How do I find the value of tan(7pi/12)?

1 Answer
Nghi N.
Aug 16, 2015

Find tan ((7pi)/12)

Ans: - (2 + sqrt3)

Explanation:

Call tan ((7pi)/12) = t
tan 2t = tan ((14pi)/12) = tan ((2pi)/12 + pi) = tan (pi/6 + pi) = tan (pi/6) = 1/sqrt3
Apply the trig identity: tan 2t = (2tan t)/(1 - tan^2 t)

tan 2t = 1/sqrt3 = 2t/(1 - tan^2 t)
1 - t^2 = 2sqrt3t -->
t^2 + 2sqrt3t - 1 = 0
D = d^2 = b^2 - 4ac = 12 + 4 = 16 --> d = +- 4
t = -2sqrt3/2 +- 4/2 -->
t = tan ((7pi)/12) = - sqrt3 +- 2

Since the arc (7pi)/12 is in Quadrant II, only the negative answer is accepted. tan((7pi)/12) = - 2 - sqrt3 = -3.732

Check by calculator
tan ((7pi)/12) = tan 105 = -3.732. OK

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