Question

How do you find the exact values of cos(11pi/12) using the half angle formula?

Answer 1

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

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

Explanation:

Call `cos ((11pi)/12) = cos t`
`cos 2t = cos ((22pi)/12) = cos ((11pi)/6) = cos ((pi)/6) = sqrt3/2`
Apply the trig identity: `cos 2t = 2cos^2 t - 1`
`cos 2t = sqrt3/2 = 2cos^2 t - 1`
`2cos^2 t = 1 + sqrt3/2 = (2 + sqrt3)/2`
`cos^2 t = (2 + sqrt3)/4`
`cos t = cos ((11pi)/12) = +- sqrt(2 + sqrt3)/2.`
Since the arc ((11pi)/12) is located in Quadrant II, only the negative answer is accepted.
`cos ((11pi)/12) = - sqrt(2 + sqrt3)/2`

Check by calculator.
`Arc ((11pi)/12) = 165` deg-> `cos ((11pi)/12) = cos 165 = - 0.97.`
`- (sqrt(2 + sqrt3)/2) = - 0.97`. OK