7.6 The Inverse Trigonometric Function

Objective To find values of the inverse trigonometric functions.

The Inverse Trigonometric FunctionWhen does a function have an inverse?

It means that the function is one-to-one.

One-to-one means that every x-value is assigned no more than one y-value AND every y-value is assigned no more than one x-value.

How do you determine if a function has an inverse?

Use the horizontal line test (HLT).

Let’s take a look at the graphs of all trigonometric functions

Inverse Sine Function

y

2

1

1

x

y = sin x

sin x has an inverse function on this interval.

Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test.

f(x) = sin x does not pass the Horizontal Line Test

and must be restricted to find its inverse.

–/2 /2

–/2 /2

The Inverse Trigonometric Function

The inverse sine function is defined byy = arcsin x if and only if sin y = x.

Angle whose sine is x

The domain of y = arcsin x is [–1, 1].

Example 1:1a. arcsin2 6

1 is the angle whose sine is .6 2

1 3b. sin2 3

3sin3 2

This is another way to write arcsin x.

The range of y = arcsin x is [–/2 , /2].

The Inverse Trigonometric Function

Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).

The other inverse trig functions are generated by using similar restrictions on the domain of the trig function. Consider the cosine function:

Inverse Cosine Function

cos x has an inverse function on this interval.

f(x) = cos x must be restricted to find its inverse.y

2

1

1

x

y = cos x

The Inverse Trigonometric Function

0

The inverse cosine function is defined byy = arccos x if and only if cos y = x.

Angle whose cosine is x

The domain of y = arccos x is [–1, 1].

Example 2: 1a.) arccos2 3

1 is the angle whose cosine is .3 2

1 3 5b.) cos2 6

35cos6 2

This is another way to write arccos x.

The range of y = arccos x is [0 , ].

The Inverse Trigonometric Function

Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).

The Inverse Trigonometric FunctionThe other trig functions require similar restrictions on their domains in order to generate an inverse.

Like the sine function, the domain of the section of the

tangent that generates the arctan is , .

2 2

The inverse tangent function is defined byy = arctan x if and only if tan y = x.

Angle whose tangent is x

Example 3: 3a.) arctan

3 6 3 is the angle whose tangent is .

6 3

1b.) tan 33 tan 3

3

This is another way to write arctan x.

The domain of y = arctan x is (-,) .

The range of y = arctan x is (–/2 , /2).

The Inverse Trigonometric Function

Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).

The Inverse Trigonometric FunctionThe inverse cotangent function is defined by

y = arccot x if and only if cot y = x. Angle whose tangent is x

Example 4:

a.) arccot14 is the angle whose cotangent is 1.

4

1b.) cot 36 cot 3

6

This is another way to write arctan x.

The domain of y = arccot x is (-,) .

The range of y = arccot x is (0, ).

Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).

The Graph of Inverse Sine

The Graph of Inverse Cosine

What is the relation between arcsin(x) and arccos(x) ?

arccos(x) = (-1)arcsin(x) + /2

arcsin(x) + arccos(x) = /2

The Graph of Inverse Tangent and Cotangent

What is the relation between arctan(x) and arcot(x) ?

arccot(x) = (-1)arctan(x) + /2

arctan(x) + arccot(x) = /2

Graphing Utility: Graph the following inverse functions.

a. y = arcsin x

b. y = arccos x

c. y = arctan x

–1.5 1.5

–

–1.5 1.5

2

–

–3 3

–

Set calculator to radian mode.

Graphing Utility: Approximate the value of each expression.

a. cos–1 0.75 b. arcsin 0.19

c. arctan 1.32 d. arcsin 2.5

Set calculator to radian mode.

Composition of Functions:f(f –1(x)) = x and (f –1(f(x)) = x.

If –1 x 1 and – /2 y /2, thensin(arcsin x) = x and arcsin(sin y) = y.

If –1 x 1 and 0 y , thencos(arccos x) = x and arccos(cos y) = y.

If x is a real number and –/2 < y < /2, thentan(arctan x) = x and arctan(tan y) = y.

Example 5: tan(arctan 4) = 4

Inverse Properties:

If x is a real number and 0 < y < , thencot(arccot x) = x and arccot(cot y) = y.

1a. find sin 0.8 with a calculator.

0.93 radians or 53.1 1b. find cos 0.5 without a calculator.

1cos 0.5 means that cos 0.5 and 0x x x

1 2so, cos 0.5

3

Example 6:

0.83

1 2a. Find cos tan with a calculator

3

1 2b. Find cos tan without a calculator.

3

inverse tangent is restricted to , quadrants I and IV2 2

x

tangent is negative in quadrant IV

2 2 23 2

9 4 13 13

r

r

1 2 3 3 13cos tan cos

3 1313

Example 7:

Example 8:

a. sin–1(sin (–/2)) = –/2

1 5b. sin sin3

53 does not lie in the range of the arcsine function, –/2 y /2.

y

x

53

3

5 23 3 However, it is coterminal with

which does lie in the range of the

arcsine function.

1 15sin sin sin sin3 3 3

does not lie in the range of the arcsin function, –/2 y

/2.

32

Example 9:

a. sin–1(sin (–3/2)) = /2

1 5b. tan sin2

y

x

42

2

3 22 2 However, it is coterminal with

which does lie in the range of the

arcsin function.

1 13sin sin sin sin2 2 2

5sin sin 12 2 1 15tan sin tan ( 1)

2 4

Example 10: 2Find the exact value of tan arccos .3

x

y

3

2

adj2 2Let = arccos , then cos .3 hyp 3

u u

2 23 2 5

opp 52tan arccos tan3 adj 2

u

u

[Solution]

Finally, we encounter the composition of trig functions with inverse trig functions. The following are pretty straightforward compositions. Try them yourself before you click to the answer.

?2

3sinsin 1

First, what do we know about

?

We know that is an angle whose sine is .2

3

so2

3sin

2

3sinsin 1

Did you suspect from the beginning that this was the answer because that is the way inverse functions are SUPPOSED to behave? If so, good instincts but….

Consider a slightly different setup:

2arcsin sin

3

This is also the composition of two inverse functions but…3

arcsin .2 3

Did you suspect the answer was going to be 2/3? This problem behaved differently because the first angle, 2/3, was outside the range of the arcsin. So use some caution when evaluating the composition of inverse trig functions.

The remainder of this presentation consists of practice problems, their answers and a few complete solutions.

Find the exact value of each expression without using a calculator. When your answer is an angle, express it in radians. Work out the answers yourself before you click.

2

3cos.9

1sin.8

3arctan.7

2

1cos.6

0arcsin.5

3

1arctan.4

1tan.3

1arccos.2

2

1sin.1

1

1

1

1

1

2

1cossin.16

3cosarccos.15

2

1arccostan.14

270sinarcsin.13

2sinarcsin.12

2

1arccos.11

2sec.10

1

1

Use a calculator. For 17-20, round to the nearest tenth of a degree.

)8787.arccos(.20

)1234.arcsin(.19

)345.2arctan(.18

)6666(.cos.17 1

Use a calculator. For 21-24, express your answers in radians rounded to the nearest hundredth.

)7878arctan(..24

2345.cos.23

)7878arcsin(..22

585.3tan.21

1

1

On most calculators, you access the inverse trig functions by using the 2nd function option on the corresponding trig functions. The mode button allows you to choose whether your work will be in degrees or in radians. You have to stay on top of this because the answer is not in a format that tells you which mode you are in.

Use a calculator. When your answer is an angle, express it in radians rounded to the hundredth’s place. When your answer is a ratio, round it to four decimal places, but don’t round off until the very end of the problem.

2345.arccostan.30

5758.sincos.29

5758.arccoscos.28

)34.2arctan(sin.27

1cosarcsin.26

58.3sinarcsin.25

1

Answers appear in the following slides.

21sin.8

33arctan.7

42

1cos.6

00arcsin.5

63

1arctan.4

41tan.3

1arccos.2

62

1sin.1

1

1

1

1

6

5

2

3cos.9 1

Answers for problems 1 – 9.

Negative ratios for arccos generate angles in Quadrant II.

y

x

12

3

The reference angle is

so the answer is

6

6

5

66

6

6

2

3

3

2sin

2

1cossin.16

32

1arccos

3cosarccos.15

33

2tan

2

1arccostan.14

2901arcsin270sinarcsin.13

22sinarcsin.12

4

3

2

1arccos.11

32/1cos2sec.10

1

11

3

y

x

-1

23

14.

x

3

1

2

y15.

5.151)8787.arccos(.20

1.7)1234.arcsin(.19

9.66)345.2arctan(.18

2.48)6666(.cos.17 1

67.0)7878arctan(..24

81.12345.cos.23

91.0)7878arcsin(..22

30.1585.3tan.21

1

1

1455.4...3341.1tan2345.arccostan.30

8175.0...6136.0cos5758.sincos.29

5758.5758.arccoscos.28

62.0...7184.0arctan)34.2arctan(sin.27

57.0...5403.0arcsin1cosarcsin.26

44.0...4245.0arcsin58.3sinarcsin.25

1

Answers for 17 – 30.

Assignment

P. 289 #1 – 8, 11 – 14 (only EXACT value),

19 – 21 (only T/F, NO counterexample)