(pi)=-1}}} and {{{sin(pi)=0}}}. Also at the angle {{{2pi}}}, there is a point on the unit circle *[Tex \LARGE \left(1,0\right)]. So this tells us that {{{cos(2pi)=1}}} and {{{sin(2pi)=0}}}.


{{{-((0)(-1)+(1)(0))/((1)(-1)-(0)(0))}}} Take the cosine of {{{pi}}} to get -1. Take the sine of {{{pi}}} to get 0. Take the cosine of {{{2pi}}} to get 1. Take the sine of {{{2pi}}} to get 0. 



{{{-(0+0)/(-1+0)}}} Multiply



{{{-(0)/(-1)}}} Add



{{{0}}} Reduce



So {{{tan(-3pi)=0}}}