Basic Trigonometry || Class 10 || by #cmrohityadav #edata

Applying Pythagoras theorem for the given right-angled triangle, we have:

(Perpendicular)2+(Base)2=(Hypotenuese)2

⇒(P)2+(B)2=(H)2

The Trigonometric properties are given below:

S.no	Property	Mathematical value
1	sin A	Perpendicular/Hypotenuse
2	cos A	Base/Hypotenuse
3	tan A	Perpendicular/Base
4	cot A	Base/Perpendicular
5	cosec A	Hypotenuse/Perpendicular
6	sec A	Hypotenuse/Base

Relation Between Trigonometric Identities:

S.no	Identity	Relation
1	tan A	sin A/cos A
2	cot A	cos A/sin A
3	cosec A	1/sin A
4	sec A	1/cos A

Trigonometric Identities:

1. sin2A + cos2A = 1
2. tan2A + 1 = sec2A
3. cot2A + 1 = cosec2A

NUMBER SYSTEM || BASIC MATHS FOR ENTRANCE EXAM || BY #CMROHITYADAV #eData

Introduction

The collection of numbers is called the Number system. These numbers are of different types such as natural numbers, whole numbers, integers, rational numbers and irrational numbers. Let us see the table below to understand with the examples.

Natural Numbers	N	1, 2, 3, 4, 5, ……
Whole Numbers	W	0,1, 2, 3, 4, 5….
Integers	Z	…., -3, -2, -1, 0, 1, 2, 3, …
Rational Numbers	Q	p/q form, where p and q are integers and q is not zero.
Irrational Numbers		Which can’t be represented as rational numbers

Note: every natural number is an integer and 0 is a whole number which is not a whole number.

Natural Numbers

All the numbers starting from 1 till infinity are natural numbers, such as 1,2,3,4,5,6,7,8,…….infinity. These numbers lie on the right side of the number line and are positive.

Whole Numbers

All the numbers starting from 0 till infinity are whole numbers such as 0,1,2,3,4,5,6,7,8,9,…..infinity. These numbers lie on the right side of the number line from 0 and are positive.

Integers

Integers are the whole numbers which can be positive, negative or zero. They cover rational numbers also but not the irrational numbers.

Example: 2, 33, 0, -67, 9.777, are integers.

Rational Numbers

A number which can be represented in the form of p/q is called a rational number. For example, 1/2, 4/5, 26/8, etc.

Irrational Numbers

A number is called an irrational number if it can’t be represented in a p/q form, where p and q are integers.

Example: √3, √5, √11, etc.

Real Numbers

The collection of all rational and irrational numbers is called real numbers. Real numbers are denoted by R.

Every real number is a unique point on the number line and also every point on the number line represents a unique real number.

Difference between Terminating and Recurring Decimals

Terminating Decimals	Repeating Decimals
If the decimal expression of a/b terminates. I.e comes to an end, then the decimal so obtained is called Terminating decimals.	A decimal in which a digit or a set of digits repeats repeatedly periodically is called a repeating decimal.
Example: ¼ =0.25	Example: ⅔ = 0.666…

Some Special Characteristics of Rational Numbers:

• Every Rational number is expressible either as a terminating decimal or as a repeating decimal.
• Every Terminating decimal is a rational number.
• Every repeating decimal is a rational number.

Irrational Numbers

• The non-terminating, non repeating decimals are irrational numbers.

Example: 0.0100100001001…

• Similarly, if m is a positive number which is not a perfect square, then √m is irrational.

Example: √3

• If m is a positive integer which is not a perfect cube, then 3√m is irrational.

Example: 3√2

Properties of Irrational Numbers

• These satisfy the commutative, associative and distributive laws for addition and multiplication.
• Sum of two irrationals need not be irrational.

Example: (2 + √3) + (4 – √3) = 6

• Difference of two irrationals need not be irrational.

Example: (5 + √2) – (3 + √2) = 2

• Product of two irrationals need not be irrational.

Example: √3 x √3 = 3

• The quotient of two irrationals need not be irrational.

2√3/√3 = 2

• Sum of rational and irrational is irrational.
• The difference of a rational number and an irrational number is irrational.
• Product of rational and irrational is irrational.
• Quotient of rational and irrational is irrational.

Real Numbers

A number whose square is non-negative is called a real number.

• Real numbers follow Closure property, associative law, commutative law, the existence of an additive identity, existence of additive inverse for Addition.
• Real numbers follow Closure property, associative law, commutative law, the existence of a multiplicative identity, existence of multiplicative inverse, Distributive laws of multiplication over Addition for Multiplication.

Rationalisation

If we have an irrational number, then the process of converting the denominator to a rational number by multiplying the numerator and denominator by a suitable number, is called rationalisation.

Example:

3/ √2 = 3/ √2 x √2/ √2 = 3 √2/2

Laws of Radicals:

Let a>0 be a real number, and let p and q be rational numbers, then we have:

i) (ap x aq) = a(p+q)

ii) (ap)q = apq

iii)ap /aq = a(p-q)

iv) ap x bp = (ab)p

Example: Simplify (36)½

Solution: (62)½ = 6(2 x ½) = 61 = 6

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Laws of Exponents || BASIC MATHS FOR ENTRANCE EXAMS || by #cmrohityadav #edata

Laws of Exponents || BASIC #MATHS FOR #ENTRANCE #EXAMS || by #cmrohityadav #edata

Laws of Exponents

The laws of exponents are demonstrated based on the powers they carry.

• Bases – multiplying the like ones – add the exponents and keep base same. (Multiplication Law)
• Bases – raise it with power to another – multiply the exponents and keep base same.
• Bases – dividing the like ones – ‘Numerator Exponent – Denominator Exponent’ and keep base same. (Division Law)

Let ‘a’ is any number or integer (positive or negative) and ‘m’,  ‘n’ are positive integers, denoting the power to the bases, then;

Multiplication Law

As per the multiplication law of exponents, the product of two exponents with the same base and different powers equals to base raised to sum of the two powers or integers.

           am × an  = am+n

Division Law

When two exponents having same bases and different powers are divided, then it results in base raised to difference of the two powers.

          am ÷ an  = am / an  = am-n

Negative Exponent Law

Any base if has a negative power, then it results in reciprocal but with positive power or integer to the base.

            a-m  = 1/am 

Exponents and Powers Rules

The rules of exponents are followed by the laws. Let us have a look on them with a brief explanation.

Suppose ‘a’ & ‘b’ are the integers and ‘m’ & ‘n’ are the values for powers, then the rules for exponents and powers are given by:

i) a0 = 1

As per this rule, if the power of any integer is zero, then the resulted output will be unity or one.

Example: 50 = 1

ii) (am)n = a(mn)

‘a’ raised to the power ‘m’ raised to the power ‘n’ is equal to ‘a’ raised to the power product of ‘m’ and ‘n’.

Example: (52)3 = 52 x 3

iii) am × bm =(ab)m

The product of ‘a’ raised to the power of ‘m’ and ‘b’ raised to the power ‘m’ is equal to the product of ‘a’ and ‘b’ whole raised to the power ‘m’.

Example: 52 × 62 =(5 x 6)2

iv) am/bm = (a/b)m

The division of ‘a’ raised to the power ‘m’ and ‘b’ raised to the power ‘m’ is equal to the division of ‘a’ by ‘b’ whole raised to the power ‘m’.

Example: 52/62 = (5/6)2

Exponents and Power Solved Questions

Example 1: Write 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 in exponent form.

Solution:

In this problem 7s are written 8 times, so the problem can be rewritten as an exponent of 8.

7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 78.

Example 2: Write below problems like exponents:

1. 3 x 3 x 3 x 3 x 3 x 3
2. 7 x 7 x 7 x 7 x 7
3. 10 x 10 x 10 x 10 x 10 x 10 x 10

Solution: 

1. 3 x 3 x 3 x 3 x 3 x 3 = 36
2. 7 x 7 x 7 x 7 x 7 = 75
3. 10 x 10 x 10 x 10 x 10 x 10 x 10 = 107 

Example 3: Simplify 253/53 

Solution: 

 Using Law: am/bm = (a/b)m

253/53  can be written as (25/5)3  = 53 = 125.

Exponents and Powers Applications

Scientific notation uses the power of ten expressed as exponents, so we need a little background before we can jump in. In this concept, we round out your knowledge of exponents, which we studied in previous classes.

The distance between the Sun and the Earth is 149,600,000 kilometers. The mass of the Sun is 1,989,000,000,000,000,000,000,000,000,000 kilograms. The age of the Earth is 4,550,000,000 years. These numbers are way too large or small to memorize in this way.  With the help of exponents and powers these huge numbers can be reduced to a very compact form and can be easily expressed in powers of 10.

Now, coming back to the examples we mentioned above, we can express the distance between the Sun and the Earth with the help of exponents and powers as following:

Distance between the Sun and the Earth 149,600,000 = 1.496× 10 × 10 × 10 × 10 × 10× 10 × 10 = 1.496× 108 kilometers.

Mass of the Sun: 1,989,000,000,000,000,000,000,000,000,000 kilograms = 1.989 × 1030 kilograms.

Age of the Earth:  4,550,000,000 years = 4. 55× 109 years

Some Very Very Standard Algebraic Identities for class 5 to PG ( #class #10 ) by #cmrohityadav #eData

Some Standard Algebraic Identities list are given below:

 1:  (a + b)2 = a2 + 2ab + b2

 2: (a – b)2 = a2 – 2ab + b2

 3: a2 – b2= (a + b)(a – b)

4: (a + b)3 = a3 + b3 + 3ab (a + b)

 5: (a – b)3 = a3 – b3 – 3ab (a – b)

 6: x3 + y3= (x + y) (x2 – xy + y2)

7: x3 – y3 = (x – y) (x2 + xy + y2)

 8: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

Similarly

• (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
• (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
• (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz

 9: a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)

 8: (x + a)(x + b) = x2 + (a + b) x + ab

    Similarly

• (x + a)(x – b) = x2 + (a – b)x – ab
• (x – a)(x + b) = x2 + (b – a)x – ab
• (x – a)(x – b) = x2 – (a + b)x + ab

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