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

1 Answer
Nghi N.
Mar 15, 2017

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

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