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## Example Questions

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

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

**Possible Answers:**

Angle measures where the tangent cannot be calculated.

Angle measures where the sine is 0, such as .

Angle measures where the cosine is 0, such as

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

**Correct answer:**

Angle measures where the cosine is 0, such as

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?

**Possible Answers:**

...

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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:

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