TrigEqns InveTrigFunc Presentation-2x3

8/8/2019 TrigEqns InveTrigFunc Presentation-2x3

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Inverse Trigonometry

Inverse Trigonometric functions

Vidyalankar Classes

Rankers Batch, IIT JEE 2011

Author, Another Inverse Trigonometry

Foundations

Outline


FoundationsMappings and FunctionInverse of a function

Outline

1 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



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



Outline

1 FoundationsMappings and FunctionInverse of a function



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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


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


Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs

Outline

2 Trigonometric functionsInverse Trigonometric functionsDefinition - Inverse Trig-functionsGraphs




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







1 Draw graph of y = sin x

2 Get graph of y = x











Truncation of domain of sine is required as the graph of inverse isnot a function



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Outline

2 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


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




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




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




Problems

Example

1 If u= cot1[

cos2] tan1[cos2] then show thatcscu= cot2



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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




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




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




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



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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




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




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




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




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





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



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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





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

TrigEqns InveTrigFunc Presentation-2x3

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