Question

How do I find the value of sec pi/12?

Answer 1

Find: `sec (pi/12)`

Explanation:

`sec (pi/12) = 1/(cos pi/12)`. Find `cos (pi/12).`
Call `cos (pi/12) = cos a` -->`cos 2a = cos ((2pi)/12) = cos (pi/6) = sqrt3/2`
Apply the trig identity: `cos 2a = 2cos^2 a - 1.`
`cos 2a = sqrt3/2 = 2cos^2 a - 1`
`cos^2 a = 1 = sqrt3/2 = (2 + sqrt3)/2`
`2cos^2 a = (2 + sqrt3)/4`
`cos a = +- sqrt(2 + sqrt3)/2`
Since the arc (pi/12) is in Quadrant I, its cos is positive.
Therefor, `cos (pi/12) = cos a = sqrt(2 + sqrt3)/2`
`sec (pi/12) = 1/(cos a) = 2/sqrt(2 + sqrt3)`.
Check by calculator:
`cos (pi/12) = cos 15 = 0.97`
`sqrt(2 + sqrt3)/2 = 1.93/2 = 0.97`. OK