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

1 Answer
Bio
Mar 5, 2016

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)

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