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Section 5.4 - Inverse Trigonometric Functions

The Inverse Sine Function

Consider the graph of the sine function ( ) sin( )f x x .

The function is a periodic function. That means that the functions repeats its values in

regular intervals, which we call the period.

Is it one to one?

If the function is not one-to-one, we run into problems when we consider the inverse of

the function. What we want to do with the sine function is to restrict the values for sine.

When we make a careful restriction, we can get something that IS one-to-one.

If we limit the function to the interval ,2 2

, the graph will look like this:

Restricted Sine function

Domain: ,2 2

Range: 1,1

On this limited interval, we have a one-to-one sine function.

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Being an one-to-one function implies that the inverse of it does exist.

Notation for the inverse of sine: ).arcsin()(or )(sin)( 1 xxfxxf

Restricted Sine Function (Blue) Inverse Sine Function (Red)

Domain: ,2 2

Domain: 1,1

Range: 1,1 Range: ,2 2

(quadrants 1 and 4)

To find the value of )(sin 1 x, find the number in the interval ,

2 2

whose sine is x.

Example: 11 1sin sin

6 2 2 6

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The Inverse Cosine Function

Let’s do the same thing with the cosine function ( ) cos( )f x x , which is not one-to-one.

Here’s the graph of ( ) cos( )f x x .

If we limit the function to the interval 0, , however, the function IS one-to-one.

Here’s the graph of the restricted cosine function.

Restricted Cosine Function

Domain: 0,

Range: 1,1

On this limited interval, we have a one-to-one cosine function.

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Being an one-to-one function implies that the inverse of it does exist.

Notation for the inverse of cosine: ).arccos()(or )(cos)( 1 xxfxxf

Restricted Cosine Function (Blue) Inverse Cosine Function (Red)

Domain: 0, Domain: 1,1

Range: 1,1 Range: 0, (quadrants 1 and 2)

To find the value of )(cos 1 x, find the number in the interval 0, whose cosine is x.

Example: 12 2cos cos

4 2 2 4

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The Inverse Tangent Function

Here’s the graph of tangent function ( ) tan( )f x x , which is not one-to-one:

If we restrict the function to the interval ,2 2

, then the function IS one-to-one.

Restricted Tangent Function

Domain: ,2 2

Range: ,

On this limited interval, we have a one-to-one tangent function.

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Being an one-to-one function implies that the inverse of it does exist.

Notation for the inverse of tangent: ).arctan()(or )(tan)( 1 xxfxxf

Restricted Tangent Function (Blue) Inverse Tangent Function (Red)

Domain: ,2 2

Domain: ,

Range: , Range: ,2 2

(quadrants 1 and 4)

To find the value of )(tan 1 x, find the number in the interval ,

2 2

whose tangent is x.

Example: 1tan 1 tan 14 4

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The Inverse Cotangent Function

Here’s the graph of cotangent function )cot()( xxf , which is not one-to-one:

If we restrict the function to the interval ),0( , then the function IS one-to-one.

Restricted Cotangent Function

Domain: ),0(

Range: ,

On this limited interval, we have a one-to-one cotangent function.

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Being an one-to-one function implies that the inverse of it does exist.

Notation for the inverse of cotangent: ).cot()(or )(cot)( 1 xarcxfxxf

Restricted Cotangent Function (Blue) Inverse Cotangent Function (Red)

Domain: ),0( Domain: ,

Range: , Range: ),0( (quadrants 1 and 2)

To find the value of )(cot 1 x

, find the number in the interval ),0( whose cotangent is x.

Example: 36

cot6

3cot 1

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The Inverse Secant Function

Here’s the graph of secant function )sec()( xxf , which is not one-to-one:

If we restrict the function to the interval

,

22,0 , then the function IS one-to-one.

Restricted Secant Function

Domain:

,

22,0

Range: ,11,

On this limited interval, we have a one-to-one secant function.

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Being an one-to-one function implies that the inverse of it does exist.

Notation for the inverse of secant: ).sec()(or )(sec)( 1 xarcxfxxf

Restricted Secant Function (Blue) Inverse Secant Function (Red)

Domain:

,

22,0 Domain: ,11,

Range: ,11, Range:

,

22,0 (quadrants 1 and 2)

To find the value of )(sec 1 x, find the number in the interval

,

22,0

whose secant is x.

Example: 23

sec3

2sec 1

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The Inverse Cosecant Function

Here’s the graph of cosecant function )csc()( xxf , which is not one-to-one:

If we restrict the function to interval

2,00,

2

, then the function IS one-to-one.

Restricted Tangent Function

Domain:

2,00,

2

Range: ,11,

On this limited interval, we have a one-to-one cosecant function.

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Being an one-to-one function implies that the inverse of it does exist.

Notation for the inverse of cosecant: ).csc()(or )(csc)( 1 xarcxfxxf

Restricted Cosecant Function (Blue) Inverse Cosecant Function (Red)

Domain:

2,00,

2

Domain: ,11,

Range: ,11, Range:

2,00,

2

(quadrants 1 and 4)

To find the value of )(csc 1 x, find the number in the interval

2,00,

2

whose cosecant is x.

Example: 26

csc6

2csc 1

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Note: We always give inverse trigonometric angles in radians.

Example 1: Compute each of the following:

a) 1 1sin

2

b) 1tan 3

c) arccos(0)

d) 1 2sin

2

.

e) 1 1sin

2

f)

2

3cos 1 .

g) arctan( 1).

h) ).2(sec 1

i) )0(csc 1

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NOTE: Domains of inverse trig functions:

)(sin)( 1 xxf ; [-1,1]

)(cos)( 1 xxf ; [-1,1]

)(tan)( 1 xxf ; ),(

)(cot)( 1 xxf ; ),(

)(sec)( 1 xxf ; ),1[]1,(

)(csc)( 1 xxf ; ),1[]1,(

For example; )2(sin 1 or 2cos 1 are not defined.

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Composition of Trigonometric Functions with their Inverses and viceversa

Recall that if f and g are inverse functions then the following hold:

xxgf ))(( for each x in the domain of g .

xxfg ))(( for each x in the domain of f .

When a trigonometric function and its inverse are composed, we need to be careful about

giving an answer that is in the range of the inverse trig function.

xx coscos 1 if ],0[ x

xx sinsin 1 if

2,

2

x

xx tantan 1 if

2,

2

x

Examples: 88

sinsin 1

but 8

7

8

7sinsin 1

88

coscos 1

but 8

9

8

9coscos 1

88

tantan 1

but 8

7

8

7tantan 1

If the inverse trigonometric function is the inner function, then our job is easier.

xx ]cos[cos 1 for any number x such that 11 x .

xx ]sin[sin 1 for any number x such that 11 x .

xx ]tan[tan 1 for any number x.

Examples:

5

1

5

1sinsin 1

7

2

7

2coscos 1

4

1]

4

1tan[tan 1

5]5tan[tan 1 .

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Example 2: Find the exact value of the following:

a) 1 7sin sin .

6

b) 1 4cos cos .

3

c) 1 3tan tan .

4

d) 5

arcsin sin3

e) 1 2tan tan

3

Example 3: Find the exact value of the following:

a) 1 1cos cos .

6

b) .13

5sincos 1

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c) 1 2tan cos .

5

d) 1 2tan cot

5

e) 1 1sin cos

4

f) 1 4tan sin

5

g) 2sectan 1

Example 4: Simplify 1

cos arctan4

x

where 0x .

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Working with Graphs of Inverse Trigonometric Functions

Here are the graphs of the trigonometric functions:

We can use graphing techniques learned in previous lectures to graph transformations of the

basic inverse trig functions.

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Example 5: Which of the following points is on the graph of )1arctan()( xxf ?

A)

0,

4

B)

4,0

C)

4,0

D)

4,2

Example 6: Which of the following can be the function whose graph is given below?

A) )1(cos)( 1 xxf

B) )1(sin)( 1 xxf

C) )1(cos)( 1 xxf

D) )1(sin)( 1 xxf

E) )1(tan)( 1 xxf

Example 7: Which of the following can be the function whose graph is given below?

A) )2(cos)( 1 xxf

B) )2(sin)( 1 xxf

C) )2(cos)( 1 xxf

D) )2(sin)( 1 xxf

E) )2(tan)( 1 xxf

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Modeling Using Sinusoidal Functions

Sine and cosine functions model many real-world situations. Physical phenomenon such as

tides, temperatures and amount of sunlight are all things that repeat themselves, and so are easily

modeled by sine and cosine functions (collectively, they are called “sinusoidal functions”).

Here are some other situations that can be modeled by a sinusoidal function:

Suppose you are on a Ferris wheel at a carnival. Your height (as you are sitting in your

seat) varies sinusoidally.

Suppose you are pushing your child as s/he sits in a swing. Your child’s height varies

sinusoidally.

The motion of a swinging pendulum varies sinusoidally.

Stock prices sometimes vary sinusoidally.

We’ll work a couple of examples involving sinusoidal variation.

Recall: Given the functions DCBxAxf )sin()( or DCBxAxf )cos()( ;

The period is: B

2

The amplitude is: A

The vertical shift is: D (up if positive)

The horizontal shift is B

C (to the right if positive)

Example 8: Determine the equation of the sine function which has amplitude is 5, the phase shift

is 4 to the left, the vertical shift is 3 down, and the period is 2.

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Example 9: The number of hours of daylight in Boston is given by

12)79(365

2sin3)(

xxf

where x is the number of days after January 1. What is the:

a. amplitude?

b. period?

c. maximum value of f(x)?

Example 10: The function )2sin(40120)( ttP models the blood pressure (in millimeters

of mercury) for a person who has a blood pressure of 160/90 (which is high); t represents

seconds. What is the period of this function? What is the amplitude?

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Example 11: Determine the function of the form )sin()( BxAxf given the following graph:

Example 12: Determine the function of the form DCBxAxf )sin()( given the graph:

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Example 13: Assume that you are aboard a research submarine doing submerged training

exercises in the Pacific Ocean. At time t = 0 you start purposing (alternately deeper and then

shallower). At time t = 4 min you are at your deepest, y = – 1000 m. At time t = 9 min you next

reach your shallowest, y = –200 m. Assume that y varies sinusoidally with time. Find an

equation expressing y as a function of t.

A) 20045

cos600)(

ttf

B) 60045

cos400)(

ttf

C) 40045

cos200)(

ttf

D) 40045

cos600)(

ttf

E) A) 600915

cos400)(

ttf

(Extra) Example: A signal buoy in the Gulf of Mexico bobs up and down with the height h of

its transmitter (in feet) above sea level modeled by 5)sin()( BtAth . During a small squall its

height varies from 1 ft to 9 ft and there are 4 seconds from one 9-ft height to the next. What are

the values of the constants A and B?