How do you find the exact values of cos(11pi/12) using the half angle formula?

1 Answer
Nghi N.
Aug 27, 2015

Find #cos ((11pi)/12)#

Ans: #- sqrt(2 + sqrt3)/2#

Explanation:

Call #cos ((11pi)/12) = cos t#
#cos 2t = cos ((22pi)/12) = cos ((11pi)/6) = cos ((pi)/6) = sqrt3/2#
Apply the trig identity: #cos 2t = 2cos^2 t - 1 #
#cos 2t = sqrt3/2 = 2cos^2 t - 1#
#2cos^2 t = 1 + sqrt3/2 = (2 + sqrt3)/2#
#cos^2 t = (2 + sqrt3)/4#
#cos t = cos ((11pi)/12) = +- sqrt(2 + sqrt3)/2.#
Since the arc ((11pi)/12) is located in Quadrant II, only the negative answer is accepted.
#cos ((11pi)/12) = - sqrt(2 + sqrt3)/2#

Check by calculator.
#Arc ((11pi)/12) = 165# deg-> #cos ((11pi)/12) = cos 165 = - 0.97.#
#- (sqrt(2 + sqrt3)/2) = - 0.97#. OK

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