Chapter 10 - NCPN

The tangent function behaves asymptotically because y = tan θ is
undefined when θ = π ± a π where a is an integer and a ≠ 0. Recall that
y
2
tanθ
= =
sinθ
and is therefore undefined whenever cos θ = 0.
x cosθ
Characteristics of Tangent Functions
The generalized tangent function y = atan bθ, with
a ≠ 0, b > 0, and θ in radians has the following
properties:
• one cycle occurs in the interval – ≠
2b to ≠
2b .
• the period of the function is ≠ b .
• there are vertical asymptotes between each repeated
cycle of the graph.
Example 3 Graphing Tangent Functions
Graph the function y = tan πθ over the interval –1.5 ≤ θ ≤ 1.5.
Solution
Because b = π, the period of the function is ≠ =1 and one cycle occurs in
the interval −
π
= −
1
to ≠ 2π
2
=
1. So there will be a vertical asymptote at
2≠
2
x = − 1 2 , x = 1 , and at each additional unit. Choose appropriate scales for
2
the horizontal and vertical axes and graph the function.
Ongoing Assessment
Graph the function y = tan ≠ θ over the interval –6 ≤ θ ≤ 6.
4
458 Chapter 10 Trigonometric Functions and Identities