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8/8/2019 TrigEqns InveTrigFunc Presentation-2x3
1/11
Inverse Trigonometry
Inverse Trigonometric functions
Vidyalankar Classes
Rankers Batch, IIT JEE 2011
Author, Another Inverse Trigonometry
Foundations
Outline
Author, Another Inverse Trigonometry
FoundationsMappings and FunctionInverse of a function
Outline
1 FoundationsMappings and FunctionInverse of a function
Author, Another Inverse Trigonometry
FoundationsMappings and FunctionInverse of a function
Function
Definition
A Mapping is a relation relating elements from set X to set Y
Definition
A function y = f(x) is defined as a mapping f: X
Y such that
x X there is a unique y Y
Author, Another Inverse Trigonometry
FoundationsMappings and FunctionInverse of a function
Function
Definition
A Mapping is a relation relating elements from set X to set Y
Definition
A function y = f(x) is defined as a mapping f: X Y such thatx X there is a unique y Y
Author, Another Inverse Trigonometry
FoundationsMappings and FunctionInverse of a function
Outline
1 FoundationsMappings and FunctionInverse of a function
Author, Another Inverse Trigonometry
8/8/2019 TrigEqns InveTrigFunc Presentation-2x3
2/11
Inverse of a function
DefinitionInverse of a function y = f(x) is defined as x = f1(y) and theinverse mapping should also be a function.
Above mappings are functions if one-to-one or many-to-one. Of themonly one-to-one and onto have inverse
Author, Another Inverse Trigonometry
Trigonometric functions
Inverse of a function
DefinitionInverse of a function y = f(x) is defined as x = f1(y) and theinverse mapping should also be a function.
Above mappings are functions if one-to-one or many-to-one. Of themonly one-to-one and onto have inverse
Author, Another Inverse Trigonometry
Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs
Outline
2 Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs
Author, Another Inverse Trigonometry
Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs
Inverse Trigonometric functions
To draw Inverse of a function
A function and its inverse are mirror images of each other in y = x
Exercise to draw Graph sin1 xusing above property
1 Draw graph of y = sin x2 Get graph of y = x
3 Get reflection of y = sin x
Author, Another Inverse Trigonometry
Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs
Inverse Trigonometric functions
To draw Inverse of a function
A function and its inverse are mirror images of each other in y = x
Exercise to draw Graph sin1 xusing above property
1 Draw graph of y = sin x
2 Get graph of y = x
3 Get reflection of y = sin x
Author, Another Inverse Trigonometry
Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs
Inverse Trigonometric functions
To draw Inverse of a function
A function and its inverse are mirror images of each other in y = x
Exercise to draw Graph sin1 xusing above property
1 Draw graph of y = sin x
2 Get graph of y = x
3 Get reflection of y = sin x
Truncation of domain of sine is required as the graph of inverse isnot a function
Author, Another Inverse Trigonometry
8/8/2019 TrigEqns InveTrigFunc Presentation-2x3
3/11
Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs
Inverse Trigonometric functions
To draw Inverse of a function
A function and its inverse are mirror images of each other in y = x
Exercise to draw Graph sin1 xusing above property
1 Draw graph of y = sin x
2 Get graph of y = x
3 Get reflection of y = sin x
Author, Another Inverse Trigonometry
Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs
Outline
2 Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs
Author, Another Inverse Trigonometry
Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs
Inverse Trig-Functions
Definition
1 y = sin1 x x = siny where1 x 1 and 2
y 2
2 y = cos1 x x = cosy where 1 x 1 and 0 y 3 y = tan1 x x = tany where x R and
2< y 0cot1 x , for x< 0
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
sin1
1
x
Prove : sin1
1
x
= csc1 x, x (,1] [1,)
Let = csc1 x where x (,1] [1,)= csc = x = sin = 1
xwhere
2,
2 and x [1,1]= = sin11
x
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
cos1
1
x
Prove : cos1
1
x
= sec1 x,x (,1] [1,)
Let = sec1 x= sec = x (by definition)=
cos =
1
x
(as1
x [
1,1] and
2
,
2
= = cos1
1
x
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
tan1
1
x
Prove : tan11
x
=cot1 x ,x> 0
cot1 x ,x< 0Let = cot1 xFor x> 0, cot = x (Since x> 0 hence
0,
2
)
= tan = 1x
(1
x> 0 and
0,
2
)
= = tan1
1
x
For x< 0, cot = x (
2,
=
2,0
)
= tan = 1x
= tan() = 1x
= = tan1 1x
= cot1 x= tan1 1x
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Problems
Example
1 If u= cot1[
cos2] tan1[cos2] then show thatcscu= cot2
Author, Another Inverse Trigonometry
8/8/2019 TrigEqns InveTrigFunc Presentation-2x3
9/11
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Outline
3 Property-IInverse trig over negationComposition f f1(x) and f1 f(x)
4 Properties IIInverse Of ReciprocalsInvolving tan(A B)One Inverse function to another
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Inverse of tan
tan1
x+ tan1
y
tan1 x+ tan1 y =
tan1
x+ y
1 xy
, if xy< 1
tan1
x+ y
1 xy
+ , if x> 0,y> 0 and xy> 1
tan1
x+ y
1 xy
, if x< 0,y< 0 and xy> 1
Find the value of
tan1 2 + tan1 3tan1
1
2+ tan1
1
3
tan11
2+ tan1 3
cot1 2 cot1(3)
Find the value of
tan1(2) + tan1(3)tan1(2) + tan1(3)tan1
1
2+ tan1(3)
tan1 2 + tan11
3
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
tan1 x tan1y
tan1 x tan1y : transformation y y above
tan1 x tan1 y =
tan1
xy1 + xy
, if 1 + xy> 0
tan1
xy1 + xy
+ , if 1 + xy< 0, x> 0 & y< 0
tan1
xy1 + xy
, if 1 + xy< 0, x< 0 & y> 0
Find value of
tan1 2 tan1 3tan1
1
2 tan1 3
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Problems
Example
1 (Problem #22 : page 66)
tan1
1
3
+ tan1
1
7
+ + tan1
1
n2 + n + 1
+ to
=
12
3, 0 ,
2and
4
2 Prove that tan112
tan2A+ tan1 (cot A) + tan1(cot3 A)
=
0 , if
4< A 1,0 < x,y 1) , if (x2 + y2 > 1,1 x,y< 0
Author, Another Inverse Trigonometry
8/8/2019 TrigEqns InveTrigFunc Presentation-2x3
10/11
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
cos1 x+ cos1y
cos1 cos1yLet cos1(xy1 x2
1y2) =
cos1 x+ cos1y =
, if |x|, |y| 1,x+y 02 , if |x|, |y| 1,x+y 0
cos1 x cos1y = , if |x|, |y| 1,xy , if |x|, |y| 1,xy
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Problems
Example1 Problem #22, Page 64 : Find all possible values of p and q for
which
cos1
p+ cos1
1 p+ cos1
1 q= 34
2 Problem #18, Page 66 :
cos1
1
2x2 +
1 x2
1 x
2
4
= cos1
x
2cos1 x holds for
1 |x| 12 x R3 0 x 14 1 x 0
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Outline
3 Property-IInverse trig over negationComposition f f1(x) and f1 f(x)
4 Properties IIInverse Of ReciprocalsInvolving tan(A B)One Inverse function to another
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Movement from one to another
Movement from one from to another
1 sin1 x = cos1
1 x2 = tan1 x1 x2 = cot
1
1 x2x
=
sec11
1 x2 = csc1 1
x
2 cos1 x = sin1
1 x2 = tan1
1 x2x
= cot1x
1 x2 =
sec1 1
x = csc1 1
1 x23 tan1 x = sin1
x1 + x2
= cos11
1 + x2= cot1
1
x=
sec1
1 + x2 = csc1
1 + x2
x
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Double Angle & Triple Angle
Involving sin 2
2sin1 x =
sin1
2x
1 x2
, 1
2 x 1
2 sin1 2x
1 x2, 1
2 x 1
sin1 2x
1 x2, 1 x 12
Involving sin 3
3sin1 x =
sin1 3x4x3, 12
x 12
sin1 3x4x3, 12
x 1 sin1 3x4x3, 1 x 1
2
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Double Angle & Triple Angle
Involving cos 2
2cos1 x =
cos1(2x2 1) ,0 x 12 cos1(2x2 1) ,1 x 1
Involving cos 3
3cos1 x =
cos1(4x3 3x) , 12
x 12 cos1(4x3 3x) ,1
2 x 1
2
2+ cos1(4x3 3x) ,1 x 12
Author, Another Inverse Trigonometry
8/8/2019 TrigEqns InveTrigFunc Presentation-2x3
11/11
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Double Angle & Triple Angle
Involving tan 2
2tan1 x =
tan1
2x
1 x2
,1 x 1
+ tan1
2x
1 x2
,x> 1
+ tan1
2x
1 x2
,x< 1
Involving tan 3
3tan1 x =
tan1
3x x31 3x2
, 1
3< x
13
+ tan1
3xx31 3x2
,x< 1
3
Author, Another Inverse Trigonometry
Property-IProperties II
Inverse Of ReciprocalsInvolving tan(AB)One Inverse function to another
Double Angle & Triple Angle
Involving sin 2 in terms of tan
2tan1 x =
sin1
2x
1 + x2
,1 x 1
sin1
2x
1 + x2
,x> 1
sin1
2x
1 + x2
,x
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