4.6Graphs of Other

Trigonometric Functionscsc, sec, tan, & cot

Graphing csc and sec is done by graphing its reciprocal function first.

Graph y = 3 + 2 csc ⎟⎠

⎞⎜⎝

⎛ +4

πx

v.s. = A =P =

h.s. =

32

πππ2

1

22==

n

4

π−

+ sin x

4

π− 4

7π4

9π−

Find the midpoint to locate y-axis.

4

3π

Draw vert. asy.

3

5

1

x-axis

4

π

What is the range and domain of y?

Range:

€

−∞,1( ] 5,∞[ )

Domain: To find the domain of csc x , set what comes

after csc = Ζ∈kk ,π

ππkx =+

4

Ζ∈−≠ℜ kkxD ,4

,:π

π

y = 2 – 4 sec (4x - ) πy = 2 – 4 sec 4(x - )

4

π

4

π

P = 2

π

4

3π4

π−

2

6

-2

v.s = 2 h.s. =4

π

A = 4

x-axis

Range: ( ] [ )∞−∞− ,62,

To find the domain, set what comes after sec =

Ζ∈+ kk ,2

ππ

πππ kx +=−2

4 ππkx +=

2

34

Ζ∈+≠ℜ kk

xD ,48

3,:

ππ

y = tan x

4

π2

π6

π3

π

1

-1

2

π−

4

π−

y = - tan x

Range: ℜDomain: set whatever is after the tan =

Ζ∈

+

k

k ,2

ππ

ππ==

nP

y = -3 tan 2x ππkx +≠

22 Ζ∈+≠ k

kx ,

24

ππ

Plug in –1, 0 and 1 for k to find vertical asymptotes.

v.a.’s at 4

3,

4,

4

πππ−

4

π−

4

π4

3π2

π

Range: ℜ

=P2

π

Range: ℜDomain: set whatever is after the cot = Ζ∈kk ,π

Period n

π= ππ

==1

y = cot x

Plug in -1, 0, and 1 for k to find the v.a.’s

Extra Sinusoid