Question

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

Answer 1

Find exact value of `cot ((pi)/12)`

Ans: `(2 - sqrt3)`

Explanation:

`cot ((pi)/12) = 1/(tan ((pi)/12)`. First find `tan ((pi)/12)`
Call `tan ((pi)/12) = t`
`tan (2t) = tan ((pi)/6) = 1/sqrt3`
Apply the trig identity: `tan 2a = (2tan a)/(1 - tan^2 a)`
We get:
`1/sqrt3 = (2t)/(1 - t^2)`
`1 - t^2 = 2sqrt3t`. Solve the quadratic equation in t.

`t^2 + 2sqrt3t - 1 = 0`
`D = d^2 = b^2 - 4ac = 12 + 4 = 16`--> `d = +- 4`
There are 2 real roots:
`tan ((pi)/12) = t = - 2sqrt3/2 +- 4/2 = -sqrt3 +- 2`
Since the arc (pi/12) is in Quadrant I, its tan is positive, then
`tan ((pi)/12) = (-sqrt3 + 2).`
Check by calculator:
`tan ((pi)/12) = tan 15 = 0.27.`
`(2 - sqrt3) = 0.27`. OK