Question

How do you find the exact value of `sec165` using the half angle formula?

Answer 1

`- (2)/(sqrt(2 + sqrt3)`

Explanation:

Use the trig identity:
`2cos^2 a = 1 + cos 2a.` (1)
In this case a = 165, and 2a = 330.
sec 330 = 1/(cos 330). First, find cos (330).
`cos (330) = cos (- 30 + 360) = cos (-30) = cos 30 = sqrt3/2`.
Substitute in equation (1), we get:
`2cos^2 (165) = 1 + cos (330) = 1 + sqrt3/2 = (2 + sqrt3)/2`
`cos^2 (165) = (2 + sqrt3)/4`
`cos (165) = +- sqrt(2 + sqrt3)/2`.
Since 165 is in Quadrant II, then, we take the negative value.
`sec (165) = 1/(cos) = - 2/(sqrt(2 + sqrt3) =`