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Chapter 3

TRIGONOMETRIC FUNCTIONS

Chapter-3 Trigonometric Function of class 11 CBSE (ncert book) contains all the main important points of the chapter Trigonometric Function of class 11 ncert-cbse. It all about revision-notes of cbse-class-11-12-maths-notes or mathematics Notes If you are preparing for government jobs like NDA SSR /AA Air force X and Y group Officer level mathematics Navy/Air force/army, This all revision-notes of cbse-class-11-12-maths-notes is enough for you to prepare yourself in a better way. Revision-notes could be revising you everything about the chapter. Everything is define here shortly. After reading this you will be a great knowledge of Trigonometric Function. Trigonometric Function is the one most important chapter of cbse-class-11-12-maths. It plays very important role in government-objective-type-questions. Chapter Trigonometric Function is the most important chapter of ncert-class-11 book.

 

• Angle

Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex. If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative.

• Degree Measure

If a rotation from the initial side to terminal side is (1/360)th of a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is divided into 60 minutes, and a minute is divided into 60 seconds. One sixtieth of a degree is called a minute, written as 1¢, and one sixtieth of a minute is called a second, written as 1².

Thus, 1° = 60¢, 1¢ = 60²

 

• Radian Measure

There is another unit for measurement of an angle, called the radian measure. Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.

 

• Relation between degree and radian

2p radian = 360° or

p radian = 180°

 

• 1 radian = (180°/p) = 57° 16¢ approximately.

1° = (p/180)radian = 0.01746 radian approximately.

 

• The relation between degree measures and radian measure of some common angles are given in the following image:

 

Trigonometric Functions

If in a circle of radius r, an arc of length l subtends an angle of q radians, then

l = r q

 

Radian measure = p/180 * Degree measure

Degree measure=180/p * Radian measure

Cos²x + sin²x = 1

1 + tan²x = sec² x

1 + cot²x = cosec²x

 
cos (2np+ x) = cos x

sin (2np+ x) = sin x

sin (– x) = – sin x

cos (– x) = cos x

cos (x + y) = cos x cos y – sin x sin y

cos (x – y) = cos x cos y + sin x sin y

cos (p/2 +x) =  – sin x
cos (p - x) =  – cos x

cos (p + x) =  – cos x 

cos (2p - x) = cos x

sin (p/2 + x) = cos x

sin (p - x) = sin x 

sin (p + x) = – sin x

sin (2p – x) = – sin x

tan (x + y) = (tan x + tan y) / (1 – tan x tan y)
tan (x - y) = (tan x - tan y) / (1 + tan x tan y)
cot (x + y) = (cot x cot y - 1) / (cot y + cot x)

cot (x - y) = (cot x cot y + 1) / (cot y - cot x)

 

cos 2x = cos² x – sin² x
cos 2x = 2 cos² x – 1
cos 2x = 1 – 2 sin² x

cos 2x = (1 – tan² x) / (1 + tan² x)

 

sin 2x = 2 sin x cos x
sin 2x = (2 tan x) / (1 + tan² x)
tan 2x = (2tan x) / (1 – tan² x)
 
sin 3x = 3sinx – 4sin³ x
cos 3x = 4cos³ x – 3cos x

tan 3x = (3tan x – tan³ x) / (1 – 3tan² x)

 

cos x + cos y = 2cos [(x + y)/2] cos [(x - y)/2]
cos x - cos y = - 2sin [(x + y)/2] sin [(x - y)/2]
sin x + sin y = 2sin [(x + y)/2] cos [(x - y)/2]
sin x - sin y = 2cos [(x + y)/2] sin [(x - y)/2]

 

2cos x cos y = cos ( x + y) + cos ( x – y)
– 2sin x sin y = cos (x + y) – cos (x – y)
2sin x cos y = sin (x + y) + sin (x – y)

2 cos x sin y = sin (x + y) – sin (x – y)

 

sin x = 0 gives x = np, where n Î Z
cos x = 0 gives x = (2n + 1) p/2, where n Î Z
sin x = sin y implies x = np + (– 1)ⁿ y, where n Î Z
cos x = cos y, implies x = 2np ± y, where n Î Z

tan x = tan y implies x = np + y, where n Î Z

 

 

 

revision-notes | cbse-class-11-12-maths-notes

Chapter 1 SETS  

Chapter-1 sets of class 11 CBSE (ncert book) contains all the main important points of the chapter sets of class 11 ncert-cbse. It all about revision-notes of cbse-class-11-12-maths-notes or mathematics Notes If you are preparing for government jobs like NDA SSR /AA Air force X and Y group Officer level mathematics Navy/Air force/army, This all revision-notes of cbse-class-11-12-maths-notes is enough for you to prepare yourself in a better way. Revision-notes could be revising you everything about the chapter. Everything is define here shortly. After reading this you will be a great knowledge of sets. Sets is the one most important chapter of cbse-class-11-12-maths. It plays very important role in government-objective-type-questions. Chapter Sets is the most important chapter of ncert-class-11 book.

 

·         A set is a well-defined collection of objects.

 

(i) Objects, elements and members of a set are synonymous terms.

(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.

(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.

 

We give below a few more examples of sets used particularly in mathematics.

N : the set of all natural numbers

Z : the set of all integers

Q : the set of all rational numbers

R : the set of real numbers

Z+ : the set of positive integers

Q+ : the set of positive rational numbers, and

R+ : the set of positive real numbers.

There are two methods of representing a set:

(i) Roster or tabular form

(ii) Set-builder form.

In roster form, the order in which the elements are listed is immaterial. Thus, the above set can also be represented as {1, 3, 7, 21, 2, 6, 14, 42}. It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct. For example, the set of letters forming the word ‘SCHOOL’ is { S, C, H, O, L} or {H, O, L, C, S}.

 

In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. For example, in the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property. Denoting this set by V, we write

V = {x : x is a vowel in English alphabet}

 

·         A set which does not contain any element is called empty set. The empty set is denoted by the symbol  f or { }.

 

·         A set which consists of a definite number of elements is called finite set. Consider some examples :

(i) Let W be the set of the days of the week. Then W is finite.

(ii) Let S be the set of solutions of the equation x²–16 = 0. Then S is finite.

(iii) Let G be the set of points on a line. Then G is infinite. Otherwise, the set is called infinite set.

All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

•  Two sets A and B are said to be equal if they have exactly the same elements.

A set does not change if one or more elements of the set are repeated. For example, the sets A= {1, 2, 3} and B = {2, 2, 1, 3, 3} are equal, since each element of A is in B and vice-versa. That is why we generally do not repeat any element in describing a set.

 

 

·         A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of R. The symbol Ì stands for ‘is a subset of’ or ‘is contained in’.

In other words, A Ì B if whenever a Î A, then a Î B. It is often convenient to use the symbol “Þ” which means implies. Using this symbol, we can write the definition of subset as follows:

A Ì B if a Î A Þ a Î B

 

 

·         A power set of a set A is collection of all subsets of A. It is denoted by P(A).

The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set. If A = { 1, 2 }, then

P( A ) = { f,{ 1 }, { 2 }, { 1,2 }}

Also, note that n [ P (A) ] = 4 = 2²

In general, if A is a set with n(A) = m, then it can be shown that

n [ P(A)] = 2^m.

 

·         Universal Set This basic set is called the “Universal Set”. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc. For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers. For another example, in human population studies, the universal set consists of all the people in the world.

 

·         The union of two sets A and B is the set of all those elements which are either in A or in B.

 

·         The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B.

 

·         The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A.

·         For any two sets A and B, (A È B)¢ = A¢ Ç B¢ and ( A Ç B )¢ = A¢ È B¢

 

·         If A and B are finite sets such that A Ç B = f, then

n (A È B) = n (A) + n (B).

If A Ç B ¹ f, then

n (A È B) = n (A) + n (B) – n (A Ç B)