Question

What is the exact value of tan 7pi/12? Thank you!

Answer 1

Remember that

`tan((7\pi)/(12))=tan(105^o)=-tan(75^o)=-tan(45^o+30^o)`

and

`tan(A+B)=(tanA+tanB)/(1-tanA*tanB)`

therefore

`tan(45^o+30^o)=(tan45^o+tan30^o)/(1-tan45^o*tan30^o)`

substitute given values

`tan(45^o+30^o)=(1+(√(3))/(3))/(1-(1)((√(3))/(3)))=((3+√(3))/(3))/((3-√(3))/(3))=(3+√(3))/(3-√(3))*(3+√(3))/(3+√(3))=(9+6√(3)+3)/(6)=(12+6√(3))/(6)=2+√(3)`

therefore

The answer is

`tan((7\pi)/(12))=-(2+√(3))`

The answer is the second option