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Trigonometry : Graphs of Inverse Trigonometric Functions

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Example Questions

Example Question #1 : Graphs Of Inverse Trigonometric Functions

True or False: The inverse of the function $\sin (x)$ is also a function.

Possible Answers:

True

False

Correct answer:

False

Explanation:

Consider the graph of the function sin(x) . It passes the vertical line test, that is if a vertical line is drawn anywhere on the graph it only passes through a single point of the function. This means that sin(x) is a function.

Screen shot 2020 08 27 at 11.59.46 am

Now, for its inverse to also be a function it must pass the horizontal line test. This means that if a horizontal line is drawn anywhere on the graph it will only pass through one point.

Screen shot 2020 08 27 at 12.00.18 pm

This is not true, and we can also see that if we graph the inverse of sin(x) ( $sin^{-1}(x)$ ) that this does not pass the vertical line test and therefore is not a function. If you wish to graph the inverse of sin(x) , then you must restrict the domain so that your graph will pass the vertical line test.

Screen shot 2020 08 27 at 12.00.40 pm

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Example Question #1 : Graphs Of Inverse Trigonometric Functions

Which of the following is the graph of the inverse of sin(x) with $D: \frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ ?

Possible Answers:

Screen shot 2020 08 27 at 10.47.06 am

Screen shot 2020 08 27 at 10.45.10 am

Screen shot 2020 08 27 at 10.46.15 am

Screen shot 2020 08 27 at 10.47.13 am

Correct answer:

Screen shot 2020 08 27 at 10.46.15 am

Explanation:

Note that the inverse of sin(x) is not $\frac{1}{sin(x)}$ , that is the reciprocal. The inverse of sin(x) is $sin^{-1}(x)$ also written as arcsin(x) . The graph of sin(x) with $D: \frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ is as follows.

Screen shot 2020 08 27 at 10.45.10 am

And so the inverse of this graph must be the following with $D: -1 \leq x \leq 1$ and $R: \frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$

Screen shot 2020 08 27 at 10.46.15 am

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Example Question #3 : Graphs Of Inverse Trigonometric Functions

Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?

Possible Answers:

The and values are switched so where y = cos(x) for the parent function, x = cos(y) for the inverse function.

Let $x = \frac{1}{x}$ so where y = cos(x) for the parent function, $y = cos(\frac{1}{x})$ for the inverse function.

The values are swapped with $\frac{1}{y}$ so where y = cos(x) for the parent function, $x = cos(\frac{1}{y})$ for the inverse function.

Let $y = \frac{1}{y}$ so where y = cos(x) for the parent function $y = \frac{1}{cos(x)}$ for the inverse function.

Correct answer:

The and values are switched so where y = cos(x) for the parent function, x = cos(y) for the inverse function.

Explanation:

To find an inverse function you swap the and values. Take cos(x) for example, to find the inverse we use the following method.

y = cos(x)

x = cos(y) (swap the and values)

$cos^{-1}(x) = y$ (solving for )

$y = cos^{-1}(x)$

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Example Question #2 : Graphs Of Inverse Trigonometric Functions

Which of the following represents the graph of $tan^{-1}(x)$ with $R: \frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$ ?

Possible Answers:

Screen shot 2020 08 28 at 9.01.00 am

Screen shot 2020 08 28 at 9.02.30 am

Screen shot 2020 08 28 at 9.02.37 am

Screen shot 2020 08 28 at 9.00.48 am

Correct answer:

Screen shot 2020 08 28 at 9.01.00 am

Explanation:

If we are looking for the graph of $tan^{-1}(x)$ with $R: \frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ , that means this is the inverse of tan(x) with $D:\frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ . The graph of tan(x) with $D:\frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ is

Screen shot 2020 08 28 at 9.00.48 am

Switching the and values to graph the inverse we get the graph

Screen shot 2020 08 28 at 9.01.00 am

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Example Question #3 : Graphs Of Inverse Trigonometric Functions

Which of the following is the graph of $cos^{-1}(x) + 2$ with $D: -1 \leq x \leq 1$ ?

Possible Answers:

Screen shot 2020 08 27 at 2.14.41 pm

Screen shot 2020 08 27 at 2.14.55 pm

Screen shot 2020 08 27 at 2.13.29 pm

Screen shot 2020 08 27 at 2.14.48 pm

Correct answer:

Screen shot 2020 08 27 at 2.13.29 pm

Explanation:

We first need to think about the graph of the function $cos^{-1}(x)$ .

Screen shot 2020 08 27 at 2.13.25 pm

Using the formula y = D + Acos(B(x - C)) where is the vertical shift, we have to perform a transformation of moving the function $cos^{-1}(x)$ up two units on the graph.

Screen shot 2020 08 27 at 2.13.29 pm

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Example Question #6 : Graphs Of Inverse Trigonometric Functions

Which of the following is the correct graph and range of the inverse function of 2sin(x) with $D:\frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ ?

Possible Answers:

$R:-\pi \leq y \leq \pi$

Screen shot 2020 08 27 at 3.51.45 pm

$R: \frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$

Screen shot 2020 08 27 at 3.51.45 pm

$R:-\pi \leq y \leq \pi$

Screen shot 2020 08 27 at 3.51.52 pm

$R: \frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$

Screen shot 2020 08 27 at 3.51.52 pm

Correct answer:

$R: \frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$

Screen shot 2020 08 27 at 3.51.52 pm

Explanation:

First, we must solve for the inverse of 2sin(x)

y = 2sin(x)

x = 2sin(y)

$\frac{x}{2} = sin(y)$

$sin^-1(\frac{x}{2}) = y$

So now we are trying to find the range of and plot the function $y = sin^{-1}(\frac{x}{2})$ . Let’s start with the graph of $sin^{-1}(x)$ . We know the domain is [-1,1] .

Screen shot 2020 08 27 at 3.51.45 pm

Now using the formula $y = D + Acos^{-1}(B(x-C))$ where $\frac{2\pi}{B}$ = Period, the period of $y = sin^{-1}(\frac{x}{2})$ is $4\pi$ . And so we perform a transformation to the graph of $sin^{-1}(x)$ to change the period from $2\pi$ to $4\pi$ .

Screen shot 2020 08 27 at 3.51.52 pm

We can see that the graph has a range of $[\frac{-\pi}{2},\frac{\pi}{2}]$

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Example Question #7 : Graphs Of Inverse Trigonometric Functions

True or False: The domain for arctan(x) will always be all real numbers no matter the value of or any transformations applied to the tangent function.

Possible Answers:

False

True

Correct answer:

True

Explanation:

This is true because just as the range of tan(x) is all real numbers due to the vertical asymptotes of the function, the function arctan(x) extends to all values of but is limited in its values of . No matter the transformations applied, all values of will still be reached.

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Example Question #8 : Graphs Of Inverse Trigonometric Functions

Which of the following is the graph of $\frac{1}{2}cos^{-1}(x) + 1$ ?

Possible Answers:

Screen shot 2020 08 27 at 1.54.54 pm

Screen shot 2020 08 27 at 1.55.06 pm

Screen shot 2020 08 27 at 1.55.00 pm

Screen shot 2020 08 27 at 1.57.17 pm

Correct answer:

Screen shot 2020 08 27 at 1.55.06 pm

Explanation:

First, we must consider the graph of $cos^{-1}(x)$ .

Screen shot 2020 08 27 at 1.54.54 pm

Using the formula $D + Acos^{-1}(B(x-C))$ we can apply the transformations step-by-step. First we will transform the amplitude, so $A = \frac{1}{2}$ so we must shorten the amplitude to $\frac{1}{2}$ .

Screen shot 2020 08 27 at 1.55.00 pm

Now we must apply a vertical shift of one unit since D=1 . This leaves us with our answer.

Screen shot 2020 08 27 at 1.55.06 pm

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Trigonometry : Graphs of Inverse Trigonometric Functions

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Graphs Of Inverse Trigonometric Functions

Example Question #1 : Graphs Of Inverse Trigonometric Functions

Example Question #3 : Graphs Of Inverse Trigonometric Functions

Example Question #2 : Graphs Of Inverse Trigonometric Functions

Example Question #3 : Graphs Of Inverse Trigonometric Functions

Example Question #6 : Graphs Of Inverse Trigonometric Functions

Example Question #7 : Graphs Of Inverse Trigonometric Functions

Example Question #8 : Graphs Of Inverse Trigonometric Functions

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