The document provides an explanation of the inverse cosine function. It begins by reviewing prerequisite concepts like trigonometric functions and one-to-one functions. It then defines the inverse cosine function as cos^-1(x), where x is in the domain [-1,1] and the output y is in the range [0,π]. Examples are provided to demonstrate how to find the principal value of the inverse cosine of different inputs. The graph of the inverse cosine function is described and examples are worked through. Key points are summarized such as the domain, range, and relationship between the graphs of cos(x) and cos^-1(x).

2. CONCEPT
INVERSE COSINE 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 = cosx

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 Cosine Function,
cos-1 : [-1, 1] →[0, π ]

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;
cos0, cosΠ/6, cosΠ/3, cosΠ/2 , cos 2Π/3 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 Cosine
Function’.

13. THE TOPIC
 Topic ‘The Inverse sine Function’ will be written on the
board as centre heading:
‘THE INVERSE COSINE FUNCTION’
y=Cos-1(x)
or
cos-1 : [-1, 1] →[0, π ]

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

15. DEVELOPMENT …contd
 The Student will be asked to complete the given table
f1 with respective cosine:
 f1=
 Expected Ans:
x 0 Π/6 Π/3
Π/2
2Π/3 3Π/4 Π
y - - - - - - -
 




































 o,
2
π
,-.7
4
3π
,-.5
3
2π
,.5
3
π
,.8
6
π
0,1f1

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

17. 17
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

18. DEVELOPMENT …contd
Q. Student will be asked identify the graph whose
horizontal line cut its only once?
A. Only from
Q. This part of the graph will be shown to the
class?
to0 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:
    































6
5π
.8,-
3
2π
.5,-0,
3
π
.5,
6
π
.8,1,0f2

20. View of graph of inverse.

21. Graph of the inverse cosine
function.
The inverse cosine function cos-1 : [-1, 1] →[0, π ],
receives a real number x in the interval
[−1, 1] as an input and gives a real
number y in the interval [0, π ]as an output
(an angle in radian measure). Let us find some
points (x, y) using the equation y = cos-1 x and
plot them in the
xy -plane.

22. Contd……….
 Note that the values of y decrease
from π to 0 as x increases from -1 to 1.
The inverse cosine function is
decreasing and continuous in the
domain. By connecting the points by a
smooth curve, we get the graph of
 y = cos-1 x as shown in Fig.

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

24. DEVELOPMENT …contd
Very well, this is known as y = cos-1x
Caution:
Student remember that
cos-1x does not mean 1/cosx

25. Definition of cos-1 x
 For -1 ≤ x ≤ 1, define cos-1 x as the
unique number y in [0, π] such that
cos y = x . In other words, the inverse
cosine function cos-1 : [-1, 1] → [0, π]
is defined by cos-1 (x) = y if and only
if cos y = x and y ∈[0, π ].

26. Important note.
 The graph of the function y = cos-1 x is
also obtained from the graph y = cos x by
interchanging x and y axes.
For the function y = cos-1 x , the x -
intercept is 1 and the y -intercept is π/2 .
The graph is not symmetric with respect
to either origin or y -axis. So, y = cos-1 x is
neither even nor odd function.

27. Example;01.
Find the principal value of cos-1 ( √3 / 2 ) .
Solution
 Let cos-1 (√3 / 2 ) = y . Then, cos y = √3 / 2.
 The range of the principal values of y = cos-1 x is
[0, π ].
 So, let us find y in [0, π ] such that cos y = √3 / 2
 But, cos π/6 = √3/2 and π/6 ∈ [0,π ].
Therefore, y = π /6
 Thus, the principal value of cos-1 (√3/2 ) is π/6 .


28. Remark.
 It is known that cos-1 x : [-1, 1] → [0, π ]
is given by
 cos-1 x = y if and only if x = cos y for
 -1 ≤ x ≤ 1 and 0 ≤ y ≤ π .
 Whenever we talk about the inverse cosine
function, we have cos x : [0, π ] → [−1, 1]
and cos−1 x : [−1, 1] → [0, π ]
The restricted domain [0, π ] is called the principal domain of cosine function
and the values of y = cos−1 x , −1 ≤ x ≤ 1, are known as principal values of
the function y = cos−1 x .

29. Example 02
 . Find the principal value of cos−1 (1/2)
 Let cos- 1 ( 1 / 2 ) = y
c o s y = 1 / 2 ( i )
b u t w e k n o w cos π/6 =1/2. (ii) π/6 ∈ [0,π ].
from (i) and (ii) we have noted that.
y= π/6.
Hence cos-1 (1 / 2 ) = π/6 .Ans.

30. Example;3
 Find the principal value of cos-1 (- √3 / 2 ) .
 Solution
 Let cos-1 (-√3 / 2 ) = y . Then, cos y = -√3 / 2.
 The range of the principal values
 of y = cos-1 x is [0, π ].
 So, let us find y in [0, π ] such that
 cos y =- √3 / 2
 But, cos 5π/6 = -√3/2 and π/6 ∈ [0,π ].
Therefore, y = 5π /6
 Thus, the principal value of cos-1 (-√3/2 ) is 5π/6 .

31. DEVELOPMENT …contd
Q.Student will be asked to find the value of cos-1(1)?
Solution as a model will be done?
A.Student, we have to find the angle whose cosine is 1
let that angle be y, then
0
0







(1)costhus
y(ii)&(i)from
(ii)1cos0but
(i)1cosy
y(1),cosy
1
1
∈ [0, π ]

32. DEVELOPMENT …contd
Q. Student will be asked to find
(i)
(ii)
(iii)
 





 





 



2
1
cos
2
1
cos
/23cos
1
1
1

33. LESSON SUMMARY
y = cos-1x or arc cosx
1. y = cos-1x iff x=cosy, where
2. Domain of cos-1(x) is -1 ≤ x ≤ 1
3. Range of cos-1x is
4. The graph of cos-1x
5. Combine graph of cos-1x and cosx.
0 ≤ y ≤ π .

34. LESSON SUMMARY…contd
6. If x is +ive, cos-1x will lie in
cosx will be –ive in
6. Caution: cos-1x
7. Find cos-1(1) ?
cosx
1

0 ≤ x ≤ π/2
π/2 < x ≤ π

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

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

37. Homework
Q.1 Evaluate without using calculator.
(i)
(ii)
(iii)
 /23cos 1







2
1
cos 1






2
1
cos 1

38. CONCLUSION
 Today we have discussed the procedure of finding the
inverse sine function i.e y=cos-1(x).
 Next time we will discuss the inverse of tangent
function.

39. THANK YOU