How do you evaluate Cos(pi/12) + Cos((5pi)/12)?

1 Answer
Nghi N.
Apr 1, 2016

(sqrt(2 + sqrt3)/2) + (sqrt(2 - sqrt3)/2)

Explanation:

S = cos (pi/12) + cos ((5pi)/12)
cos ((5pi)/12) = cos (-pi/2 + pi/2) = sin (pi/12).
Reminder: cos (pi/2 - a) = sin a --> Complementary arcs.
Evaluate sin (pi/12) and cos (pi/12) by using the trig identity:
cos 2a = 2cos^2 a - 1 = 1 - 2sin^2 a.
a. cos (pi/6) = sqrt3/2 = 2cos^2 (pi/12) - 1
2cos^2 (pi/12) = 1 + sqrt3/2 = (2 + sqrt3)/2
cos^2 (pi/12) = (2 + sqrt3)/4
cos (pi/12) = (sqrt(2 + sqrt3)/2) --> cos (pi/12) is positive.
b. cos (pi/6) = sqrt3/2 = 1 - 2sin^2 (pi/12)
2sin^2 (pi/12) = 1 - sqrt3/2 = (2 - sqrt3)/2
sin (pi/12) = (sqrt(2 - sqrt3)/2) --> sin (pi/12) is positive.

S = (sqrt(2 + sqrt3)/2) + (sqrt(2 - sqrt3)/2)

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