Compendium ClassXIIMaths

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




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




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.




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}




(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




(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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Compendium ClassXIIMaths