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8/8/2019 Compendium ClassXIIMaths
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Compendium/Math/Class XII 1 © 2010 Vidyamandir Classes
Dear Students,
Vidyamandir Classes academic team knows that you are rigourously studying to cover the entire prescribedsyllabus. As the Final Exams approach, this is the time when you need to revisit the concepts you havelearned. At this time, you have to be very focused and directed in your approach.
To make your learning process precise, effective and enjoyable, we at Vidyamandir Classes conceptualizedthe compendium series, strategically designed to help you in scoring high grades in examination. TheCompendium is primarily intended to present the concepts of chapter in a concise manner. All key definitions,diagrams and formulae have been integrated for a quick revision of the chapter.
To help you to easily master complicated concepts, definitions, diagrams and formulae, we have addedinteresting tips, mnemonics, maps and matrices. Let us take a look at the elements of the Compendium andhow to use them.
Knowing these features will make it easier for you to assimilate complex information.
Icon Description How it can help you
Concept mapTo directly recapitulate main concepts of the
chapter.
Drawing Tips
Drawing Tips
To help you draw and remember diagrams,
we have thoughtfully developed some
mnemonics to help you to memoriseinformation
Compare
Contrast MatrixTo help you in comparing different concepts
Memory TipsTo make your learning process effective, easy
tips have been provided.
In this compendium, we have also incorporated:• CBSE Blue Print: Type of questions asked and the weightage of different forms of questions.
• Analysis of Previous Years CBSE questions: The topic wise analysis of previous years question alongwith the marks allocated.
• We are confident that this Compendium will prove very helpful in achieving excellent result in yourexams.
All the very best for your exams!
Vidyamandir Classes Academic Team
ABOUT THE COMPENDIUM
8/8/2019 Compendium ClassXIIMaths
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Compendium/Math/Class XII 2 © 2010 Vidyamandir Classes
A bout I nverse Tri gonomet ri c F uncti ons
Since this chapter was introduced in class XIIth in the year 2008, we have only last year’s question
paper as reference.
Types of Questions Very short answer
( 1 mark)
Short answer
( 4 marks)
Long answer
(6 marks)
Option I Number of questions 1 1 -
Option II Number of questions - - 1
As you can see, the questions asked were mainly conceptual and based on the properties of
inverse trigonometric functions. You need to have a clear idea of principal values of all six inverse
trigonometric functions. Also, questions based on properties need to be practised rigourously .
CBSE BLUEPRINT
Inverse Trigonometric Functions
1. Evaluate :1 1
sin sin3 2
π
. 1 mark
2. Prove the following:1 1 1 11 1 1 1
tan tan tan tan3 5 7 8 4
π 4 mark
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Compendium/Math/Class XII 3 © 2010 Vidyamandir Classes
INVERSE TRIGONOMETRIC FUNCTIONS
Key Formulae
Function1
co s y x
Domain 1,1 x
Range
(principal value
branch)
0, π
Function1
si n y x
Domain 1,1 x
Range
(principal value
branch)
,2 2
π π
Function1ta n y x
Domain , x
Range
(principal value
branch)
,2 2
π π
Funct ion1c ot y x
Doma in , x
R ange
(principal value
branch )
0, π
Both sin–1 x and cos–1 x
have the same domain.
Graphs of sin–1 x
and cos–1 x have a phase
difference of
Both tan–1 x and cot–1 x
have the same domain.
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Compendium/Math/Class XII 4 © 2010 Vidyamandir Classes
Both sec–1 x and
cosec –1 x have the
same domain.
Graphs of sec–1 x and
cosec–1 x have a phase
difference of
Range of cosec–1 x is
the same as that of
sin–1 x except the
element 0.
Range of sec–1 x is the
same as that of cos–1 x
except the element
.
Key Properties
(i) 1 1sin
x
= 1cosec , 1 or 1 x x x
(ii) 1 1cos x
= 1sec , 1 or 1 x x x
(iii) 1 1tan
x
= 1cot , 0 x x
Trigonometric Inverse Functions of Reciprocals
Function 1cosec y x
Domain ( , 1] [1, ) x
Range
(principal
value
branch)
, {0}2 2
π π
Function1
sec y x
Domain ( , 1] [1, ) x
Range
(principal
value
branch)
[0, π ] – { π /2}
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Compendium/Math/Class XII 5 © 2010 Vidyamandir Classes
(i)1 1
sin cos x x = , [ 1,1]
2
π x
(ii)1 1
tan cot x x = , R
2
π x
(iii)1 1
cosec sec x x = , 1
2
π x
Trigonometric Inverse of Negative Argument Functions
Trigonometric Inverse of Trigonometric Functions and Trigonometric Value of Trigonometric
Inverse Function
Important properties which are used extensively in solving questions based on inverse
trigonometric functions.
Note the Pattern
In the first table, we get
the same inverse function
with negative sign
whereas in the second
table an additional
appears with a negative
inverse function.
Note the
Pattern
In the first
table, inverse
of a
trigonometric
function gives
the angle
whereas in the
second table,
trigonometric
function of an
inverse
function gives
the value of
the function.
(i) 1sin ( ) x
= 1sin , [ 1,1] x x
(ii) 1tan ( ) x
= 1tan ( ), R x x
(iii) 1cosec ( ) x
= 1cosec , 1 x x
–1sin sin θ θ , ,
2 2
π π θ
1sin sin x = x , 1,1 x
–1cos cos θ θ , 0,θ π 1
cos cos x = x , 1,1 x
–1tan tan θ θ , ,
2 2
π π θ
1
tan tan x = x , x R
–1cosec cosec θ θ , , ; 0
2 2
π π θ θ
1cosec cosec x
= x ,
( , 1] [1, ) x
–1sec sec θ θ , 0, ;
2
π θ π θ 1
sec sec x = x , ( , 1] [1, ) x
–1cot cot θ θ , 0,θ π 1
cot cot x = x , x R
(i) 1cos ( ) x
= 1cos , [ 1,1]π x x
(ii) 1sec ( ) x
= 1s ( ), 1π ec x x
(iii) 1cot ( ) x
= 1cot , Rπ x x
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(i)
tan–1
x + tan–1
y =1
tan , 11
x y
xy xy
(ii)tan
–1 x – tan
–1 y = 1 –
tan , 11
x y xy
xy
(iii)2tan
–1 x = 1
2
2tan , 1
1
x x
x
(iv)2tan
–1 x = 1
2
2sin , 1
1
x x
x
(v)2tan–1 x =
2
12
1cos , 0
1
x x
x
(vi)2tan
–1 x = 1
2
2tan , 1 1
1
x x
x
Properties of Tangent Inverse Function
These properties are used
when two tangent inverse
functions are added or
subtracted. Also, tan–1 x
can be written in terms of
sin –1 x or cos–1 x using
these properties.
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