Question

How do I find the value of tan pi/12?

Answer 1

`tan (pi/12) = 2 - sqrt3`

Explanation:

Use trig identity:
`tan 2a = (2tan a)/(1 - tan^2 a)`
In this case, trig table gives:
`tan (pi/6) = (2tan (pi/12))/(1 - tan^2 (pi/12)) = 1/sqrt3`
Cross multiply:
`2tan (pi/12) = 1 - tan^2 (pi/12)`
`tan^2 (pi/12) + 2sqrt3tan (pi/12) - 1 = 0`
Solve this quadratic equation for tan (pi/12):
`D = d^2 = b^2 - 4ac = 12 + 4 = 16` --> `d = +- 4`
There are 2 real roots:
`tan (pi/12) = - b/(2a) +- d/(2a) = - (2sqrt3)/2 +- 4/2 = - sqrt3 +- 2`
Since tan (pi/12) is positive, there fore:
`tan (pi/12) = 2 - sqrt3`