`165^@ = 135^@ + 30^@` 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(135+30)=sin(135)cos(30)+cos(135)sin(30)`

`sin(135+30)=(sqrt2/2)(sqrt3/2)+(-sqrt2/2)(1/2)`

`sin(135+30)=sqrt6/4-sqrt2/4=sqrt2/4(sqrt3-1)`

 

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

`cos(135+30)=cos(135)cos(30)-sin(135)sin(30)`

`cos(135+30)=(-sqrt2/2)(sqrt3/2)-(sqrt2/2)(1/2)`

`cos(135+30)=-sqrt6/4-sqrt2/4=-sqrt2/4(sqrt3+1)`

 

`tan(135+30)=(tan(u)+tan(v))/(1-tan(u)tan(v))`

` ` `tan(135+30)=(tan(135)+tan(30))/(1-tan(135)tan(30))`

`tan(135+30)=(-1+sqrt3/3)/(1-(-1)sqrt3/3)=((-3+sqrt3)/3)/((3+sqrt3)/3)=(-3+sqrt3)/(3+sqrt3)`

After rationalizing the denominator the answer is `-2+sqrt3`

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