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GRAPHS OF OTHER TRIG FUNCTIONS

credit: Shawna Haider

GRAPHS OF OTHER

TRIG FUNCTIONS

xxf tan xxf cot

xxf sec xxf cosec

6

We are interested in the graph of y = f(x) = tan x

Start with a "t" chart and let's choose values from our unit

circle and find the tangent values. Tangent has a period

of so it will repeat every .

x y = tan x

3

2

undefined

73.13

4

1

6

58.0

3

3

x

y

2

3

2

3

6

would mean there is

a vertical asymptote

here

6

y = tan xLet's choose more values.

x y = tan x

6

0 0

73.13

4

1

3

58.03

3

x

y

2

3

2

3

6

would mean there is a

vertical asymptote here2

undefined

Since we went from we have one complete period 2

to2

Let's see what the graph would look like for y = tan x

for 3 complete periods.

The vertical lines

are not part of

the graph but are

the asymptotes.

If you use your

graphing

calculator it will

probably put

those in as well

as showing the

graph.

Transformations apply as usual. Let’s try one.

4tan2

xy

reflect about x-axis

right /4

up 2

xy tan

xy tan

4tan

xy

4tan2

xy

Since the period of tangent is ,

the period of tan x is:

T

xy 2tan The period would be /2

y = tan x

y = tan 2x

3

6

What about the graph of y = f(x) = cot x?

This would be the reciprocal of tangent so let's take our

tangent values and "flip" them over.

x tan x y = cot x

3

2

undefined

3

4

1

6

3

1

x

y

2

2

3

6

0

58.03

1

1

73.13

3

6

x tan x y = cot x

6

0 0

3

4

1

3

3

1

x

y

2

2

3

6

undefined

58.03

1

1

73.13

y = cot xLet's choose more values.

2

undefined 0

We need to see

more than one

period to get a

good picture of

this.

y = cot x

Again the vertical lines are

not part of the graph but

are the asymptotes.

Let's look at the tangent graph

again to compare these.

Notice vertical

asymptotes of one are

zeros of the other.

y = tan x

For the graph of y = f(x) = cosec x we'll take the

reciprocals of the sine values.

x sin x y = cosec x

6

0 0

2

1

2

1

6

5

When we graph these rather than plot points

after we see this, we'll use the sine graph as a

sketching aid and then get the cosecant graph.

x

y

1

- 1

undefined

2

1

2

12

y = f(x) = cosec xchoose more values

x sin x y = cosec x

6

7

0

2

1

2

31

6

11

2

1

We'll use the sine graph as the sketching aid.

x

y

1

- 12 0

6

2

undefined

2

1

2

undefined

When the sine

is 0 the

cosecant will

have an

asymptote.

Again the vertical lines are not part of the graph but are where

the cosecant is undefined (which is where the sine was 0

although this graphing program seemed to draw them a little to

the right. They should cross the x-axis where the sine is 0)

Let's add in the

graph of the sine

function so you

can see how if

you graph it, you

can then easily

use it to graph

the cosecant.

Let's look over a few periods at the graph of y = cosec x

For the graph of y = f(x) = sec x we'll take the

reciprocal of the cosine values.

x cos x y = sec x

3

0 1

2

1

2

0

3

22

1

x

y

1

- 1 6

1

2

undefined

2

y = f(x) = sec x Choose more values.

x cos x y = sec x

3

4

1

2

1

2

30

3

5

2

1

Again the cosine graph will help graph the secant graph.

x

y

1

- 1 6

2 1

1

2

undefined

2

1

Again the vertical lines are not part of the graph but are where

the secant is undefined (which is where the cosine was 0)

Let's look over a few periods at the graph of y =sec x

Let's add in the graph of the cosine function so you can

see how if you graph it, you can then easily use it to

graph the secant.

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