Question

Find 2 common angles that sum to (17pi/12) 2. Evaluate tan(17pi/12) using the sum identity for tangent.

Answer 1

Note that

`(17 \pi )/(12) = (3 \pi )/(12) + (14 \pi )/(12) = ( \pi )/(4) + (7 \pi )/(6)`

Note that

`x= ( \pi )/(4):\,\, sin(x) =cos(x)= (1)/( √(2) ),\,tan(x)=1\\x= (7 \pi )/(6) :\,\,sin(x)=- (1)/(2) ,\,\,cos(x)=- ( √(3) )/(2) ,\,\,tan(x)= (1)/( √(3) )`

Use the identity

`tan(x+y)= (tan(x)+tan(y))/(1-tan(x)tan(y))`

Therefore

`tan( (17 \pi )/(12) )= (1+ (1)/( √(3) ) )/(1- (1)/( √(3) ) ) = ( √(3)+1 )/( √(3)-1) = (( √(3)+1 )^(2))/(( √(3)-1 )( √(3)+1 )) = (3+1+2 √(3))/(3-1) =2+ √(3)`

Answer: 2 + √3