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by cos(t) causes a reversal of the inequality so that sec(t) = sec(t) ≤ −1. In this case, as cos(t) → 0− ,
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sec(t) = cos(t) ≈ very small (−) ≈ very big (−), so that as cos(t) → 0− , we get sec(t) → −∞. Since
f (t) = cos(t) admits all of the values in [−1, 1], the function F (t) = sec(t) admits all of the values
in (−∞, −1] ∪ [1, ∞). Using set-builder notation, the range of F (t) = sec(t) can be written as
{u : u ≤ −1 or u ≥ 1}, or, more succinctly,7 as {u : |u| ≥ 1}.8 Similar arguments can be used
to determine the domains and ranges of the remaining three circular functions: csc(t), tan(t) and
cot(t). The reader is encouraged to do so. (See the Exercises.) For now, we gather these facts into
the theorem below.
7 Using Theorem 2.3 from Section 2.4.
Notice we have used the variable ‘u’ as the ‘dummy variable’ to describe the range elements. While there is no
mathematical reason to do this (we are describing a set of real numbers, and, as such, could use t again) we choose
u to help solidify the idea that these real numbers a...

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