College Trigonometry, 2011a

802 Foundations of Trigonometry
all of these properties are direct results of them being reciprocals of the cosine and sine functions,
respectively.
Theorem 10.24. Properties of the Secant and Cosecant Functions
ˆ The function F (x) = sec(x)
– has domain { x : x ≠ π 2 + πk, k is an integer} =
– has range {y : |y| ≥1} =(−∞, −1] ∪ [1, ∞)
– is continuous and smooth on its domain
– is even
– has period 2π
ˆ The function G(x) = csc(x)
– has domain {x : x ≠ πk, k is an integer} =
– has range {y : |y| ≥1} =(−∞, −1] ∪ [1, ∞)
– is continuous and smooth on its domain
– is odd
– has period 2π
∞⋃
k=−∞
∞⋃
k=−∞
( (2k +1)π
,
2
(kπ, (k +1)π)
)
(2k +3)π
2
In the next example, we discuss graphing more general secant and cosecant curves.
Example 10.5.4. Graph one cycle of the following functions. State the period of each.
1. f(x) =1− 2 sec(2x)
Solution.
2. g(x) =
csc(π − πx) − 5
3
1. To graph y =1− 2 sec(2x), we follow the same procedure as in Example 10.5.1. First, we set
the argument of secant, 2x, equal to the ‘quarter marks’ 0, π 2 , π, 3π 2
and 2π and solve for x.
a 2x = a x
0 2x =0 0
π
2
2x = π π
2 4
π
π 2x = π
3π
2
2x = 3π 3π
2 4
2π 2x =2π π
2