Important Formulae & Rules- Part - 2

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Trigonometry

Important Formulae & Rules- Part - 2

Author: Leonardodvin

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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

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

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

The point of intersection of the altitudes.

Periodic Function

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)

for all x.

Pigeonhole Principle

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

Power Mean Inequality

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

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

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

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 Square-Arithmetic Mean Inequality

For positive numbers $x_1, x_2,\cdots,x_n,$

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

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

Schur's Inequality

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

The region enclosed by a circle and two radii of the circle.

Stewart's Theorem

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

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

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:

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

ViÃ¨te's Theorem

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,

$x_1+x_2+\cdots +x_n=-\frac {a_{n-1}}{a_n};$
$x_1x_2+\cdots +x_ix_j+x_{n-1}x_n=\frac {a_{n-2}}{a_n};$
$\overset {.}{\underset {.}{.}}$
$x_1x_2\cdots x_n=(-1)^n \frac {a_0}{a_n}.$