GRAPHS OF OTHER TRIG FUNCTIONS

xxf tan xxf cot

xxf sec xxf csc

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 here

2

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 over 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) = csc x we'll take the reciprocals of the sine values.

x sin x y = csc 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) = csc xchoose more values

x sin x y = csc x

6

7 0

2

1

2

31

6

112

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 = csc 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

52

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.