Question

How do you simplify using the half angle formula `cos(-(7pi)/12)`?

Answer 1

`- sqrt(2 - sqrt3)/2`

Explanation:

`cos ((-7pi)/12) = cos ((-pi)/12 - (6pi)/12) = sin ((-pi)/12) =`
`= - sin ((pi)/12)`
To find `sin ((pi)/12)` use trig half angle identity:
`sin (t/2) = +- sqrt((1 - cos t)/2)`
In this case, `cos t = cos ((2pi)/12) = cos ((pi)/6) = sqrt3/2`
We get, with `(pi/12)` being in Quadrant 1,
`sin ((pi)/12) = sqrt(1 - sqrt3/2)/2 = sqrt(2 - sqrt3)/2`
Finally,
`cos ((-7pi)/12) = - sin (pi/12) = - sqrt(2 - sqrt3)/2`
Check by calculator.
`- sin (pi/12) = - sin 15^@ = - 0.258`
`- sqrt(2 - sqrt3)/2 = - 0.517/2 = - 0.258`. Proved.