Graphing Tangent and Cotangent - Trigonometry
Card 1 of 32
Which of the following is not a solution to the following equation?
Which of the following is not a solution to the following equation?
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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.
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
The only such choice is
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Which of the following best describes where the asymptotes are located on a tangent graph?
Which of the following best describes where the asymptotes are located on a tangent graph?
Tap to reveal answer
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.
In trigonometry,
It may also be thought of as
This is because
This means that whenever cosine is 0, tangent is undefined, because it would be evaluated by dividing by 0.
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The following is a graph of which function?
The following is a graph of which function?
Tap to reveal 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:
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
Hence the best choice is:
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Which of the following is the graph of 
?
Which of the following is the graph of 
Tap to reveal answer
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 
To derive the graph of 
and the graph of 
Vertical asymptotes will occur in the graph of 
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
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Which of the following is the graph of 
?
Which of the following is the graph of 
Tap to reveal answer
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 
.
We will begin by considering the general graph of 
The general equal of a tangent transformation equation is 
Period =
Period =
So the period is shortened from 
Now, we will consider 
And we are left with the graph of 
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Which of the following is the graph of 
?
Which of the following is the graph of 
Tap to reveal answer
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
To derive the graph of 
Now we will simply invert the tangent graph to get the cotangent graph
And we are left with our cotangent graph
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Which of the following is the graph of 
.
Which of the following is the graph of 
Tap to reveal answer
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 
.
First, we will consider the graph of 
The general form of a cotangent transformation function is 
This leaves us with our graph of 
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True or False: The period of tangent and cotangent function is 
.
True or False: The period of tangent and cotangent function is 
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This is because 
and 
causing the tangent function to be undefined at these points and forming a vertical asymptote. This is also true for the cotangent function because 
so wherever 
is zero or undefined, cotangent will be as well.
This is because 
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Which of the following is not a solution to the following equation?
Which of the following is not a solution to the following equation?
Tap to reveal 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.
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
The only such choice is
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Which of the following best describes where the asymptotes are located on a tangent graph?
Which of the following best describes where the asymptotes are located on a tangent graph?
Tap to reveal answer
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.
In trigonometry,
It may also be thought of as
This is because
This means that whenever cosine is 0, tangent is undefined, because it would be evaluated by dividing by 0.
← Didn't Know|Knew It →
The following is a graph of which function?
The following is a graph of which function?
Tap to reveal 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:
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
Hence the best choice is:
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True or False: The period of tangent and cotangent function is .
True or False: The period of tangent and cotangent function is
Tap to reveal answer
This is because and causing the tangent function to be undefined at these points and forming a vertical asymptote. This is also true for the cotangent function because so wherever is zero or undefined, cotangent will be as well.
This is because
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Which of the following is the graph of ?
Which of the following is the graph of
Tap to reveal answer
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
To derive the graph of
and the graph of
Vertical asymptotes will occur in the graph of
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
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Which of the following is the graph of ?
Which of the following is the graph of
Tap to reveal answer
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 .
We will begin by considering the general graph of
The general equal of a tangent transformation equation is
Period =
Period =
So the period is shortened from
Now, we will consider
And we are left with the graph of
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Which of the following is the graph of ?
Which of the following is the graph of
Tap to reveal answer
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
To derive the graph of
Now we will simply invert the tangent graph to get the cotangent graph
And we are left with our cotangent graph
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Which of the following is the graph of .
Which of the following is the graph of
Tap to reveal answer
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 .
First, we will consider the graph of
The general form of a cotangent transformation function is
This leaves us with our graph of
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Which of the following is not a solution to the following equation?
Which of the following is not a solution to the following equation?
Tap to reveal 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.
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
The only such choice is
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Which of the following best describes where the asymptotes are located on a tangent graph?
Which of the following best describes where the asymptotes are located on a tangent graph?
Tap to reveal answer
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.
In trigonometry,
It may also be thought of as
This is because
This means that whenever cosine is 0, tangent is undefined, because it would be evaluated by dividing by 0.
← Didn't Know|Knew It →
The following is a graph of which function?
The following is a graph of which function?
Tap to reveal 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:
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
Hence the best choice is:
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Which of the following is the graph of ?
Which of the following is the graph of
Tap to reveal answer
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
To derive the graph of
and the graph of
Vertical asymptotes will occur in the graph of
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
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