This document provides an overview of a lesson on inverse trigonometric functions. The lesson aims to teach students the inverse sine function y=sin-1(x). It begins with reviewing prerequisite concepts like trigonometric functions. Then it introduces the topic and develops the concept that y=sin-1(x) if x=siny. Examples are used to illustrate inverse functions and their graphs. Students complete practice problems finding inverse sines and their ranges. The lesson concludes with a summary of key points and homework assignments.

2. CONCEPT
INVERSE
TRIGONOMETRIC
FUNCTIONS
By
Mr SADIQ HUSSAIN
Fazaia Inter College,
Shaheen Camp, Peshawar
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 = sinx

4. THE AIM
 To teach the student ‘The Inverse sine Function’

5. 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;
sin0, sinΠ/6, sinΠ/3, sinΠ/2 etc
Example:
The following examples will be shown the class
a
b
c
1
2
3
yx f

6. 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

7. 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

8. 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.

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

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

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

12. DEVELOPMENT …contd
 The Student will be asked to complete the given table
f1 with respective sine:
 f1=
 Expected Ans:
x -Π/2 -Π/3 -Π/6 0 Π/6 Π/3 Π/2
y - - - - - - -
( )( )






























−





−





−= ,1
2
π
,.8
3
π
,.5
6
π
0,0.5,
6
π-
.8,
3
π-
1,
2
π-
f1

13. 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

14. 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?
2
π
to
2
π-

15. DEVELOPMENT …contd
Q. The student will be asked to interchange
the ordered pairs of f1?
( )( )






























−





−





−=
2
π
1,
3
π
.8,
6
π
.5,0,0
6
π-
.5,
3
π-
.8,
2
π-
1,f2
Q. The student will be
asked to depict these
ordered pairs on the graph.
A. A graph will be shown to
the students:

16. DEVELOPMENT …contd
Q. The student will be asked that what
conclusion you have drawn from the graph f1
and f2?
A. f2 is the reflection of f1.
f1 f2

17. DEVELOPMENT …contd
Very well, this is known as y = sin-1
x
sin-1
x
sinx
y=x
Caution:
Student remember that
sin-1
x does not mean 1/sinx
i.e sin-1(x)
sinx
1
≠

18. DEVELOPMENT …contd
Q. Student will be asked to find the value of sin-
1
(1)?
Solution as a model will be done?
A. Student, we have to find the angle whose sine is 1
let that angle be y, then
2
π
(1)sinthus
2
π
y(ii)&(i)from
(ii)1
2
π
sinbut
(i)1siny
2
π
,
2
π
yε(1),siny
1
1
=
=
=
=⇒



−
=
−
−

19. DEVELOPMENT …contd
Q. Student will be asked to find
(i)
(ii)
(iii)
( )





 −





 −
−
−
−
2
1
sin
2
1
sin
/23sin
1
1
1

20. LESSON
SUMMARY
y = sin-1
x or arc sinx
1. y = sin-1
x iff x=siny, where
2. Domain of sin-1
(x) is
3. Range of sin-1
x is
4. The graph of sin-1
x
5. Combine graph of sin-1
x and
sin x
[ ]11,xε
2
π
,
2
π
yε −




−
[ ]11,−





−
2
π
,
2
π

21. LESSON SUMMARY…contd
6. If x is +ive, sin-1
x will lie in
7. If x is –ive, sin-1
x will lie in
8. Caution: sin-1
x
9. Find sin-1
(1)






2
π
0,





−
0,
2
π
sinx
1
≠

22. 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 sin-1
(x)
 Also, we have learnt the graph of sin-1
(x)
 The student will be asked:

Was there anything you didn’t comprehend well?

Anything you would like to ask?

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

24. Homework
Q.1 Evaluate without using calculator.
(i)
(ii)
(iii)
( )/23sin 1
−−





−
2
1
sin 1





−
2
1
sin 1

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

26. THANK YOU