How do you find the exact value of cos 11π/12?

1 Answer
Oct 24, 2015

Find exact value of #cos ((11pi)/12)#

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

Explanation:

On the trig unit circle,
cos ((11pi)/12) = cos (-pi/12 + 2pi) = cos (-pi/12) = cos (pi/12).
First, find #cos (pi/12)#. Call cos #(pi/12) = cos x#
Apply the trig identity: #cos 2x = 2cos^2 x - 1#
cos 2x = cos (2pi)/12 = cos (pi/6) = sqrt3/2.
#sqrt3/2 = 2cos^2 x - 1.#
#cos^2 x = (2 + sqrt3)/4#
#cos x = cos (pi/12) = sqrt(2 + sqrt3)/2#.
The negative answer is rejected because the arc #(pi/12)# is in Quadrant I.
Check by calculator.
#cos pi/12 = cos 15^@ = 0.97#
sqrt(2 + sqrt3)/2 = 1.93/2 = 0.97 .OK