Question

How do you evaluate `sec^-1( 2/ sqrt3)`?

Answer 1

`pi/6`

Explanation:

Ask yourself: "secant of what angle gives me `2/sqrt3`?"

Since `sectheta=1/costheta`, an easier question to ask would be, "cosine of what angle gives me `sqrt3/2`?"

We know that `cos(pi/6)=sqrt3/2`, so `sec(pi/6)=2/sqrt3`.

While secant is also positive in quadrant four, and there are an infinite amount of coterminal angles where secant is `2/sqrt3`, the domain of `sec^-1(x)` is restricted from `(0,pi)`, so `pi/6` is the only valid angle.

Answer 2

`30^@` or `330^@`

Explanation:

`sec^-1(2/sqrt(3))`
`sectheta=2/sqrt(3)`
`1/costheta=2/sqrt(3)`
`costheta=sqrt(3)/2`
`theta=cos^-1(sqrt(3)/2)`
`theta=30^@`

However, since `2/sqrt(3)` is positive, there is more than one answer because according to the CAST rule, `cos` is positive in more than one quadrant:

![http://mathonline.wikidot.com/cast-rule]()

`cos` is positive in quadrants `1` and `4`.

To find the other angle that would also give an answer of `2/sqrt(3)`, subtract `30^@` from `360^@`. This ensures that your principal angle is in quadrant `4` :

`360^@-30^@`
`=330^@`

`:.`, `sec^-1(2/sqrt(3))` is `30^@` or `330^@`.