PPT: Inverse Trigonometric Functions | PPTs for JEE Main & Advanced PDF Download

 Page 1


7.6
Inverse Trig Functions
Objective:  In this section, we will look at the definitions and 
properties of the inverse trigonometric functions.  We will recall 
that to define an inverse function, it is essential that the function 
be one-to-one.
Page 2


7.6
Inverse Trig Functions
Objective:  In this section, we will look at the definitions and 
properties of the inverse trigonometric functions.  We will recall 
that to define an inverse function, it is essential that the function 
be one-to-one.
7.6 Inverse Trigonometric Functions 
and Trig Equations
) arctan( ) ( tan
1
x x y = =
-
) arcsin( ) ( sin
1
x x y = =
-
) arccos( ) ( cos
1
x x y = =
-
?
?
?
?
?
?
-
2
,
2
? ?
Domain: [–1, 1]
Range:
Domain: [–1, 1]
Range: [0, p]
?
?
?
?
?
?
-
2
,
2
? ?
Domain: 
Range:
?
Page 3


7.6
Inverse Trig Functions
Objective:  In this section, we will look at the definitions and 
properties of the inverse trigonometric functions.  We will recall 
that to define an inverse function, it is essential that the function 
be one-to-one.
7.6 Inverse Trigonometric Functions 
and Trig Equations
) arctan( ) ( tan
1
x x y = =
-
) arcsin( ) ( sin
1
x x y = =
-
) arccos( ) ( cos
1
x x y = =
-
?
?
?
?
?
?
-
2
,
2
? ?
Domain: [–1, 1]
Range:
Domain: [–1, 1]
Range: [0, p]
?
?
?
?
?
?
-
2
,
2
? ?
Domain: 
Range:
?
Let us begin with a simple question:
x x f
x x f
=
=
-
) (
) (
1
2
What is the first pair of inverse functions that pop 
into YOUR mind?
This may not be your pair but 
this is a famous pair . But 
something is not quite right 
with this pair . Do you know 
what is wrong?
Congratulations if you guessed that the top function 
does not really have an inverse because it is not 1-1 
and therefore, the graph will not pass the horizontal 
line test.
Page 4


7.6
Inverse Trig Functions
Objective:  In this section, we will look at the definitions and 
properties of the inverse trigonometric functions.  We will recall 
that to define an inverse function, it is essential that the function 
be one-to-one.
7.6 Inverse Trigonometric Functions 
and Trig Equations
) arctan( ) ( tan
1
x x y = =
-
) arcsin( ) ( sin
1
x x y = =
-
) arccos( ) ( cos
1
x x y = =
-
?
?
?
?
?
?
-
2
,
2
? ?
Domain: [–1, 1]
Range:
Domain: [–1, 1]
Range: [0, p]
?
?
?
?
?
?
-
2
,
2
? ?
Domain: 
Range:
?
Let us begin with a simple question:
x x f
x x f
=
=
-
) (
) (
1
2
What is the first pair of inverse functions that pop 
into YOUR mind?
This may not be your pair but 
this is a famous pair . But 
something is not quite right 
with this pair . Do you know 
what is wrong?
Congratulations if you guessed that the top function 
does not really have an inverse because it is not 1-1 
and therefore, the graph will not pass the horizontal 
line test.
Consider the graph of 
.
2
x y =
-??? ???
???
???
???
???
????
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 (
1 1
- = =
- -
f or f
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.
Page 5


7.6
Inverse Trig Functions
Objective:  In this section, we will look at the definitions and 
properties of the inverse trigonometric functions.  We will recall 
that to define an inverse function, it is essential that the function 
be one-to-one.
7.6 Inverse Trigonometric Functions 
and Trig Equations
) arctan( ) ( tan
1
x x y = =
-
) arcsin( ) ( sin
1
x x y = =
-
) arccos( ) ( cos
1
x x y = =
-
?
?
?
?
?
?
-
2
,
2
? ?
Domain: [–1, 1]
Range:
Domain: [–1, 1]
Range: [0, p]
?
?
?
?
?
?
-
2
,
2
? ?
Domain: 
Range:
?
Let us begin with a simple question:
x x f
x x f
=
=
-
) (
) (
1
2
What is the first pair of inverse functions that pop 
into YOUR mind?
This may not be your pair but 
this is a famous pair . But 
something is not quite right 
with this pair . Do you know 
what is wrong?
Congratulations if you guessed that the top function 
does not really have an inverse because it is not 1-1 
and therefore, the graph will not pass the horizontal 
line test.
Consider the graph of 
.
2
x y =
-??? ???
???
???
???
???
????
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 (
1 1
- = =
- -
f or f
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.
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:
x x f
x for x x f
=
? =
-
) (
0 ) (
1
2
Note how each graph reflects across the line y = x onto 
its inverse.
x y =
??? ??? ??? ???
x
4
y=x
2
x y =
2