Question

Using the unit circle, how do you find the value of the trigonometric function: sec(-227pi/4) ?

Answer 1

`sec(-227pi/4) = -sqrt2`

Explanation:

First let's work that value within the secant to reveal the biggest integer amount of `pi`s, so we can eliminate them by looking at the period:
`sec(-227pi/4) = sec(-56pi -3pi/4)`

The period of the secant is `2pi` so
`sec(-56pi -3pi/4) = sec(-28(2pi)-3pi/4) = sec(-3pi/4)`

The secant is an even function, that is, `f(-x) = f(x)`
So, `sec(-3pi/4) = sec(3pi/4)`

Now, we can either just look at the unit circle or continue using formulas. Since you specifically asked for the unit circle, we should look for `cos(3pi/4)`

We see that `cos(3pi/4) = -sqrt2/2` or `cos(3pi/4) = -1/sqrt2`
So, `sec(3pi/4) = 1/cos(3pi/4) = -sqrt2`