# Trigonometry : Graphing Tangent and Cotangent

## Example Questions

### Example Question #1 : Graphing Tangent And Cotangent

Which of the following best describes where the asymptotes are located on a tangent graph?

Angle measures where the cosine is 0, such as

Angle measures where the sine and cosine are equal, such as .

Angle measures where the sine is 0, such as .

Angle measures where the tangent cannot be calculated.

Angle measures where the cosine is 0, such as

Explanation:

In trigonometry,

.

It may also be thought of as .

This is because

and , so

This means that whenever cosine is 0, tangent is undefined, because it would be evaluated by dividing by 0.

### Example Question #2 : Graphing Tangent And Cotangent

Which of the following is not a solution to the following equation?

...

Explanation:

We can factor the original expression as follows:

So from this equation we conclude either that:

or

So any number that is not some integer multiple of  away from these two solutions is not a solution to the original equation.

The only such choice is  , which is  ; n is not an integer, therefore it is not a solution.

### Example Question #3 : Graphing Tangent And Cotangent

The following is a graph of which function?

Explanation:

The graph looks to have infinite range, but multiple vertical asymptotes. That means we can limit our choices to tangent and cotangent graphs.

Furthermore, we observe that the graph starts at the bottom and increases from left to right, consistent with tangent graphs. So we narrow our focus to the choices involving tangents.

To decide between the remaining two graphs, observe that y-intercept (where x=0) of our graph is (0,1).

Now   evaluated at  is , which means that we need a vertical shift of  unit.

Hence the best choice is:

### Example Question #4 : Graphing Tangent And Cotangent

Which of the following is the graph of ?

Explanation:

To derive the graph of  , recall that .  The graph of   is

and the graph of is

Vertical asymptotes will occur in the graph of   whenever .  This is because the denominator of the tangent function will be equal to zero whenever the cosine function is equal to zero and then the entire function will be undefined at those points.  Wherever cosine crosses the x-axis a vertical asymptote will occur.  If we overlay the sine and cosine graphs we see the following:

So our tangent graph will follow the same form as the sine and cosine graphs when they are increasing, but will have vertical asymptotes wherever cosine crosses the x-axis.

And we are left with our graph of

### Example Question #5 : Graphing Tangent And Cotangent

Which of the following is the graph of ?

Explanation:

We will begin by considering the general graph of  and apply transformations step by step to produce a graph of .  The graph of  is

The general equal of a tangent transformation equation is .  A is the amplitude of the graph of tangent.  Here,  so we do not need to apply a transformation here.  Next, we will consider the period.  The period of the tangent function is equal to .  So the period of our graph would be

Period =

Period =

So the period is shortened from  to .

Now, we will consider  is the phase shift of our graph.  So we will shift our graph  units to the left.  This does not change our graph visually due to the period now being .  Lastly, we will consider  is the vertical shift of our graph, and so we must shift our graph 1 unit up.

And we are left with the graph of

### Example Question #6 : Graphing Tangent And Cotangent

Which of the following is the graph of ?

Explanation:

To derive the graph of  recall that .  So the tangent and cotangent graphs are reciprocals of one another.  We will consider the tangent graph since it is one we are more familiar with:

Now we will simply invert the tangent graph to get the cotangent graph

And we are left with our cotangent graph

### Example Question #7 : Graphing Tangent And Cotangent

Which of the following is the graph of .

Explanation:

First, we will consider the graph of  and apply transformations step-by-step.  The graph of  is

The general form of a cotangent transformation function is .  For our function ,  and so we need to increase the amplitude 4 units.

here so we do not need to make any changes to the period of this graph.  , giving us a negative phase shift of  units.

This leaves us with our graph of .

### Example Question #1 : Graphing Tangent And Cotangent

True or False: The period of tangent and cotangent function is .