Question

How do you find the exact functional value tan 7pi/12 using the cosine sum or difference identity?

Answer 1

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

Ans: `(1 + sqrt3)/(1 - sqrt3)`

Explanation:

`sin ((7pi)/12) = sin (pi/3 + pi/4) =`
`= sin (pi/3).cos (pi/4) + sin (pi/4).cos (pi/3) =`
`(sqrt3/2)(sqrt2/2) + (sqrt2/2)(1/2) = ((sqrt2)/4)(1 + sqrt3)`
`cos ((7pi)/12) = cos (pi/3 + pi/4) =`
`= cos (pi/3).cos (pi/4) - sin (pi/3).sin (pi/4) =`
`= (1/2)(sqrt2/2) - (sqrt2/2)(sqrt3/2) = (sqrt2/4)(1 - sqrt3)`

`tan ((7pi)/12) = sin/(cos) = (1 + sqrt3)/(1 - sqrt3)`
Check by calculator:
`tan ((7pi)/12) = tan 105 = -3.73`
`(1 + sqrt3)/(1 - sqr3) = 2.73/(-0.73) = - 3.73`. OK