What is the exact value of the following expression? cos (−5π/12)

Use the sum and difference identities to determine the exact value.

1 Answer
Jun 5, 2018

-(sqrt(2 + sqrt3)/2)(2+32)

Explanation:

cos ((-5pi)/12) = cos (pi/12 - pi) = - cos (pi/12)cos(5π12)=cos(π12π)=cos(π12)
Find cos (pi/12)cos(π12) by using trig identity:'
cos 2a = 2cos^2 a - 1cos2a=2cos2a1.
In this case
cos 2a = cos (pi/6) = sqrt3/2cos2a=cos(π6)=32
2cos^2 (pi/12) = 1 + sqrt3/2 = (2 + sqrt3)/22cos2(π12)=1+32=2+32
cos^2 (pi/12) = (2 + sqrt3)/4cos2(π12)=2+34
cos (pi/12) = sqrt(2 + sqrt3)/2cos(π12)=2+32 (because cos (pi/12) is positive)
Finally,
cos ((-5pi)/12) = - cos (pi/12) = - sqrt(2 + sqrt3)/2cos(5π12)=cos(π12)=2+32