CHAPTER 4
LESSON 6
Teacher’s Guide
The Functions y  tan x, y  csc x, y  sec x, and y  cot x
AW 4.6 & 4.7
MP 4.6
Objectives:
 To define the tangent function and the three reciprocal trigonometric functions as
functions of real numbers.
 To draw and analyze graphs of the tangent function and the three reciprocal
trigonometric functions as functions of real numbers.
Review
1. Using a calculator set in radian mode complete the following table for y  tan x .
Approximate tangent values to two decimal places.
Quadrant 1
0
x
tan x 0

4
1
Quadrant 2
Quadrant 3
Quadrant 4

3

2
2
3
3
4

5
4
4
3
3
2
1.73

-1.73
-1
0
1
1.73

5
3
7 2
4
-1.73 -1
0
2. Use the table from #1 above.
a)
In which quadrants is the tangent positive?
I and III
b)
In which quadrants is the tangent negative?
II and IV
c)
Where is the tangent undefined?
d)
What is the cotangent at the values where the tangent is undefined?
±
π 3π 5π
,± ,± ,L
2
2
2
_____0_____
3. Use a calculator set in radian mode. Determine the value to 4 decimal places of the
following:
a)
csc 3.2
17.1309
c) cot 4.5
0.216
b)
sec 1.6
–34.2471
d) csc 

4. a) For which values of x is the function y  sin x equal to 0?
n
b) What is the value of the function y  csc x at these values?
Undefined or Ø
c) How are these x-values indicated on the graph of y  csc x ? With vertical asymptotes
Example 1
Graph y  tan x ,
where 
3
5
x
2
2
Hint: First determine where the function y  tan x is undefined, there will be a vertical
asymptote at each such value of x.
Properties of the function y  tan x

Period
π

Domain
all real numbers except L -

Range
all real numbers

x-intercepts
0,±π,±2π, L

y-intercepts
0

asymptotes
x=±
3π π π 3π
,- , ,
,L
2
2 2 2
π 3π 5π
,±
,±
,L
2
2
2
Since y  tan x is undefined for all multiples of

, it has no amplitude.
2
Example 2
Graph y  tan 2 x where   x   . Hint: Since x is multiplied by 2, the graph has a horizontal
1
stretch by a factor of
compared to y  tan x , use this to determine the period of the new
2
graph.
Example 3
Graph, y  cot x where   x  3 .
The cosecant, secant, and cotangent functions of a real number are defined as the reciprocals of
the sine, cosine and tangent functions, respectively.
1
i.e.
csc x = sin x , provided sin x  0
1
sec x = cos x , provided cos x  0
1
cot x = tan x , provided tan x  0
The period of the reciprocal function is the same as the period of its primary trigonometric
function. Thus, the period of: y = csc x is 2 , y = sec x is 2 , and y = cot x is y  csc x  .
Example 4
Graph the function y  sec x , where 
3
5
x
.
2
2
a) What is the period of the function?
2π
b) State the domain and the range of the function.
-3π -π π 3π 5π
, , ,
,
Domain:
All real numbers except
2 2 2 2 2
Range:
y £ -1 or y ³ 1
c) What are the equations of the asymptotes of the function?
nπ
x=
where n is an odd integer.
2
Example 5
Graph, y  csc x where   x  3 .