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Chapter 35. Inverse Circular Functions. Prepared by : Tan Chor How (B.Sc). Some fundamental concepts. Let. y = sinx. then we have. or. i.e. is the inverse function of. Iff y is the 1-1 function!. doesn’t mean. Also doesn’t mean. 1. 0. -1. In this region, - PowerPoint PPT Presentation
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04/19/23 By Chtan FYHS-Kulai 1
Prepared by : Tan Chor How (B.Sc)
04/19/23 By Chtan FYHS-Kulai 2
04/19/23 3By Chtan FYHS-Kulai
Let y = sinx
then we have ysin -1x
2
130sin or
2
1
6sin
i.e.62
1sin 1
04/19/23 4By Chtan FYHS-Kulai
ysin -1xis the inverse function of
xy sin
Iff y is the 1-1 function!
04/19/23 5By Chtan FYHS-Kulai
ysin -1x
doesn’t mean sin
yx
Also doesn’t mean
ysin
1x
04/19/23 6By Chtan FYHS-Kulai
x
y1
-1
0
xy sin
2
2-
-
In this region, ,y is 1-1 function.
22
x
04/19/23 7By Chtan FYHS-Kulai
Now, if you flip the previous graph,
x
y
02
2-
xy -1sin
1
-1Principal valuesThe principal values of y is defined as that value lying between .
2
04/19/23 8By Chtan FYHS-Kulai
Similarly, check the cosine graph
x
y1
-1
0 2
2-
-
xy cos In this region, ,y is 1-1 function.
x0
04/19/23 9By Chtan FYHS-Kulai
Now, if you flip the previous graph,
x
y
2
2- 1
-1 0
Principal valuesThe principal values of y is defined as that value lying between 0 and ∏ .
xy -1cos
04/19/23 10By Chtan FYHS-Kulai
x
y
0 2
2-
-
xy tan
04/19/23 11By Chtan FYHS-Kulai
Graph of xy 1tan
x
y
2
2-
0
xy -1tan
Principal valuesThe principal values of y is defined as that value lying between .
2
04/19/23 12By Chtan FYHS-Kulai
Some books write as .
Domain of y is
Range of y is
xy -1sinxy arcsin
1,1
2,2
04/19/23 13By Chtan FYHS-Kulai
62
1sin 1
2
1
6sin)
2
1sin(sin 1
04/19/23 14By Chtan FYHS-Kulai
In general, we have
xx )sin(sin 1
1,1x
04/19/23 15By Chtan FYHS-Kulai
xx 11 sinsin
We also have :
1,1x
04/19/23 By Chtan FYHS-Kulai 16
e.g. 1
Evaluate . 2
2sin 1
Soln :
2
2
4sin
42
2sin 1
04/19/23 By Chtan FYHS-Kulai 17
e.g. 2
Evaluate .
2
3-sin 1
Soln :
2
3-
3-sin
3-
2
3-sin 1
04/19/23 By Chtan FYHS-Kulai 18
e.g. 3 Evaluate .
3
2sinsin 1
Soln :
1,13
2x
3
2
3
2sinsin 1
04/19/23 By Chtan FYHS-Kulai 19
Soln :
e.g. 4Evaluate .
4sinsin 1
Let 4
sin 1 x
4sin
x
44sinsin 1
04/19/23 20By Chtan FYHS-Kulai
Now, let see Same as .
Domain of y is
Range of y is
xy -1cosxy arccos
1,1
,0
04/19/23 21By Chtan FYHS-Kulai
In general, we have
xx )cos(cos 1
1,1x
04/19/23 22By Chtan FYHS-Kulai
xx 11 coscos
We also have :
1,1x
04/19/23 By Chtan FYHS-Kulai 23
e.g. 5
Evaluate .
1800 andSoln :Between
gives .
2
3-sin 1
6
5cos150cos
or
2
3
04/19/23 24By Chtan FYHS-Kulai
Now, let see Same as .
Domain of y is
Range of y is
xy -1tanxy arctan
,
22-
,
04/19/23 25By Chtan FYHS-Kulai
Now, let see Same as .
Domain of y is
Range of y is
xy -1cotxarcy cot
,
,0
04/19/23 By Chtan FYHS-Kulai 26
xx )tan(tan 1
xx )cot(cot 1
,x
04/19/23 By Chtan FYHS-Kulai 27
e.g. 6Evaluate .
2
3sintan 1
Soln :
602
3sin 1
360tan
04/19/23 By Chtan FYHS-Kulai 28
e.g. 7Evaluate .
5
4sincos 1
Soln :Let
5
4sin 1
5
4sin
3
54
5
3cos
5
4sincos 1
04/19/23 By Chtan FYHS-Kulai 29
e.g. 8Find the value of the following Expression :
2
1cos
5
3sinsin 11
04/19/23 By Chtan FYHS-Kulai 30
Soln : Let and 5
3sin 1a
2
1cos 1b
2
1cos,
5
3sin ba
ba3
5
4 1
23
04/19/23 By Chtan FYHS-Kulai 31
2
1cos
5
3sinsin 11
ba sinbaba sincoscossin
2
3
5
4
2
1
5
3
10
343
04/19/23 By Chtan FYHS-Kulai 32
e.g. 9Find the value of the following Expression :
2tan3tan 11
04/19/23 By Chtan FYHS-Kulai 33
Soln : Let 2tan,3tan 11 ba
2tan3tan banda
ba
baba
tantan1
tantantan
161
5
04/19/23 By Chtan FYHS-Kulai 34
There are 2 possible answers.
135451tan 1 or
4
3135
orba
[because a and b are both positive values, a+b must be positive value.]
04/19/23 By Chtan FYHS-Kulai 35
04/19/23 By Chtan FYHS-Kulai 36
2cossin 11
xx
1,1x
Identity (1)
04/19/23 By Chtan FYHS-Kulai 37
2cottan 11
xx
x
Identity (2)
04/19/23 By Chtan FYHS-Kulai 38
Let prove the identity #1
To prove :2
cossin 11 xx
Same as to prove :
xx 11 cos2
sin
A
04/19/23 By Chtan FYHS-Kulai 39
LHS of A : xx 1sinsinCheck slide #14
RHS of A : xxx
11 coscoscos2
sin
xx 11 cos2
sinsinsin
We have,
and
2sin
2- 1-
x
x-1cos0 [ x(-1) ]
B
04/19/23 By Chtan FYHS-Kulai 40
0-cos- -1 x
02
cos-2
-2
1-
x
2cos-
22- 1-
x C
B and C state that both and are .
x1sin
x1cos-2
2,2
04/19/23 By Chtan FYHS-Kulai 41
xx 11 cos2
sin
i.e.
2cossin 11
xx
04/19/23 By Chtan FYHS-Kulai 42
Let prove the identity #2
To prove :2
cottan 11 xx
Same as to prove :
xx 11 cot2
tan
A
04/19/23 By Chtan FYHS-Kulai 43
xx 1tantan
xxxc
11 cotcotot-2
tan
xxc 11 tantanot-2
tan
But 2
tan2
1 x
and x1cot0 [ x(-1) ]
04/19/23 By Chtan FYHS-Kulai 44
0cot 1 x
2cot
221
x
Both and x1tan x1cot2
2,2
04/19/23 By Chtan FYHS-Kulai 45
e.g. 10Prove that
11
7tan
4
1tan
3
1tan 111
04/19/23 By Chtan FYHS-Kulai 46
Soln :BA
4
1tan,
3
1tan 11
Let
then 4
1tan,
3
1tan BA
BA
BABA
tantan1
tantantan
11
7
41
31
1
41
31
04/19/23 By Chtan FYHS-Kulai 47
4,0,
BA
2,0
BA
i.e.11
7tan 1BA
11
7tan
4
1tan
3
1tan 111
04/19/23 By Chtan FYHS-Kulai 48
e.g. 11Prove that
3
2tan2
13
5cos 11
Soln :BA
3
2tan,
13
5cos 11
Let
LHS:13
5
13
5coscos 1
04/19/23 By Chtan FYHS-Kulai 49
RHS: B2cos3
2tan2cos 1
1cos2 2 B
B2
3
13
13
51
13
92
2,02,
BA
3
2tan2
13
5cos 11
04/19/23 By Chtan FYHS-Kulai 50
04/19/23 By Chtan FYHS-Kulai 51
Do keep in mind :
Equation Range of solution
The only solution
x1sin
x1cos
x1tan
x1cot
2,2
,0
2,2
,0
sinx
cosx
tanx
cotx
04/19/23 By Chtan FYHS-Kulai 52
e.g. 12Solve the equation .
61sin 1 x
Soln :
2,26
6sin1
x
2
1
2
3 x
04/19/23 By Chtan FYHS-Kulai 53
e.g. 13Solve the equation .
4tan 21
x
Soln :
4tan2
x
12 x
1 x
04/19/23 By Chtan FYHS-Kulai 54
e.g. 14Solve the equation , assuming that all the inverse tangents are positive acute angles.
2tan2tantan 111 xx
04/19/23 By Chtan FYHS-Kulai 55
Soln : 2tantan2tantantan 111 xx
Let xBxA 2tan,tan 11
2tantan1
tantan
BA
BA
xBxA 2tan,tan
04/19/23 By Chtan FYHS-Kulai 56
221
22
x
xx
02322 2 xx
02122 xx
2,22
1 xx
[reject]
04/19/23 By Chtan FYHS-Kulai 57
04/19/23 By Chtan FYHS-Kulai 58
Differentiation of an inverse function :
If 53)( xxfy
then 3dx
dy
The inverse function is : xy 35
3
5y
x
04/19/23 By Chtan FYHS-Kulai 59
3
5)(1
yxyf
3
1
dy
dx
i.e.
dxdydy
dx 1
04/19/23 By Chtan FYHS-Kulai 60
So, in general, )(xfy
0)(' xf)(1 yfx is the inverse
function of )(xfy
dxdydy
dx 1
04/19/23 By Chtan FYHS-Kulai 61
Differentiation of an inverse circular function :
xy 1sin 1,1x
Its inverse function :
yx sin
2,2
y
04/19/23 By Chtan FYHS-Kulai 62
0cos ydy
dx
22
y
dydxdx
dy 1
yy 2sin1
1
cos
1
21
1
x
04/19/23 By Chtan FYHS-Kulai 63
So,
2
1
1
1sin
xx
dx
d
04/19/23 By Chtan FYHS-Kulai 64
Another way to derive this formula :
xy 1sin yx sin
ydx
dx
dx
dsin
dx
dyy
dy
dsin1
04/19/23 By Chtan FYHS-Kulai 65
dx
dyycos1
ydx
dy
cos
1
y2sin1
1
21
1
x
04/19/23 By Chtan FYHS-Kulai 66
Similarly,
2
1
1
1cos
xx
dx
d
21
1
1tan
xx
dx
d
21
1
1cot
xx
dx
d
04/19/23 By Chtan FYHS-Kulai 67
e.g. 153
sin 1 xy
Find the differentiation of y.Soln :
3
31
12
x
dx
d
xdx
dy
29
1
x
04/19/23 By Chtan FYHS-Kulai 68
e.g. 16Find dy/dx if .
x
xy
1
1tan 1
Soln :
x
x
dx
d
xxdx
dy
1
1
11
1
12
04/19/23 By Chtan FYHS-Kulai 69
222
2
1
)1()1(1
11
1
x
xx
xx
x
22 1
1
22
2
xx
04/19/23 By Chtan FYHS-Kulai 70
04/19/23 By Chtan FYHS-Kulai 71
(1)cx
x
dx
1
2sin
1
(2)cx
x
dx
1
2tan
1
04/19/23 By Chtan FYHS-Kulai 72
In a general format :(1)
ca
x
xa
dx
1
22sin
(2)
ca
x
axa
dx
1
22tan1
04/19/23 By Chtan FYHS-Kulai 73
e.g. 17Evaluate . 241 x
dx
Soln :
22
414
1
41 x
dx
x
dx
2
2
214
1
x
dx
04/19/23 By Chtan FYHS-Kulai 74
cx
21
tan
211
4
1 1
cx 2tan2
1 1
04/19/23 By Chtan FYHS-Kulai 75
Home works :
Mathematics 3 (Further Mathematics)
Ex 15a, Ex 15d, Misc Ex.