![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](https://image.slidesharecdn.com/sadiqhussain-151028171637-lva1-app6891/85/INVERSE-TRIGONOMETRIC-FUNCTIONS-by-Sadiq-hussain-20-320.jpg)
INVERSE TRIGONOMETRIC FUNCTIONS by Sadiq hussain
- 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