Precalculus : Graphing Inverse Trig Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Graphing Inverse Trig Functions

State the domain and range of sin(x) and arcsin(x).

Possible Answers:

sin(x) domain: 

sin(x) range: 

arcsin(x) domain: 

arcsin(x) range: 

sin(x) domain: 

sin(x) range: 

arcsin(x) domain: 

arcsin(x) range: 

sin(x) domain: 

sin(x) range: 

arcsin(x) domain: 

arcsin(x) range: 

sin(x) domain: 

sin(x) range: 

arcsin(x) domain: 

arcsin(x) range: 

Correct answer:

sin(x) domain: 

sin(x) range: 

arcsin(x) domain: 

arcsin(x) range: 

Explanation:

The two graphs help provide a visual. The domain is all possible x values, and the range is all possible y values. The arcsin function turns the sine function on its side, but, in order to continue being a function (by passing the horizontal line test), the arcsine function's range is restricted to . It is not a coincidence that the range of the sine function is equal to the domain of the arcsin function because of the nature of inverse relationships.

y=sin x

Sinx

y=arcsin x

Arcsinx

 

sin(x) domain: 

sin(x) range: 

arcsin(x) domain: 

arcsin(x) range: 

Example Question #81 : Graphs And Inverses Of Trigonometric Functions

True or false: None of the inverses of the trigonometric functions are functions.

Possible Answers:

True

False

Correct answer:

True

Explanation:

This is true. Each of the inverse trig functions fails to pass the vertical line test, and is therefore not a function. For this reason, you will also see graphs of the inverse trig functions that restrict the domain such that is can be graphed as a function.

Example Question #1 : Graphing Inverse Trig Functions

Find the inverse of the function y = sin(x). Then graph both functions.

Possible Answers:

y = tan(x)

Graphs:

y = sin(x)

Sinx

y = tan(x)


Tan x

y = sin(x)

Graphs:

y = sin(x)

Sinx

y = sin(x)


Sinx

y = sin-1(x)

Graphs:

y = sin (x)

Sinx

y = sin-1(x)


Arcsinx

y = cos(x)

Graphs:

y = sin(x)

Sinx

y = cos(x)


Cosine

y = cos-1(x)

Graphs:

y = sin (x)

Sinx

y = cos-1(x)


Arccos

Correct answer:

y = sin-1(x)

Graphs:

y = sin (x)

Sinx

y = sin-1(x)


Arcsinx

Explanation:

To find the inverse of a function, exchange the values of x and y, then solve for y.

Change y = sin(x) to x = sin(y). Then solve for y:

x = sin(y) 

sin-1(x) = sin-1(sin(y))

sin-1(x) = y

 

Now graph each function:

y = sin(x)

Sinx

y = sin-1(x)

Arcsinx

If you look at the upper graph and restrict the domain to the interval [-/2, /2], you can more clearly see the relationship between the two graphs. 

Screen shot 2020 06 01 at 1.16.29 pm

To further illustrate the relationship between the two functions, examine the inputs and outputs of the two functions on the interval [-/2, /2]: 

y = sin(x)

Sintable

y = sin-1(x)

Arcsin table

Example Question #1351 : Pre Calculus

Which quadrant could arccot (−½) fall in?

Possible Answers:

II and IV

I and IV

III and IV

II and III

Correct answer:

II and IV

Explanation:

The cotangent function is negative in quadrants II and IV, so arccot (−½) could fall in either of these quadrants. The below image shows where each function is positive. Any that are not noted are negative. Since cotangent is positive in Quadrants I and III, it is negative in Quadrants II and IV.

Alg2 trig graphik 66

Example Question #81 : Graphs And Inverses Of Trigonometric Functions

This graph could be which of the following functions?

Arcsec

Possible Answers:

arcsin(x)

arcsec(x)

arccos(x)

arctan(x)

arccsc(x)

Correct answer:

arcsec(x)

Explanation:

This is the graph of arcsec(x). Think of a few points of the function sec(x): (-, -1), (0,1), (, -1). Now, interchange the x and y values to yield: (-1, -), (1, 0), (-1, ). Notice that (-1, -) and (1, 0) both exist on this graph. (-1, ) does not exist on this graph because in order to make this a function, the domain was required to be restricted such that it would pass the vertical line test. You can see how the graph of arcsec(x) relates to the graph of sec(x) by comparing the two:

Secx

Example Question #1351 : Pre Calculus

Which quadrant could arcsin (−½) fall in?

Possible Answers:

I and II

I and IV

II and III

III and IV

I and III

Correct answer:

III and IV

Explanation:

The sine function is negative in quadrants III and IV, so arcsin (−½) could fall in either of these quadrants. The below image shows where each function is positive. Any that are not noted are negative. Since sine is positive in Quadrants I and II, it is negative in Quadrants III and IV.

Alg2 trig graphik 66

Example Question #82 : Graphs And Inverses Of Trigonometric Functions

This graph could be the graph of which of the following functions?

Arccot

Possible Answers:

arctan(x)

arccsc(x)

tan(x)

arccot(x)

cot(x)

Correct answer:

arccot(x)

Explanation:

This is the graph of arccot(x). Think of a few characteristics of the function cot(x): it has a point at (-/2, 0), and vertical asymptotes at x=-, x=0, and x=. Notice that in the graph above, we have the point (0, -/2), which flips the x and y values of the point on cot(x), and we can also see horizontal asymptotes at both y=0 and y=. We don't also have another horizontal asymptote at y=- because we needed to restrict the domain of arccot(x) in order for it to be a function and pass the vertical line test. You can look at the graph of cot(x) below and compare the two and see their similarities:


Cotx

Example Question #2 : Graphing Inverse Trig Functions

Evaluate the following expression, assuming that all angles are in Quadrant I:

Screen shot 2020 06 01 at 11.00.58 am

Possible Answers:

12/5

5/13

13/5

12/13

5/12

Correct answer:

12/5

Explanation:

To solve Screen shot 2020 06 01 at 11.00.58 am, first, let . Then, Screen shot 2020 06 01 at 11.09.09 am. Because this value is positive, we know that this must be in Quadrant I or Quadrant II, but given the instructions, we can assume A is in Quadrant I.

Screen shot 2020 06 01 at 11.23.28 am

Next, recall that .

Using the Pythagorean Theorem, you can solve for the following:

x2 + y2 = r2

x2 + 122 = 132

x = 5

Therefore, . Now, solve: 

Screen shot 2020 06 01 at 11.00.58 amScreen shot 2020 06 01 at 11.46.42 am

Therefore, Screen shot 2020 06 01 at 11.00.58 amScreen shot 2020 06 01 at 11.47.50 am

Example Question #1351 : Pre Calculus

Evaluate the following expression, assuming that all angles are in Quadrant I:

Possible Answers:

Correct answer:

Explanation:

To solve , first, let . Then, . Because this value is positive, we know that this must be in Quadrant I or Quadrant IV, but given the instructions, we can assume A is in Quadrant I.

Next, recall that . Therefore

By definition, , assuming that , since the cosine function can only output values within this range.

Therefore, .

Example Question #1 : Graphing Inverse Trig Functions

State the domain and range of  and .

Possible Answers:

 domain: 

 range: 

 domain: 

 range: , such that 

 domain: 

 range: 

domain: 

 range:  

 domain: , such that 

 range: 

 domain: 

 range:  

 domain: 

 range: 

 domain: 

 range: 

 domain: 

 range: 

 domain: 

 range:  

Correct answer:

 domain: , such that 

 range: 

 domain: 

 range:  

Explanation:

The two graphs help provide a visual. The domain is all possible x values, and the range is all possible y values. The arctan function exchanges all x and y values of the tangent function. However, in order to continue being a function (by passing the horizontal line test), the arctangent function's range is restricted to . It is not a coincidence that the range of the tangent function is equal to the domain of the arctangent function because of the nature of inverse relationships.

Screen shot 2020 08 21 at 3.28.17 pm

Screen shot 2020 08 21 at 3.28.22 pm

 domain: , such that 

 range: 

 domain: 

 range:  

You can see that while the range and the  domain match, the  domain and the  range are not exact matches; this is so that  can be a function (by being restricted).

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