Question

How do you use the angle sum or difference identity to find the exact value of `cos((-11pi)/12)`?

Answer 1

`- sqrt(2 + sqrt3)/2`

Explanation:

`cos ((-11pi)/12) = cos (pi/12 - (12pi)/12) =`
`= cos (pi/12 - pi) = - cos (pi/12)`
Find `cos (pi/12)` by using trig identity:
`2cos^2 a = 1 + cos 2a`
Trig table of special arcs -->
`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 (Quadrant I), take the positive result.
Finally,
`cos ((-11pi)/12) = - cos (pi/12) = - sqrt(2 + sqrt3)/2`