Question

How do you find the exact value of `sec((5pi)/12)`?

Answer 1

`2/(sqrt(2 - sqrt3)`

Explanation:

`sec ((5pi)/12) = 1/cos`. Find `cos ((5pi)/12)`
On the trig unit circle,
`cos ((5pi)/12) = cos ((6pi)/12 - pi/12) = cos (pi/2 - (pi)/12)`=
`= sin (pi/12)`. (complementary arcs)
Find `sin (pi/12)` by using trig identity:
`cos (pi/6) = sqrt3/3 = 1 - 2sin^2 (pi/12)`
`sin^2 (pi/12) = 1 -sqrt3/2 = (2 - sqrt3)/4`
`sin (pi/12) = sqrt(2 - sqrt3)/2`--> (sin (pi/12) is positive.
We have:
`cos ((5pi)/12) = sin (pi/12) = sqrt(2 - sqrt3)/2`
Finally, flip the value of cos.
`sec ((5pi)/12) = 2/(sqrt(2 - sqrt3)`