4.6 Graphs of Other Trigonometric Functions
Objectives: Understand the graphs of y = tan x,
y = cot x, y = csc x and y = sec x. Graph
variations of y = tan x, y = cot x, y = csc x and
y = sec x.
The Graph of y = tan x
Period:
The tangent function is an odd function.
The tangent function is undefined at
The Tangent Curve: The Graph of y = tan x and Its Characteristics
Period: π
Domain: All real numbers except odd multiples of .
Range: All real numbers
Vertical Asymptotes: odd multiples of .
x‐intercepts: occurs midway between each pair of consecutive asymptotes.
Odd function: origin symmetry
Main Points on Graph: and of the way between consecutive asymptotes have y‐coordinates of ‐1 and 1, respectively. 1
The Tangent Curve: The Graph of y = tan x and Its Characteristics
Graphing Variations of y = tan x
Graphing Variations of y = tan x
2
Example: Graphing a Tangent Function
Graph y  3 tan 2 x for 

4
x
3
.
4
Step 1 Find two consecutive asymptotes.
Example: Graphing a Tangent Function
Graph y  3 tan 2 x for 
 4

4
x


4
2
3
.
4
3
4
Step 2 Identify an x‐intercept, midway between the consecutive asymptotes.
The Cotangent Curve: The Graph of y = cot x and Its Characteristics
Period: π
Domain: All real numbers except odd multiples of π.
Range: All real numbers
Vertical Asymptotes: odd multiples of π.
x‐intercepts: occurs midway between each pair of consecutive asymptotes.
Odd function: origin symmetry
Main Points on Graph: and of the way between consecutive asymptotes have y‐coordinates of 1 and ‐
1, respectively. 3
The Cotangent Curve: The Graph of y = cot x and Its Characteristics
Graphing Variations of y = cot x
Graphing Variations of y = cot x (continued)
4
Example: Graphing a Cotangent Function
Graph y  1 cot  x
2
2
Step 1 Find two consecutive asymptotes.
Example: Graphing a Tangent Function
1 
Graph y  cot x
2
2
Step 2 Identify an x‐intercept, midway between the consecutive asymptotes.
4.5 Graphs of Other Trigonometric Functions
Objectives: Understand the graphs of y = tan x,
y = cot x, y = csc x and y = sec x. Graph
variations of y = tan x, y = cot x, y = csc x and
y = sec x.
5
The Graphs of y = csc x and y = sec x
We obtain the graphs of the cosecant and the secant curves by using the reciprocal identities
1
1
csc x 
and sec x 
.
sin x
cos x
We obtain the graph of y = csc x by taking reciprocals of the y‐values in the graph of y = sin x. Vertical asymptotes of y = csc x occur at the x‐intercepts of y = sin x.
We obtain the graph of y = sec x by taking reciprocals of the y‐values in the graph of y = cos x. Vertical asymptotes of y = sec x occur at the x‐intercepts of y = cos x.
The Cosecant Curve: The Graph of y = csc x and Its Characteristics The Cosecant Curve: The Graph of y = csc x and Its Characteristics
6
The Secant Curve: The Graph of y = sec x and Its Characteristics
The Secant Curve: The Graph of y = sec x and Its Characteristics Example: Using a Sine Curve to Obtain a Cosecant Curve


y  sin  x  
Use the graph of to obtain the graph of 4



y  csc  x   .
4

The x-intercepts of
the sine graph correspond
to the vertical asymptotes
of the cosecant graph.
7
Example: Graphing a Secant Function
Graph y = 2 sec 2x for 
3
3
x .
4
4
Graph the reciprocal function, y = 2 cos 2x. Amplitude:
We will use quarter-periods
to find x-values for the
five key points.
period:
0
Example: Graphing a Secant Function
3
3
Graph y = 2 sec 2x for

x .
4
4
The key points for our graph of y = 2 cos 2x are: The Six Curves of Trigonometry
8
The Six Curves of Trigonometry
The Six Curves of Trigonometry
9