6.6 Graphing Other Trig Functions
We have established earlier in this chapter that tan x = sin x / cos x, sec x = 1/cos x, and
csc x = 1/sin x. Referring to the chart from section four on the values of sin x and cos x,
we are able to develop the values for the other trigonometric functions.
Complete the chart.
angle θ
0
π/6
sin θ
0
½
cos θ
1
3 /2
2 /2
½
π/2
2π/3
2 /2
3 /2
1
3 /2
3π/4
5π/6
2 /2
½
π
7π/6
0
-½
5π/4
4π/3
- 2 /2
-½
3π/2
5π/3
- 2 /2
- 3 /2
-1
- 3 /2
7π/4
11π/6
- 2 /2
-½
2π
0
2 /2
3 /2
1
π/4
π/3
0
-½
- 2 /2
- 3 /2
-1
- 3 /2
0
½
tan θ
0
3 /3
1
csc θ
und.
2
2
2
3
und.
und.
2
2 3 /3
1
0
-1
und.
und.
und.
-1
0
1
und.
Graphing y = tan x on the interval [0, 2π] we get:
Notice the length of
one period (or cycle)
of tan x is π.
Graphing y = cot x on the interval [0, 2π] we get:
Notice the length of
one period (or cycle)
of cot x is π.
sec θ
1
2 3 /3
Graphing y = sec x on the interval [0, 2π] we get:
Notice the length of
one period (or cycle)
of sec x is 2π.
By expanding the graph we will notice the length of one period to be 2π. The same holds
true for y = csc x. Graphing this function the interval [0, 2π] we get:
Notice the length of
one period (or cycle)
of csc x is 2π.
Unlike the graphs of y = sin x and y = cos x, the graphs of y = tan x, y = cot x, y = sec x,
and y = csc x do not have amplitudes. The graphs can still stretch vertically.
Example Graph y = 2tan x on [-π, 2π]
Example Graph y = ½ cot x on [-π, 2π]
The importance of these two graphs is knowing where the asymptotes lie and
remembering that the values of tan π/4 and cot π/4 are 1. The location of this point will
assist you in knowing which way the graph curves.
Graphing these trig functions with period changes is done similarly as we determined
these periods for sine and cosine.
period =
length of a normal period
coefficient on x
The normal periods for these functions are as follows:
tan x : period length of π
cot x : period length of π
sec x : period length of 2π
csc x : period length of 2π
Example
Graph y = tan (2x) on [0, 2π].
Note:
period = π/2
Try the following:
Graph on [0, 2π].
y = sec 2x
y = csc ( ½ x)
y = 2sec x
y = cot ( ½ x)
Answers:
Period : 2π/2 = π
Period : 2π/ ½ = 4π
Dotted image present
for you to see one cycle
Period : 2π/1 = 2π
Vertical stretch factor 2
Period : 2π/ ½ = 4π