Mathematics

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Important Formulae & Rules- Part - 2

by Satish

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Law of Cosines
In a triangle ABC,

|CA|^2=|AB|^2+|BC|^2-2|AB|.|BC|\cos \angle ABC,


and analogous equations hold for |AB|^2 and |BC|^2.


Median formula
This is also called the length of the median formula. Let AM be a median in triangle ABC. Then

|AM| ^ 2 =\frac {2|AB|^2+2|AC|^2-|BC|^2}{4}


Minimal Polynomial
p<>. We call a polynomial p(x) with integer coefficients irreducible if p(x) cannot be written as a product of two polynomials with integer coefficients neither of which is a constant. Suppose that the number \alpha is a root of a polynomial q(x) with integer coefficients. Among all polynomials with integer coefficients with leading coefficient 1 (i.e., monic polynomials with integer coefficients) that have \alpha as a root, there is one of smallest degree. This polynomial is the minimal polynomial of \alpha. Let p(x) denote this polynomial. Then p(x) is irreducible, and for any other polynomial q(x) with integer coefficients such that q(\alpha)=0, the polynomial p(x) divides q(x); that is, q(x) = p(x)h(x) for some polynomial h(x) with integer coefficients.


Orthocenter of a Triangle
p<>. The point of intersection of the altitudes.


Periodic Function
p<>. A function f (x) is periodic with period T > 0 if T is the smallest positive real number for which

f(x+T)=f(x)%


%{font-family:verdana}for all x.



Pigeonhole Principle
p<>. If n objects are distributed among k < n boxes, some box contains at least two objects.


Power Mean Inequality
p<>. Let a_1, a_2,\cdots , a_n be any positive numbers for which a_1+a_2+\cdots +a_n=1. For positive numbers x_1, x_2,\cdots ,x_n we define

M_{-\infty}=min\{x_1,x_2,\cdots ,x_k\},


M_{\infty}=max\{x_1,x_2,\cdots ,x_k\},
M_0=x_1^{a1}x_2^{a2}\cdots x_n^{an},
M_t=(a_1x_1^t+a_2x_2^t+\cdots +a_kx_k^t)^{1/t},%


%{font-family:verdana}where t is a non-zero real number. Then

M_{-\infty}\underline <M_s\underline <M_t\underline < M_{\infty}

for s\underline <t


Rearrangement Inequality
p<>. Let a_1 \underline <a_2\underline <\cdots \underline <a_n;b_1\underline <b_2\underline <\cdots \underline <b_n be real numbers, and let c_1,c_2,\cdots ,c_n be any permutations of b_1\underline <b_2\underline <\cdots \underline <b_n. Then
a_1b_n+a_2b_{n-1}+\cdots+a_nb_1\underline <a_1c_1+a_2c_2+\cdots+a_nc_n\underline <a_1b_1+a_2b_2+\cdots +a_nb_n,
with equality if and only if a_1=a_2=\cdots=a_n or b_1=b_2=\cdots=b_n


Root Mean Squareâ€"Arithmetic Mean Inequality
p<>. For positive numbersx_1, x_2,\cdots,x_n,

\sqrt {\frac {x_1^2+x_2^2+\cdots+x_k^2}{n}}\underline >\frac {x_1+x_2+\cdots +x_k}{n}


The inequality is a special case of the power mean inequality.



Schur’s Inequality
p<>. Let x, y, z be non-negative real numbers. Then for any r > 0,

x^r(x-y)(x-z)+y^r(y-z)(y-x)+z^r(z-x)(z-y)\underline >0.

Equality holds if and only if x = y = z or if two of x, y, z are equal and the third is equal to 0.
The proof of the inequality is rather simple. Because the inequality is symmetric in the three variables, we may assume without loss of generality that x\underline >y\underline >z. Then the given inequality may be rewritten as

(x-y)[x^r(x-z)-y^r(y-z)]+z^r(x-z)(y-z)\underline >0

and every term on the left-hand side is clearly nonnegative. The first term is positive if x > y, so equality requires x = y, as well as z^r(x-z)(y-z)=0, which gives either x = y = z or z = 0.


Sector
p<>. The region enclosed by a circle and two radii of the circle.



Stewart’s Theorem
p<>. In a triangle ABC with cevian AD, write a = |BC|, b = |CA|, c = |AB|, m = |BD|, n = |DC|, and d = |AD|. Then

d^2a+man=c^2n+b^2m.

This formula can be used to express the lengths of the altitudes and angle bisectors of a triangle in terms of its side lengths.



Trigonometric Identities
p=. \sin^2a+\cos^2 a = 1,
1+\cot^2a=csc^2a,
\tan^2 x + 1 = \sec^2 x.

  • Addition and Subtraction Formulas:

    \sin(a\pm b)=\sin a \cos b \pm \cos a \sin b,


    \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b,
    \tan(a\pm b)=\frac {\tan a \pm \tan b}{1 \mp \tan a \tan b},
    \cot(a \pm b)=\frac {\cot a \cot b \mp 1}{\cot a \pm \cot b}.


  • Double-Angle Formulas:

    \sin 2a = 2 \sin a \cos a =\frac {2 \tan a}{1 + \tan^2 a},


    \cos 2a = 2 \cos^2 a-1 = 1-2 \sin^2 a =\frac {1-\tan^2 a}{1+\tan^2 a},
    \tan 2a =\frac {2 \tan a}{1-\tan^2 a};
    \cot 2a =\frac {\cot^2 a-1}{2 \cot a}.


  • Triple-Angle Formulas:

    \sin 3a = 3 \sin a-4 \sin^3 a,


    \cos 3a = 4 \cos^3 a-3 \cos a,
    \tan 3a =\frac {3 \tan a-\tan^3 a}{1-3 \tan^2 a}.


  • Half-Angle Formulas:

    \sin^2 \frac {a}{2}=\frac {1-\cos a}{2},


    \cos^2 \frac {a}{2}=\frac {1+\cos a}{2},
    \tan \frac {a}{2}=\frac {1-\cos a}{\sin a}=\frac {\sin a}{1+\cos a},
    \cot \frac {a}{2}=\frac {1+\cos a}{\sin a}=\frac {\sin a}{1-\cos a}.


  • Sum-to-Product Formulas:

    \sin a + \sin b = 2 \sin \frac {a+b}{2}\cos \frac {a-b}{2},


    \cos a + \cos b = 2 \cos \frac {a+b}{2} \cos \frac {a-b}{2},
    \tan a + \tan b=\frac{\sin(a + b)}{\cos a \cos b}.


  • Difference-to-Product Formulas:

    \sin a-\sin b = 2 \sin \frac {a-b}{2}\cos \frac {a + b}{2},


    \cos a-\cos b = -2 \sin \frac {a-b}{2}\sin \frac {a+b}{2},
    \tan a-\tan b=\frac {\sin(a-b)}{\cos a \cos b}.


  • Product-to-Sum Formulas:

    2 \sin a \cos b = \sin(a + b) + \sin(a-b),


    2 \cos a \cos b = \cos(a + b) + \cos(a-b),
    2 \sin a \sin b = -\cos(a + b) + \cos(a-b).

Viete's Theorem
p<>. Let x_1,x_2,\cdots ,x_n be the roots of polynomial

P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0,

where a_n \not =0 and a_0,a_1,\cdots ,a_n \in C. Let s_k be the sum of the products of the x_i taken k at a time. Then

s_k=(-1)^k\frac {an-k}{an};

that is,

x_1+x_2+\cdots +x_n=-\frac {a_{n-1}}{a_n};


x_1x_2+\cdots +x_ix_j+x_{n-1}x_n=\frac {a_{n-2}}{a_n};
\overset {.}{\underset {.}{.}}
x_1x_2\cdots x_n=(-1)^n \frac {a_0}{a_n}.


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1 Comment
    Vivek M Tripathi
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    TrIpAtHi $Ur@JMon, 16 Feb 2009 09:41:32 -0000

    its good and very handy

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