Question

How do you find the exact functional value cos (-11/12 pi) using the cosine sum or difference identity?

Answer 1

Find `cos ((-11pi)/12)`

Ans: `- sqrt(2 + sqrt3)/2`

Explanation:

On the trig unit circle,

`cos ((-11pi)/12) = cos (pi/12 - (12pi)/12) = cos (pi/12 - pi) =`
`= -cos (pi/12)`
Find `cos (pi/12)`. Call `cos (pi/12) = cos t`
Apply the trig identity: `cos 2t = 2cos^2 t - 1`

`cos 2t = cos pi/6 = sqrt3/2 = 2cos^2 t -1`
`cos^2 t = (2 + sqrt3)/4`
`cos t = cos (pi/12) = sqrt(2 + sqrt3)/2`
Since the arc (--11pi)/12 is located in Quadrant III,
`cos ((-11pi)/12) = - cos (pi/12) = - sqrt(2 + sqrt3)/2`