`-(7pi)/12` Find the exact values of the sine, cosine, and tangent of the angle.

`sin((-7pi)/12)=-sin((7pi)/12)`

`=-sin(pi/3+pi/4)`

using the identity `sin(x+y)=sin(x)cos(y)+cos(x)sin(y)`

`=-(sin(pi/3)cos(pi/4)+cos(pi/3)sin(pi/4))`

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

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

rationalizing the denominator,

`=(-sqrt(2)(sqrt(3)+1))/4`

`cos((-7pi)/12)=cos((7pi)/12)`

`=cos(pi/3+pi/4)`

using the identity `cos(x+y)=cos(x)cos(y)-sin(x)sin(y)`

`=cos(pi/3)cos(pi/4)-sin(pi/3)sin(pi/4)`

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

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

rationalizing the denominator,

`=(sqrt(2)(1-sqrt(3)))/4`

`=(sqrt(2)-sqrt(6))/4`

`tan((-7pi)/12)`

`=sin((-7pi)/12)/cos((-7pi)/12)`

plug in the values evaluated above,

`=((-sqrt(2)(sqrt(3)+1))/4)/((sqrt(2)-sqrt(6))/4)`

`=(-sqrt(2)(sqrt(3)+1))/(sqrt(2)-sqrt(6))`

rationalize the denominator,

`=-((sqrt(6)+sqrt(2))(sqrt(2)+sqrt(6)))/((sqrt(2)-sqrt(6))(sqrt(2)+sqrt(6)))`

`=-(2sqrt(3)+6+2+2sqrt(3))/(2-6)`

`=-(4sqrt(3)+8)/(-4)`

`=sqrt(3)+2`