Question

How do you use the half angle identify to find the exact value of `cos((7pi)/12)`?

Answer 1

• sqrt(2 + sqrt3)/2

Explanation:

Using trig unit circle and trig table -->
`cos ((7pi)/12) = cos (pi/12 + pi) = - cos (pi/12)`.
Find cos (pi/12) by using trig identity:
`2cos^2 a = 1 + cos 2a`
`2cos^2 (pi/12) = 1 + cos (pi/6) = 1 + sqrt3/2 = (2 + sqrt3)/2`
`cos^2 (pi/12) = (2 + sqrt3)/4`
`cos (pi/12) = +- sqrt(2 + sqrt3)/2.`
Since `cos (pi/12)` is positive, only the positive value is accepted.
Finally,
`cos ((7pi)/12) = - cos (pi/12) = - sqrt(2 + sqrt3)/2`
Check by calculator.
`cos ((7pi)/12) = - cos (pi/12) = - cos 15 = - 0.97`
`- sqrt(2 + sqrt3)/2 = sqrt(3.73)/2 = 1.93/2 = 0.97`. OK