`(11pi)/12 = (3pi)/4 + pi/6` Find the exact values of the sine, cosine, and tangent of the angle.

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`sin(u+v)=sin(u)cos(v)+cos(u)sin(v)`

`sin((3pi)/4+pi/6)=sin((3pi)/4)cos(pi/6)+cos((3pi)/4)sin(pi/6)`

`sin((3pi)/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((3pi)/4+pi/6)=cos((3pi)/4)cos(pi/6)-sin((3pi)/4)sin(pi/6)`

`cos((3pi)/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((3pi)/4+pi/6)=(tan((3pi)/4)+tan(pi/6))/(1-tan((3pi)/4)tan(pi/6))=(-1+(sqrt3/3))/(1-(-1)(sqrt3/3))=(-3+sqrt3)/(3+sqrt3)`

The rationalized answer is `-2+sqrt3.`

 

 

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