How do you find the exact values of tan 165° using the half angle formula?

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Answer 1 · 2015-07-13T03:53:50

Find tan (165)

Call tan 165 = t

#tan 330 = tan 2(165) = tan (-30 + 360) = tan (-30) = - tan 30 = -1/sqrt3#

Use trig identity: #tan 2t = (2tan t)/(1 - tan^2 t).#

#tan 2t = tan 330 = - 1/sqrt3 = (2t)/(1 - t^2)#

#t^2 - 1 = 2sqrt3t#

#t^2 - 2sqrt3t - 1 = 0#. Solve this quadratic equation by the formula.

#D = d^2 = b^2 - 4ac = 12 + 4 = 16 -> d = +- 4#

#t = sqrt3 +- 2#
Since t = 165 deg, then tan 165 < 0, and the answer (sqrt3 - 2) is accepted.