Question

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

Answer 1

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