How do you use the half angle identity to find exact value of cos75?

1 Answer
Nghi N.
Aug 30, 2015

Find cos 75

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

Explanation:

Call cos 75 = cos t.
#cos 2t = cos 150 = -srqt3/2#
Apply the trig identity: #cos 2t = 2cos^2 t - 1.#
#-sqrt3/2 = 2cos^2 t - 1#
#2cos^2 t = 1 - sqrt3/2 = (2 - sqrt3)/2#
#cos^2 t = (2 - sqrt3)/4#
#cos t = +- sqrt(2 - sqrt3)/2#
Since the arc 75 is located in Quadrant I, cos 75 > 0. Then,
#cos 75 = cos t = sqrt(2 - sqrt3)/2#

Check by calculator.
cos 75 = 0.26 ; #sqrt(2 - sqrt3)/2 = 0.5176/2 =0.26#. OK

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