Inverse Trigonometric Functions

Inverse Trigonometric Functions

y=sin⁻¹ x⇒ x= sin y

y=sin y ⇒ x= sin⁻¹ y

sin(sin⁻¹ x) = x

sin⁻¹ (sin x) = x

sin⁻¹ ⅟𝖝 = cosec⁻¹ x

cos⁻¹(-x) = 𝛑 - cos⁻¹x

cos⁻¹  ⅟𝖝 = sec⁻¹x

cot⁻¹ (-x) = 𝛑 - cot⁻¹ x

tan⁻¹  ⅟𝖝 = cot⁻¹ x

sec⁻¹(-x) =  𝛑 - sec⁻¹ x

sin⁻¹ (-x) = -sin⁻¹x

tan⁻¹(-x) = -tan⁻¹x

Important Maths Symbols

Important Maths Symbols

Plus    +	Minus     ₋
Multiplied    ✖	Divided ÷
Plus or minus    ±	Not equal to ≠
Square root of  √	Percent     %
Equal to    =	Infinity    ∞
Approximately equal to   ≈	Integral    ∫
Greater than    ＞	Sum of    ∑
Less than   <	Equivalent to   ⇔
Greater than or equal    ≥	Implies    ⇒
Less than or equal    ≤	For all  ∀
Square brackets    []	Belongs to  ∈
Parentheses   ()	Pi   𝛑
Curly brackets   ❴❵	Empty set   ∅
Not belong to ∉	Cube root of   ∛
Angle   ∠	Right angle  ∟
Fourth root of   ∜

Some important points to remember in Factors and Multiples 6th grade Maths Plus

Some important points to remember in Factors and Multiples

1. A factor of a number divides the number exactly.

2. A multiple of a number is a number divisible by the given number.

3. 1 is a factor of every number.

4. Every number is a multiple of itself.

5. Every number is a factor of itself.

6. A number may have infinitely many multiples.

7. A number may have only a finite number of factors.

8. The minimum number of factors of a number is 2.

9. Every prime number has only two factors, viz. 1 and the number itself.

10. A factor of a number is always less than or equal to the number.

11. A multiple of a number is always greater than or equal to the number.

12. All even numbers are divisible by 2.

13. All odd numbers are not divisible by 2.

14. A composite number has at least three factors.

15. 1 is neither a prime nor a composite number.

16. The only prime number which is even is 2.

17. The least composite number is 4.

18. The least prime number is 2.

19. A composite number may be even or odd.

20. All prime numbers except 2 are odd.

21. The product of two odd numbers is always odd.

22. The product of two even numbers is always even.

23. The product of an even number and an odd number is always even.

24. The sum of difference of two even numbers is always even.

25. The sum of difference of two odd numbers is always even.

26. The sum or difference of tan even number and an odd number is always odd.

27. 0 is an even number.

28. If a number a is divisible by b, then a will be divisible by each factor of b.

29. If a number a is divisible by two co-prime numbers b and c, then a is divisible by their product b ✖ c.

30.  If a number a divides  by two other numbers b and c exactly, then a is divides their sum b + c and their difference b - c (b > c) esactly.

31. The H.C.F. of two or more given numbers is the greatest number which divides the given numbers exactly.

32. The L.C.M. of two or more numbers is the least number which is divisible by all the numbers.

33. The H.C.F.  of two or more given numbers is always less than or equal to the given numbers.

34. The L.C.M. of two or more given numbers is always greater than or equal to the given numbers.

35. The H.C.F. of two or more numbers always divides their L.C.M. exactly.

36. If a number a is a factor of another number b, then a is the L.C.M. of a and b.

37. The L.C.M. of two or more prime numbers is always equal to their product.

38. The L.C.M. of two co - prime numbers is always equal to their product.

39. The H.C.F.  of two or more prime numbers is always equal to 1.

40. The H.C.F. of two co - prime numbers is always equal to 1.

Difference between H.C.F and L.C.M of two or more numbers 6th grade Maths Plus

Difference between H.C.F and L.C.M

H.C.M	L.C.M
1. It is the largest of all the common factors of the given numbers.	1. It is the least of all the common multiples of the given numbers.
2. It is the greatest number which divides all the  given numbers exactly.	2. It is the least number which divisible by all the  given numbers.
3. It is always less than or equal to any of the given numbers.	3. It is always greater than or equal to any of the given numbers.
4. H.C.F. of two or more prime numbers or two co-prime numbers is always 1.	4. L.C.M. of two or more prime numbers or two co-prime numbers is equal to the product of all prime numbers or the product of two co-prime numbers.
5. H.C.F. of two or more numbers divides their L.C.M. exactly.	5. L.C.M. of two or more numbers is divisible by these numbers exactly.

Difference between factors and multiples 6th grade Maths Plus

Difference between factors and multiples

Factors	Multiples
1. A given number has a finite number of factors.	1. A given number has a infinitely many multiples.
2. The factors of a given number are all less than or equal to the given number.	2. The multiples of a given number are all more than or equal to the given number.
3. The minimum number of factors of a given number is 2, i.e.  1 and the number itself. Eg: 5 has two factors only, viz. 1 and 5	3. Every given number has infinitely many multiples and so there is no minimum  number of multiples of a given  number.
4. A factor of a given number divides the given number exactly.	4. A multiple of a given number divisible by the given number.
5. The factors of a given number can be obtained by selecting those numbers out of the numbers 1,2,3,4..... which divide the given number exactly.	5. The multiples of a given number can be obtained by multiplying the given  number successively by 1,2,3....

Easy Maths Using Vedic Maths - Properties of "0"

Easy Maths Using Vedic Maths - Properties of "0"

1. Indian mathematics invented zero(0).

2. Bhaskara 11 is the 1st mathematician, who gave the idea of infinite to a fraction having zero as its denominator. 

3. Fraction with denominator zero is only a symbol to represent infinity. 

4. If zero is added to or subtracted from any number, that number remains as it is (without any change in sign). If any number is subtracted from zero its sign changes. 

5. If zero is multiplied by any number or divided by any number, the result is zero.

6. The single digit '0' does not have any value of its own and when a place value is empty, "0" will occupy the place.

7. It is considered as even number it is neither positive not negative. 

8. Zero is the additive identity.

9. Any  number other than 1, raised to the power of zero is 1.

10. For a base number 10, when it is raised to any power, the number of zeros in the result will be equal to that number of power.

Ex: 10¹ = 10; 10² = 100; 10³ = 1000, ............. 

Tests for divisibility of numbers - 6th class

Tests for divisibility of numbers

Divisibility by 2:  A number is divisible by 2, if the digit in its unit's place is divisible by 2. Hence, this number must be even and so the digit in the unit's place of this number must be 0, 2, 4, 6 or 8. Other numbers which end with the digits 1, 3, 5, 7 or 9 will not be divisible by 2. Thus, numbers like 20592, 31296, 802598, etc. are all divisible by 2. 

Divisibility by 3: A number is divisible by 3, if the sum of the digits of this number is divisible by 3. For example, 409362, 523758, etc. are all divisible by 3. For, the sum of the digits of the 1st number is 4 + 0 + 9 + 3 + 6 + 2 = 24 which is divisible by 3. Similarly, the sum of the digits of the 2nd number is 5 + 2  + 3 + 7 + 5 + 8 = 30, which is divisible by 3.

Divisibility by 9: A number is divisible by 9, if the sum of the digits of this number is divisible by 9. For example, 8208 is divisible by 9, for, 8 + 2 + 0 + 8 = 18 is divisible by 9. 

Divisibility by 5: A number is divisible by 5, if the digit in its unit's place is 0 or 5. For example, 30590, 8935, etc. are all divisible by 5. A number whose unit's digit is other than 0 or 5 is not divisible by 5. Thus, 31574, 8932, etc. are not divisible by 5. 

Divisibility by 10: A number whose unit's digit is 0 is divisible by 10. A number whose unit's digit is not 0 is not divisible by 10. Thus, 3890, 52300, etc. are all divisible by 10, but 31925, 73212, etc. re not divisible by 10. 

Divisibility by 11: A number is divisible by 11, if the difference of the sum of its digits in odd places and the sum of its even places starting from the unit's place is divisible by 11. For example, we take the following numbers: (1) 107877 and (2) 35849. In 107877 the digits in the odd places are 7, 8, 0 and those in the even places are 7, 7 and 1. Therefore, the sum of the digits in odd places = 7 + 8 + 0 = 15 and the sum of the digits in even places = 7 + 7 + 1 = 15. Difference of the two sums = 15 -  15 = 0 which is divisible by 11. Hence, the given number 107877 is divisible by 11.In 35849,  the sum of the digits in the odd places = 9 + 8 + 3 = 20 and the sum of the digits in the even places = 4 + 5 = 9. The difference of the two sums = 20 - 9 = 11 which is divisible by 11. 

Divisibility by 4:  A number is divisible by 4, if the number formed by its digits in ten's place and unit's place is divisible by 4. Thus, the numbers like 532, 2316, 89372, etc. are all divisible by 4, since 32, 16, 72 are divisible by 4. 

Divisibility by 8:  A number is divisible by 4, if the number formed by its digits in hundred's place, ten's place and unit's place is divisible by 8. Thus, the numbers 5720, 26824, 357128, etc. are all divisible by 8, since 720, 824, 128 are all divisible by 8.

Divisibility by 6:  A number is divisible by 4, if it is divisible by both 2 and 3. Clearly, the number should be even and the sum of its digits should be divisible by 3. For example, the numbers 2256, 41568, 980352, etc, are ll divisible by 6, since all these numbers are even and the sum of the digits of each number is divisible by 3.