How do you use the half angle identity to find exact value of sin 75 degrees?

1 Answer
Nghi N.
Sep 23, 2015

Find sin (75)

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

Explanation:

Call sin 75 = sin t
Apply the trig identity: #cos 2t = 1 - 2sin^2 t#
#cos 2t = cos 150 = -sqrt3/2 = 1 - 2sin^2 t#
#2sin^2 t = 1 + sqrt3/2#
#sin^2 t = (2 + sqrt3)/4#
#sin t = +- sqrt(2 + sqrt3)/2#
Since the arc 75 is in Quadrant I, its sin is positive, therefor

#sin t = sin 75 = sqrt(2 + sqrt3)/2#

Check by calculator.
sin 75 = 0.97
#sqrt(2 + sqrt3)/2 = (sqrt3.73)/2 = (1.93)/2 = 0.97#. OK

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