How do you use the half angle identify to find the exact value of #cos((7pi)/12)#?

1 Answer
Nghi N
Nov 11, 2016
  • sqrt(2 + sqrt3)/2

Explanation:

Using trig unit circle and trig table -->
#cos ((7pi)/12) = cos (pi/12 + pi) = - cos (pi/12)#.
Find cos (pi/12) by using trig identity:
#2cos^2 a = 1 + cos 2a#
#2cos^2 (pi/12) = 1 + cos (pi/6) = 1 + sqrt3/2 = (2 + sqrt3)/2#
#cos^2 (pi/12) = (2 + sqrt3)/4#
#cos (pi/12) = +- sqrt(2 + sqrt3)/2.#
Since #cos (pi/12)# is positive, only the positive value is accepted.
Finally,
#cos ((7pi)/12) = - cos (pi/12) = - sqrt(2 + sqrt3)/2#
Check by calculator.
#cos ((7pi)/12) = - cos (pi/12) = - cos 15 = - 0.97#
# - sqrt(2 + sqrt3)/2 = sqrt(3.73)/2 = 1.93/2 = 0.97#. OK

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