How do you find the exact values of tan 7pi/8 using the half angle formula?

1 Answer
Aug 27, 2015

Find # tan ((7pi)/8)#

Ans: (1 - sqrt2)

Explanation:

Call #tan ((7pi)/8) = tan t#
#tan 2t = tan ((14pi)/8) = tan ((7pi)/4) = - tan pi/4 = - 1#
Apply the trig identity: #tan 2t = (2tan t)/(1 - tan^2 t)#
#tan 2t = - 1 = (2tan t)/(1 - tan^2 t)#
#tan^2 t - 2tan t - 1 = 0#
#D = d^2 = b^2 - 4ac = 4 + 4 = 8 #--> #d = +- 2sqrt2.#
#tan t = 2/2 +- 2sqrt2/2 = 1 +- sqrt2#
Since the arc (7pi)/8 is located in Quadrant II, then the answer is:
#tan t = tan ((7pi)/8) = 1 - sqrt2#= - 0.414
Check by calculator.
#(7pi)/8 = 157.5# deg --> tan (157.5) = -0.414. OK