Question

How do you find the domain & range for `sec theta`?

Answer 1

Domain

`{x| x = (k+1/2) pi, k in RR\\\ZZ}`

Range

`(-oo,1] uu [1,oo)`

Explanation:

Graph of `y = sec(x)`
graph{sec(x) [-20, 20, -10, 10]}
`sec theta = 1/cos theta`. As usual, division of zero is not allowed.

`cos theta = 0` when `theta = pi/2, {3pi}/2, {5pi}/2 ...`

In general, `cos theta = 0` when `theta = (k+1/2) pi`, for `k in ZZ`.

The domain for `sec(theta)` is any real number that

when subtracted `pi/2`, is not an integer multiple of `pi`.

In mathematical notations, it is

`{x| x = (k+1/2) pi, k in RR\\\ZZ}`

Note that the domain of `sec(theta)` and `tan(theta)` are identical.

The since `-1 <= cos(theta) <=1`, you can look at the graph of `y = 1/x`, and close in on the portion of `-1 <= x <=1`.
graph{1/x [-5, 5, -2.5, 2.5]}
You will see that it is either `y >= 1` or `y <= -1`. Similarly, for `sec(theta)`, it is either `sec(theta) >= 1` or `sec(theta) <= -1`.

In mathematical notations, it is

`(-oo,1] uu [1,oo)`