How do you use the half angle identity to find exact value of tan(pi/12)?

1 Answer
Nghi N.
Oct 29, 2015

Find exact value of tan (pi/12)

Ans: #2 - sqrt3#

Explanation:

Call #tan (pi/12) = tan x#
#tan 2x = tan (pi/6) = sqrt3/3 = 1/sqrt3#
Apply the trig identity: #tan 2x = (2tan x)/(1 - tan^2 x)#
#1/sqrt3 = (2tan x)/(1 - tan^2 x)#
#1 - tan^2 x = 2sqrt3tan x#
Solve the quadratic equation:
#tan^2 x + 2sqrt3tan x - 1 = 0#
#D = d^2 = b^2 - 4ac = 12 + 4 = 16 #--> #d = +- 4#
#tan x = -(2sqrt3)/2 +- 4/2 = - sqrt3 +- 2#
Since tan pi/12 is positive, then #tan x = 2 - sqrt3.#
Check by calculator.
#tan (pi/12) = tan (15^@) = 0.27#
#(2 - sqrt3) = 2 - 1.73 = 0.27#. OK

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