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

1 Answer
Nghi N.
Oct 26, 2015

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

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