The document discusses the inverse tangent function, tan-1(x). It begins by reviewing one-to-one functions and their inverses. Examples are provided to illustrate one-to-one functions. The lesson aim is to teach students the inverse tangent function. The domain of tan-1(x) is all real numbers, and the range is (-π/2, π/2). The graph of tan-1(x) is the reflection of the tangent function graph over the line y=x. Students are asked to find the values of various inverse tangent functions. Homework problems are assigned involving evaluating inverse tangent functions without a calculator.

2. CONCEPT
INVERSE TANGENT FUNCTION
By
Mr SADIQ HUSSAIN
MANJANBAZAM CADET
COLLEGE-TARBELA CANTT.
Presentation on concept Delivery

3. PRIOR KNOWLEDGE
 Trigonometric functions
 Ordered pairs
 One-to-one function
 What is the horizontal line test?
 Domain and range of y = tanx.

4.  To determine if a function has an inverse function, we
need to talk about a special type of function called a
OnetoOne Function. A onetoone function is a
function where each input (xvalue) has a unique output
(yvalue). To put it another way, every time we
plug in a value of x we will get a unique value of y, the
same yvalue will never appear more than once. A
onetoone function is special because only onetoone
functions have an inverse function.
 Note; only one-to-one function exists an inverse function
Explanation;

5. Examples ;
 Now let’s look at a few examples to help
demonstrate what a onetoone function is. Example 1:
Determine if the function
 f = {(7, 3), (8, –5), (–2, 11), (–6, 4)} is a
onetoone function.
The function f is a onetoone function because each of
the yvalues in the ordered pairs is unique; none of
the yvalues appear more than once. Since the
function f is a onetoone function, the function f must
have
an inverse.

6. Example 2:
Determine if the function
h = {(–3, 8), (–11, –9), (5, 4), (6, –9)} is a onetoone
function?
The function h is not a onetoone function because
the yvalue of –9 is not unique; the yvalue of –9
appears more than once. Since the function h is not a
onetoone function, the function h does not have an
inverse.
Remember that only onetoone function have an
inverse.

7. THE AIM
 To teach the student ‘The Inverse Tangent Function’

8. INTRODUCTION
 Some questions will be asked to check if the students
know:
 What is real valued function?
 What is the inverse of y=f(x)?
 What is the relation between f(x) and f-1(x)?
 What are the values of the following trigonometric ratio;
tan0, tanΠ/6, tanΠ/3, tanΠ/4, tanΠ/2 .etc
Example:
The following examples will be shown the class
a
b
c
1
2
3
yx f

9. INTRODUCTION…contd
Q. Student will be asked to find f-1(x)?
Good
Q. Is f-1 again a function?
A. Yes
Q. What are the reasons?
A. Because one-to-one correspondence between
domain and range in f-1(x) is established.
a
b
c
1
2
3
yx f-1

10. INTRODUCTION…contd
Q. Another relation f1 = { ( 0 , 1 ) ( -1 , 0 ) } will be
given to the class then student will be asked to
interchanged the ordered pairs:
A. f2 = { ( 1 , 0 ) ( 0 , -1 ) }
Q. Student will be asked to depict these two relations
f1 & f2 on the graph paper?
A. A graph will be shown:
Q. Student what you have
noted from the
graph of f1 and f2?
f2
-1
1
-1
1
f1
Y=x

11. INTRODUCTION…contd
A. Graph of f1 and f2 are reflection images of each
other over the line y=x
Q. So, what should be the relation between f1 and f2?
A. f2 is an inverse of f1.
Very well students
Here, teacher will clear as components of the order
pairs of a 1-1 function are interchanged for its
inverse function.

12. THE LESSON AIM
 Now the aim of the lesson will be announced, Student
today we will study the concept of ‘The Inverse
Tangent Function’.

13. THE TOPIC
 Topic ‘The Inverse sine Function’ will be written on the
board as centre heading:
‘THE INVERSE TANGENT FUNCTION’
y=Tan-1(x)

14. DEVELOPMENT
 Concept: y=tan-1(x). Iff x=tany
 DLO:
 The student will understand the concept of y=tan-1(x)
 To find the angle y whose tan is x i.e x=tany

15. DEVELOPMENT …contd
 The Student will be asked to complete the given table
f1 with respective Tan:
 f1=
 Expected Ans:
x -Π/2 -Π/3 -Π/6 0 Π/6 Π/3 Π/2
y - - - - - - -
f1= {(π/2 ,-∞) (-π/3 ,-√3) , (-π/4 ,-1) (-π/6 ,-1/(√3))(0,0,)
 (-π/2 ,-∞) (π/6 ,1/(√3)) , (π/4 ,1) (π/3 ,√3) (π/2 ,∞)}

16. DEVELOPMENT …contd
 A graph will be shown to
the class:
Q. Student will be asked
to identify the graph
f1, is it 1-1 function?
A. No
Q. What are the reasons?
A. Because horizontal line
cut the graph at many
points.
Good
We'll take tan x from -/2 to /2

17. Graph of Tanx.

18. DEVELOPMENT …contd
Q. Student will be asked identify the graph whose
horizontal line cut its only once?
A. Only from
This part of the graph of tanx
will be shown to the
class?
2
π
to
2
π-

19. DEVELOPMENT …contd
Q.The student will be asked to interchange the ordered
pairs of f1?
Q. The student will be asked
to depict these ordered
pairs on the graph.
A. A graph will be shown to
the students:
{(−∞,-
𝜋
2
) (−√3,-
𝜋
3
) , (-1,-
𝜋
4
) (-
1
√3
-
𝜋
6
)(0,0), (∞,
𝜋
2
,)
(
1
√3,
,
𝜋
6
) , (1,
𝜋
4
) (√3,
𝜋
3
) ( ∞,
𝜋
2
)}

20. DEVELOPMENT …contd
Q. The student will be asked that what conclusion
you have drawn from the graph f1 and f2?
f1 fffffffffffff
A. f2 is the reflection of f1.
A2. Opposite to Tanx.
f2

21. Comparison view of Tan and arctan

22. DEFINITION:
The inverse tangent function, denoted by tan−1( x)
or arctan x), is defined to be the inverse of the restricted
tangent function
Domain ; tan−1(x ) =R= (−∞,∞) and
Range ;tan−1(x ) = (− , π /2 , π/ 2).
Since h −1 (x) = y if and only if h(y) = x,
we have: tan−1 x = y if and only if
tan(y) = x and − π/ 2 < y < π/ 2 . Since h(h −1 (x)) = x
and h −1 (h(x)) = x, we have: tan(tan−1 (x)) = x
for x ∈ (−∞,∞) tan−1 (tan(x)) = x for x ∈( − π/ 2 , π/ 2 ).

23. 23
Inverse Tangent Function
f(x) = tan x must be restricted to find its inverse.
Tan x has an inverse
function on this interval.
y
x
2
3
2
3
2

2

y = tan x

24. DEVELOPMENT …contd
Caution:
Student remember that
Tan-1x does not mean 1/Tanx i.e
Tan-1(x)
Tanx
1


25. tan-1 x is the inverse function of tan x but again we must have restrictions to
have tan x a one-to-one function.
    
    
1 1
1 1
tan tan where
2 2
tan tan where
f f x x x x
f f x x x x
  
 
    
     
We'll take tan x from -/2 to /2








2
1
,
2
3
 1,0
 0,1
 1,0 
 0,1








2
3
,
2
1









2
3
,
2
1









2
3
,
2
1









2
1
,
2
3









2
1
,
2
3









2
1
,
2
3









2
3
,
2
1









2
2
,
2
2








2
2
,
2
2









2
2
,
2
2









2
2
,
2
2
6

4
3
2

3
2
4
3
6
5

6
7
4
5
3
4
2
3
3
5
4
7 6
11
2
0

26. DEVELOPMENT …contd
Q. Student will be asked to find the value of tan-1(1)?
Solution as a model will be done?
A. Student, we have to find the angle whose tan is 1
let that angle be y, then
.Re quiredAns
4
π
(1)tanthus
4
π
y(ii)(i)from
(ii)1
4
π
tanbut
(i)1tany
(1),tany
1
1







yƐ (− , π /2 , π/ 2).

27. DEVELOPMENT …contd
Q. Student will be asked to find
(i)
(ii)
(iii)
 
 





 




3
1
tan
tan
3tan
1
1
1
1

28. LESSON SUMMARY
y = tan-1x or arc tanx
1. y = tan-1x iff x=tany, where
2. Domain of tan-1(x) is ;
3. Range of tan-1x is (− , π /2 , π/ 2).
4. The graph of tan-1x
5. Combine graph of tan-1x and
tan( x).
( , ) 

29. 29
Composition of Functions:
f(f –1(x)) = x and (f –1(f(x)) = x.
If –1  x  1 and – /2  y  /2, then
sin(arcsin x) = x and arcsin(sin y) = y.
If –1  x  1 and 0  y  , then
cos(arccos x) = x and arccos(cos y) = y.
If x is a real number and –/2 y /2, then
tan(arctan x) = x and arctan(tan y) = y.
Example: tan(arctan 4) = 4
Inverse Properties:

30. Combine graph of Tan and arctan.

31. Combine Graph of inv.Trig.Function.

32. LESSON SUMMARY…contd
6. If x is +ive, tan-1x will lie in
7. If x is –ive, tan-1x will lie in
8. Caution: tan-1x
9. Find tan-1(-1) ?.




2
π
0,




0,
2
π
tanx
1


33. RECAPITULATION
 An oral recap will be carried out in about three minutes
which will cover the following points:
 Today we have discussed the inverse Tan function
 We have understood the domain of tan-1(x)
 Also, we have learnt the graph of tan-1(x)
 The student will be asked:
 Was there anything you didn’t comprehend well?
 Anything you would like to ask?

34. CONSOLIDATION
 What do you meant by the inverse sine function?
(Knowledge)
 What is the domain of y=tan-1(x)? (Knowledge)
 What is the range of y=tan-1(x)? (Knowledge)
 What is the difference b/w the graph of tanx and
tan-1x? (Analysis)
 Find tan-1(-1)? (Application)
 Find tan{tan-1(-1)}? (Synthesis).

35. Homework
Q.1 Evaluate without using calculator.
(i)
(ii)
(iii)
 3tan 1

 11
tan






3
1
tan 1

36. CONCLUSION
 Today we have discussed the procedure of finding the
inverse tan function i.e y=tan-1(x).
 Next time we will discuss the inverse of sec function.

37. THANK YOU