Chapter 7

7.3 Graphs of Other Trigonometric Functions

Graph of y = 10 x and y = cot x

fig 7.3-1

Domain: x ≠ ± π⁄2, ± 3π⁄2, ...
Range: ℝ
Period: π
x-intercepts: x = 0, ±&pi, ±2π, ...
Asymptotes: x = ±π⁄2, ± 3π⁄2, ...
fig 7.3-2

Domain: x ≠ 0, ±π, ±2π, ...
Range: ℝ Period: π
x-intercepts: x = ±π⁄2, ± 3π⁄2, ...
Asymptotes: x = 0, ±π, ±2π, ...
Example 7.
Draw the graphs of y = a tan bx and y = a cot bx for a, b > 0.

Solution
y = tan x → y = a tan bx,
(x, y) → (x ⁄b, ay)
fig 7.3-3

y = cot x → y = a cot bx
(x, y) → (x ⁄b, ay)
fig 7.3-4

Graph of y = sec x   and y = csc x

fig 7.3-5

Domain: x ≠ ± π⁄2, ± 3π⁄2, ...
Range: y ≤ -1 or y ≥ 1
Period: 2π
x-intercepts: x = 0, ±&pi, ±2π, ...
Asymptotes: x = ±π⁄2, ± 3π⁄2, ...
fig 7.3-6

Domain: x ≠ 0, ±π, ±2π, ...
Range: y ≤ -1 or y ≥ 1
Period: 2π
x-intercepts: x = ±π⁄2, ± 3π⁄2, ...
Asymptotes: x = 0, ±π, ±2π, ...
Note that transform y = sec x to y = a sec bxand y = csc x to y = a csc bx with (x, y) → x⁄b, ay), we get the graph of y = a csc bx and y = csc bx.

Exercise 7.3
1. Draw the graph of (a) y = 1⁄2 tan πx
Solution
Domain: x ≠ ± 1⁄2, ± 3⁄2, ...
Range: ℝ
Period: 1
x-intercepts: x = 0, ±1, ±2, ...
Asymptotes: x = ±1⁄2, ± 3⁄2, ...
fig 7.3-f1

(b) y = 2 tan 1⁄2 x.

Domain: x ≠ 0, ±π, ±3π, ±5π ...
Range: ℝ Period: 2π
x-intercepts: x = ±2π, ± 4π, ...
Asymptotes: x = 0, ±π, ±3π, 5π ...
fig 7.3-2

2. Draw the graph of (a) y = 2 cot π⁄3 x
(b) y = 3 cot 2x.
Solution
(a) y = 2 cot π⁄3 x
Domain x ≠ 0, ±3, ±6, ...
Range ℝ
Period 3
x-intercept ±3⁄2, π9⁄2, ...
Asymptote x = 0, ±3, ±6, ...
fig 7.3-3

(b) y = 3 cot 2x.
Domain x ≠ 0, ±π⁄2, ± π, ± 3π⁄2, ...
Range ℝ
Period π⁄2
x-intercepts ±π⁄4, ±3π⁄4, ...
Asymptotes x = 0, ±π⁄2, ±π, ±3π⁄2, ...
fig 7.3-4

3. Draw the graph of (a) y = 2 sec πx
Solution
(a) y = 2 sec πx
Domain x ≠ ±0.5, ±1.5, ±2.5, ...
Range y ≤ -2 of y ≥ 2
Period 2
x-intercepts ±1⁄2, ±3⁄2, ...
Asymptotes x = ±0.5, ±1.5, ±2.5, ...
fig 7.3-5

(b) y = 2 csc 1⁄2 x.
Domain x ≠ ±0, ±2π, ±4π, ±6π ...
Range y ≤ -2 of y ≥ 2
Period 4π
x-intercept 0, 2π, 4π, ...
Asymptotes x = ±0, ±2π, ±4π, ±6π, ...
fig 7.3-6

4. Show that y = a tan bx , y = a cot bx, and y = a csc bx are odd functions.
Solution
Let y = f(x) = a tan bx
f(-x) = a tan (b(-x))
= a tan (-bx)
= -a tan bx
= - f(x)
∴ y = a tan bx is an odd function.

Let y = g(x) = a cot bx
g(-x) = a cot (b(-x))
= a cot (-bx)
= -a cot bx
= - g(x)
∴ y = a cot bx is an odd function.

Let y = h(x) = a cosec bx
h(-x) = a cosec (b(-x))
= a cosec (-bx)
= -a cosec bx
= - h(x)
∴ y = a cosec bx is an odd function.

5. Show that y = a sec bx is an even function.
Solution
Let y = f(x) = a sec bx
f(-x) = a sec (b(-x))
= a sec (-bx)
= a sec bx
= f(x)
∴ y = a sec bx is an even function.