Question

Use the half-angle formulas to determine the exact values of sine, cosine, and tangent of the angle.? 11pi/12

Answer 1

`sin (pi/12) = sqrt(2 - sqrt3)/2`
`cos (pi/12) = sqrt(2 + sqrt3)/2`
`tan (pi/12) = sqrt(2 - sqrt3)/sqrt(2 + sqrt3)`

Explanation:

`cos ((11pi)/12) = cos ((-pi)/12 + (12pi)/12) = cos (-pi/12 + pi) =`
`= - cos (-pi/12) = - cos (pi/12)`
Find `cos (pi/12)` by using the trig identity:
`2cos^2 a = 1 + cos 2a`.
In this case `cos 2a = cos ((2pi)/12) = cos (pi/6) = sqrt3/2` -->
`2cos^2 (pi/12) = 1 + sqrt3/2 = (2 + sqrt3)/2`
`cos^2 (pi/12) = (2 + sqrt3)/4`
`cos (pi/12) = sqrt(3 + sqrt3)/2` (cos pi/12 is positive)
`sin^2 (pi/12) = 1 - cos^2 (pi/12) = (4 - 2 - sqrt3)/4`
`sin (pi/12) = sqrt(2 - sqrt3)/2`
`tan (pi/12) = sqrt(2 - sqrt3)/sqrt(2 + sqrt3`