math_formulaAlgebra
(a+b)2=a2+b2+2ab
(a-b)2=a2+b2-2ab
(a+b)3=a3+3a2b+3ab2+b3
(a-b)3=a3-3a2b+3ab2-b3
a3+b3=(a+b)(a2-ab+b2)
a3-b3=(a-b)(a2+ab+b2)
ex=1+x1!+x22!+x33!+x44!+x55!+.....
ln(1+x)=x1-x22+x33-x44+x55-x66+..
cosx=1-x22!+x44!-x66!+x88!-....
Complex Number
z=a+ib=reiπœƒ
i=-1
zΒ―=a-ib
|z|=a2+b2
|z|=|-z|
zzΒ―=|z|2
|z1z2|=|z1||z2|
|z1z2|=|z1||z2|
(cosπœƒ+isinπœƒ)n=cosnπœƒ+isinnπœƒ
amp(z)=arg(z)=πœƒ=tan-1ba
Cube root of unity=1,πœ”,πœ”2πœ”=(-1+i32)πœ”2=(-1-i32)
Coordinate geometry
Point, Line
Distance formulaL=(x1-x2)2+(y1-y2)2
Slope of a line when coordinates of any two points on the line are givenslope=m=y2-y1x2-x1
Two lines with slope m1andm2 are parallel if m1=m2
Two lines with slope m1andm2 are perpendicular if m1.m2=-1
Angle between two lines with slope m1andm2 tanπœƒ=m2-m11+m1m2
Equation of straight line: Point slope formulam=y-y0x-x0
Equation of straight line: Point slope formulam=y-y0x-x0
Equation of straight line: two point formulay-y1x-x1=y2-y1x2-x1
Equation of straight line: slope intercept formy=mx+cwhere c is y-intercept
Equation of straight line: intercept formxa+yb=1where, a is x-intercept and b is y-intercept
Equation of straight line: normal formxcosπœ”+ysinπœ”=pwhere, p is normal distance of line from origin and πœ” is angle which normal makes with positive x-axis.
Conic Section
Equation of circle with center (h,k) and radius r will be(x-h)2+(y-k)2=r2
Set Theory
A binary operation * on the set X is called commutative, if aβˆ—b=bβˆ—a for every a,b∈X
Given a binary operation βˆ—:AΓ—Aβ†’A , an element e∈A , if exists, if aβˆ—e=a=eβˆ—a,βˆ€a∈A
Given a binary operation βˆ—:AΓ—Aβ†’A , with the identity element e in A , an element a∈A is said to be invertible with respect to the operation * , if there exist an element b in A such that aβˆ—b=e=bβˆ—athen b is called inverse of a and is denoted by a-1
Matrix and Determinants
(AT)T=A
(kA)T=kAT
(A+B)T=AT+BT
(AB)T=BTAT
A square matrix A=[aij] is said to be symmetric if AT=Ai.e.[aij]=[aji] for all possible values of i and j
A square matrix A=[aij] is said to be skew symmetric if AT=-Ai.e.[aij]=-[aji] for all possible values of i and j . So, diagonal elements of a skew symmetric matrix are zero.
For any square matrix [A] with real number entries,A+AT is symmetric matrixA–AT is a skew symmetric matrix
Permuntation and Combination
nCr=n!(n-r)!r!
nCr=nCn-r
nPr=r!nCr
nCr+nCr-1=n+1Cr
nC1+nC2+nC3+...+nCn=2n
nC0+nC2+..=nC1+nC3...=2n-1
nPr=n!(n-r)!
nPn=n!
nPr=nΓ—n-1Pr-1
calculus Calculus Continuity
Suppose f is a real function on a subset of real numbers then c be a point in the domain of "f", then "f" is continupos at "c" if : Limx→cf(x)=f(c)
If "f" and "g" are two real functions continuous at a real number "c", then (a) f+giscontinuousatx=c(b) f-giscontinuousatx=c(c) f.giscontinuousatx=c(d) (fg)iscontinuousatx=c
Limits
Limx→0sinxx=1
Limx→0tanxx=1
Limx→axn-anx-a=naxn-1
Limx→0ex-1x=1
Limx→0(1+x)1x=e
Limxβ†’βˆž(1+ax)x=ea
Limx→01-cosxx=0
Limx→0sin-1xx=1
Limx→0ax-1x=lna=logea
Limx→0ln(1+x)x=1
Limxβ†’βˆž(1+1x)x=e
if Limx→af(x)g(x)givesform00 as f(a)=0andg(a)=0 then Limx→af(x)g(x)=f'(a)g'(a)
Differentiation
ddx(uv)=ududx+vdvdx
ddx(xn)=nxn-1
ddx(sinx)=cosx
ddx(cosx)=-sinx
ddx(tanx)=sec2x
ddx(cosecx)=-cosecx.cotx
ddx(secx)=secx.tanx
ddx(cotx)=-cosec2x
ddx(lnx)=1x
ddx(ex)=ex
ddx(ax)=axlna
ddx(sin-1x)=11-x2
ddx(logax)=1xlna
ddx(cos-1x)=-11-x2
ddx(tan-1x)=11+x2
ddx(cot-1x)=-11+x2
ddx(sec-1x)=1xx2-1
ddx(cosec-1x)=-1xx2-1
Integration
∫xndx=xn+1n+1+C
∫cosxdx=sinx+C
∫sinxdx=-cosx+C
∫sec2xdx=tanx+C
∫cosecx.cotxdx=-cosecx+C
∫secx.tanxdx=secx+C
∫cosec2xdx=-cotx+C
∫dxx=lnx+C
∫exdx=ex+C
∫axdx=axlna+C
∫dx1-x2=sin-1x+C
∫dxx1-x2=cosec-1x+C
∫dxx1-x2=-cosec-1x+C
∫dxx2-a2=12alog|x-ax+a|+C
∫dxa2-x2=12alog|a+xa-x|+C
∫dxx2+a2=1atan-1xa+C
∫dxx2-a2=log|x+x2-a2|+C
∫dxa2-x2=sin-1xa+C
∫dxx2+a2=log|x+x2+a2|+C
∫secxdx=log|secx+tanx|+C
∫cosecxdx=log|cosecx-cotx|+C
∫cotxdx=log|sinx|+C
∫tanxdx=log|secx|+C
∫x2+a2dx=12xx2+a2+a22log|x+x2+a2|+C
∫x2-a2dx=12xx2-a2-a22log|x+x2-a2|+C
∫a2-x2dx=12xa2-x2+a22sin-1xa+C
∫f(x)g(x)=f(x)∫g(x)dx-∫[f'(x)∫g(x)dx]dx
∫ex[f(x)+f'(x)]dx=exf(x)+C
trigonometry Trigonometry
πœ‹radian=180Β°
sin(-A)=-sinA
cosec(-A)=-cosecA
cos(-A)=cosA
sec(-A)=secA
tan(-A)=-tanA
cot(-A)=-cotA
sin(πœ‹2-A)=cosA
cos(πœ‹2-A)=sinA
tan(πœ‹2-A)=cotA
cosec(πœ‹2-A)=secA
sec(πœ‹2-A)=cosecA
cot(πœ‹2-A)=tanA
sin(πœ‹2+A)=cosA
cos(πœ‹2+A)=-sinA
tan(πœ‹2+A)=-cotA
cosec(πœ‹2+A)=secA
sec(πœ‹2+A)=-cosecA
cot(πœ‹2+A)=-tanA
sin(πœ‹-A)=sinA
cos(πœ‹-A)=-cosA
tan(πœ‹-A)=-tanA
cosec(πœ‹-A)=cosecA
sec(πœ‹-A)=-secA
cot(πœ‹-A)=-cotA
sin(πœ‹+A)=-sinA
cos(πœ‹+A)=-cosA
tan(πœ‹+A)=tanA
cosec(πœ‹+A)=-cosecA
sec(πœ‹+A)=-secA
cot(πœ‹+A)=cotA
sin(A+B)=sinAcosB+cosAsinB
sin(A-B)=sinAcosB-cosAsinB
cos(A+B)=cosAcosB-sinAsinB
cos(A-B)=cosAcosB+sinAsinB
tan(A+B)=tanA+tanB1-tanA.tanB
tan(A-B)=tanA-tanB1+tanA.tanB
sinA+sinB=2sinA+B2cosA-B2
sinA-sinB=2cosA+B2sinA-B2
cosA+cosB=2cosA+B2cosA-B2
cosA-cosB=-2sinA+B2sinA-B2
sin2A=2sinAcosA
cos2A=cos2A-sin2A
cos2A=2cos2A-1
cos2A=1-2sin2A
tan2A=2tanA1-tan2A
sin3A=3sinA-4sin3A
cos3A=4cos3A-3cosA
tan3A=3tanA-tan3A1-3tan2A
Principal Solution: the solution of trigonometric equation with unknown angle 'x', where0β©½xβ©½2πœ‹ , are called principal solution.
General Solution: the solution of trigonometric equation which are generalized by using periodicity are called general solution.
if sinπœƒ=0 , then πœƒ=nπœ‹
if cosπœƒ=0, then πœƒ=(2n+1)πœ‹2
if sinπœƒ=sin𝛼, then , πœƒ=nπœ‹+(-1)n𝛼
if cosπœƒ=cos𝛼 , then , πœƒ=2nπœ‹Β±π›Ό
if tanπœƒ=tan𝛼 , then , πœƒ=nπœ‹+𝛼
Trigonometry (Solution of triangle)
Law of cosinesa2=b2+c2-2bccosAb2=c2+a2-2cacosBc2=a2+b2-2abcosC
Law of sinesasinA=bsinB=csinC
Projection Rulea=bcosC+ccosBb=ccosA+acosCc=acosB+bcosA
Mollweide's formulaa+bc=cosA-B2sinC2
Napier Analogy (tangent Rule)tanB-C2=b-cb+ccotA2tanC-A2=c-ac+acotB2tanA-B2=a-ba+bcotC2