4.6
Graphs of Other
Trigonometric Functions
csc, sec, tan, & cot
Graphing csc and sec is done by graphing its
reciprocal function first.
'#
&
Graph y = 3 + 2 csc $ x + !
4"
%
+ sin x
v.s. = 3
A= 2
2
!
2
!
P=
=
= 2!
n
!
h.s. = "
4
1
5
9!
"
4
x-axis
3
!
! 1 4 3!
"
4
4
7!
4
Find the midpoint to locate y-axis. Draw vert. asy.
What is the range and domain of y?
Range:
("#,1][5,#)
Domain: To find the domain of csc x , set what comes
after csc =
k# , k " !
! x + ! = k!
4
&
D : %, x $ k& # , k " !
4
y = 2 – 4 sec (4x -
!)
!
y = 2 – 4 sec 4(x - )
4
!
"
4
6
2
x-axis
-2
!
h.s. =
4
v.s = 2
P=
!
2
A=4
!
4
3!
4
Range:
(" !,"2][6, ! )
To find the domain, set what comes after sec =
#
+ k# , k " !
2
!
3!
4 x " ! = + k!
4x =
+ k!
2
2
3% k%
D : $, x #
+
, k "!
8
4
y = tan x
!
P = =!
n
y = - tan x
Range:
1
Domain: set
whatever is
after the tan =
! !! !
6 43 2
!
!
"
"
4
2
-1
!
#
+ k# ,
2
k "!
y = -3 tan 2x
$ k$
!
,k "!
2 x " + k! x # +
4 2
2
Plug in –1, 0 and 1 for k to find vertical asymptotes.
v.a.’s at
! ! 3!
" , ,
4 4 4
!
P=
2
Range:
!
!
"
4
!
4
!
2
3!
4
y = cot x
Range:
!
Domain: set
whatever is
after the cot =
k# , k " !
Plug in -1, 0, and 1 for k to find the v.a.’s
Period
! !
= = =!
n 1
Extra Sinusoid