KNOWLEDGE

In triangle ABC, let us denote the magnitudes of the sides AB, BC, CA by c, a, b respectively and the angle BAC, angle CBA, angle ACB by A, B, C respectively. Proof :  From the cosine rules, we have cos(B) = ((c 2 + a 2 - b 2 )/2ca), cos(C) = ((a 2 + b 2 - c 2 )/2ab) Now, let us consider the right hand side expression in the equation to be proved. Therefore, b cos(C) + c cos(B) = b ( (a 2  + b 2  - c 2 )/2ab ) + c ( (c 2  + a 2  - b 2 )/2ca ) ⇒ b cos(C) + c cos(B) = ( (a 2  + b 2  - c 2 ) + (c 2  + a 2  - b 2 ) )/2a = (2a 2 )/2a = a. Therefore, a = b cos(C) + c cos(B) Similarly, we can prove that b = c cos(A) + a cos(C) and c = a cos(B) + b cos(A)

Hello, Everyone. In this, we are going to derive the formula to find the sum of the squares of the first n natural numbers. Let S = 1 2  + 2 2  + 3 2  + .......... + (n-1) 2  + n 2 (m+1) 3 - m 3 = (m 3 + 3m 2 + 3m + 1) - m 3 = 3m 2 + 3m + 1 ------------ (i) put m = 1,2,3, ....... , (n-1), n in (i) Then, 2 3  - 1 3   = 3(1) 2  + 3(1) + 1 ------------ (1) 3 3  - 2 3   = 3(2) 2  + 3(2) + 1 ------------ (2) 4 3  - 3 3   = 3(3) 2  + 3(3) + 1 ------------ (3) ................... n 3  - (n-1) 3   = 3(n-1) 2  + 3(n-1) + 1 ------------ (n-1) (n+1) 3  - n 3   = 3(n) 2  + 3(n) + 1 ------------ (n) Now, by adding equations (1),(2),(3), ...... ,(n-1) and (n), we get the following equation. (n+1) 3  - 1 3   = 3(1 2  + 2 2  + 3 2  + .......... + (n-1) 2  + n 2 ) + 3(1 + 2 + 3 + ......... + (n-1) + n) + (1 + 1 + 1 +........ n times) 1 + 2 + 3 +...

Hello, Everyone. In this, we are going to see a proof for the formula to find sum of the first n Natural numbers. Let S = 1 + 2 + 3 + 4 + ......... + (n-1) + n , where n is any natural number.  --------------- (1) The r th term from the starting of the series, in the above series is r. ------- (2) From the property of  addition of numbers, we can write S as shown below. S = n + (n-1) + (n-2) + ......... + 2 + 1 ---------- (3)  The r th term from the starting of the series, in the above series is n-(r-1) = n - r + 1 -------- (4) Let us add (1) and (3) such that the corresponding terms are added as shown in the below figure. From (2) and (4), The r th term from the starting of the series, in the resulting series obtained from addition of (1) and (3) is equal to r + (n - r + 1) = n + 1. From this, it is obvious that the r th term of the resulting series is independent of r. Therefore, as n is a constant, every term in the resulting series is equal to n + 1. Therefore, by add...

What to find :   Locus of point from which tangents drawn to a circle have a particular angle (Angle not equal to zero) between them.  Hello, Everyone. Before going to find our required solution, let us try to observe a phenomenon in the life.   Figure : Illustration of variation of the angle with perpendicular distance from the wall Assumptions :  Let us assume that a person X is standing in front of a wall shown in above figure. Assume height of the wall is more than the person. Let A, B and C be the different positions of eye of the person as the person moves on the horizontal line segment shown in the above figure. Let "O" be the point on the wall lying on the horizontal line segment drawn from the eye to the wall such that it is perpendicular to the wall. Let "E" be the one end of the wall as shown in the above figure. In the above figure, variation of the angle means variation of the a ngle between the horizontal line segment and line segment joining the eye ...

Hello Everyone. Before going to the proof, let us go into the basics of Rational and Irrational numbers. Rational Number  :   A number that can be expressed as the fraction p/q of two integers p and q, where q not equal to zero. Irrational Number  :   A number that cannot be expressed as the fraction p/q of two integers p and q,  where q not equal to zero. What is required to prove  :   Sum of a rational number and an irrational number will be an irrational number. Proof  : Let "m" be an irrational number. --------- (1) Let "n" be a rational number. ------------(2) Let us think that we don't know whether m + n is rational or irrational number. So, let us assume m + n as a rational number initially. Then, m + n = p/q where p and q are some integers and q is not equal to zero. (According to the definition) So, m + n = p/q Subtract "n" on both sides. Then, m = (p/q) - n = difference of two rational numbers Difference of two rational numbers will be a...

Hello Everyone. Before going to solve the question, let us go and understand divisibility rules of 2, 4, 8,…..     as it is an essential thing that should be known for understanding this approach of finding solution. Now, let us go to the divisibility rules of 2, 4, 8,...... Divisibility rule of 2 : A number is divisible by 2 if its units place digit is one of the digits 0, 2, 4, 6, 8. Divisibility rule of 4 : A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Divisibility rule of 8 : A number is divisible by 8 if the number formed by its last three digits is divisible by 8. Similarly, the divisibility rules of 16, 32, 64, .... follow the same pattern as we can see in the above divisibility rules. Now, we have understood the divisibility rules of 2, 4, 8, 16, 32,........ So, now let us go into the question. What to find :  Value of  "a" that satisfies   4^10 - 1 = 1048a75, by using only divisibility rules without actually c...