Inverse Trigonometric Functions

MADE BY

UMANG LAKHERACLASS-X ‘A’

ROLL NO.-33

3

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 Testand must be restricted to find its inverse.

4

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:

1a. arcsin 2 6 1 is the angle whose sine is .6 2

1 3b. sin 2 3 3sin 3 2

This is another way to write arcsin x.

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

5

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

6

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: 1a.) arccos 2 3

1 is the angle whose cosine is .3 2

1 3 5b.) cos 2 6

35cos 6 2

This is another way to write arccos x.

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

7

Inverse Tangent Functionf(x) = tan x must be restricted to find its inverse.

Tan x has an inverse function on this interval.

y

x

23

23

2

2

y = tan x

8

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

Angle whose tangent is x

Example: 3a.) arctan 3 6

3 is the angle whose tangent is .6 3

1b.) tan 3 3 tan 33

This is another way to write arctan x.

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

9

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: tan(arctan 4) = 4

Inverse Properties:

10

a. sin–1(sin (–/2)) = –/2

1 5b. sin sin 3

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

EXAMPLES:

11

2Find the exact value of tan arccos .3

x

y

3

2

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

2 23 2 5

opp 52tan arccos tan3 adj 2u

u

EXAMPLES:

Consider the graph of .2xy

x

y

Note the two points on the graph and also on the line y=4.

f(2) = 4 and f(-2) = 4 so what is an inverse function supposed to do with 4?

?2)4(2)4( 11 forf

By definition, a function cannot generate two different outputs for the same input, so the sad truth is that this function, as is, does not have an inverse.

12

So how is it that we arrange for this function to have an inverse?

We consider only one half of the graph: x > 0.

The graph now passes the horizontal line test and we do have an inverse:

xxf

xforxxf

)(

0)(1

2

Note how each graph reflects across the line y = x onto its inverse.

xy

x

4

y=x

2xy

2

13

A similar restriction on the domain is necessary to create an inverse function for each trig function.Consider the sine function.

You can see right away that the sine function does not pass the horizontal line test.

But we can come up with a valid inverse function if we restrict the domain as we did with the previous function.

How would YOU restrict the domain?

x

yy = sin(x)

y = 1/2

14

Take a look at the piece of the graph in the red frame.

x

yWe are going to build the inverse function from this section of the sine curve because:

This section picks up all the outputs of the sine from –1 to 1.

This section includes the origin. Quadrant I angles generate the positive ratios and negative angles in Quadrant IV generate the negative ratios. Lets zoom in and look at some key

points in this section.15

x

yy = sin(x)

12

23

3

22

4

21

6

0021

6

22

4

23

3

12

)(

xfx

I have plotted the special angles on the curve and the table.

16

1

2

3

30

60

45

45 1

2

2

The graphs give you the big picture concerning the behavior of the inverse trig functions. Calculators are helpful with calculations (later for that). But special triangles can be very helpful with respect to the basics.

Use the special triangles above to answer the following. Try to figure it out yourself before you click.

)2(csc

23arccos

1

21/230csc6

30

2330cos

630

becauseor

becauseor

1

2

3

30

60

45

45 1

2

2

OK, lets try a few more. Try them before you peek.

21arcsin

)3(tan

21arcsin

1

2145sin)

4(45

31360tan)

3(60

2145sin)

4(45

becauseor

becauseor

becauseor

THANK

YOU