Question

How do you use the half angle identity to find exact value of tan(pi/12)?

Answer 1

Find exact value of tan (pi/12)

Ans: `2 - sqrt3`

Explanation:

Call `tan (pi/12) = tan x`
`tan 2x = tan (pi/6) = sqrt3/3 = 1/sqrt3`
Apply the trig identity: `tan 2x = (2tan x)/(1 - tan^2 x)`
`1/sqrt3 = (2tan x)/(1 - tan^2 x)`
`1 - tan^2 x = 2sqrt3tan x`
Solve the quadratic equation:
`tan^2 x + 2sqrt3tan x - 1 = 0`
`D = d^2 = b^2 - 4ac = 12 + 4 = 16`--> `d = +- 4`
`tan x = -(2sqrt3)/2 +- 4/2 = - sqrt3 +- 2`
Since tan pi/12 is positive, then `tan x = 2 - sqrt3.`
Check by calculator.
`tan (pi/12) = tan (15^@) = 0.27`
`(2 - sqrt3) = 2 - 1.73 = 0.27`. OK