a2-b2=(a+b)(a-b)
a3+b3=(a+b)(a2-ab+b2)
a3-b3=(a-b(a2+ab+b2)
Triangle inequality
|a+b|≤|a|+|b|
|a-b|≤|a|+|b|
| a |≤b & lt; = & gt-b≤a≤b
|a-b|≥|a|-|b|
-|a|≤a≤|a|
Solution of quadratic equation in one variable
-b+√(b2-4ac)/2a
-b-√(b2-4ac)/2a
Relationship between root and coefficient
X 1+X2=-b/a
X 1*X2=c/a
Note: Vieta theorem.
discriminant
b2-4ac=0
Note: The equation has two equal real roots.
b2-4ac >0
Note: The equation has two unequal real roots.
B2-4ac & lt; 0
Note: The equation has no real root, but has a plurality of yokes.
formulas of trigonometric functions
Two-angle sum formula
sin(A+B)=sinAcosB+cosAsinB
sin(A-B)=sinAcosB-sinBcosA
cos(A+B)=cosAcosB-sinAsinB
cos(A-B)=cosAcosB+sinAsinB
tan(A+B)=(tanA+tanB)/( 1-tanA tanB)
tan(A-B)=(tanA-tanB)/( 1+tanA tanB)
ctg(A+B)=(ctgActgB- 1)/(ctg B+ctgA)
ctg(A-B)=(ctgActgB+ 1)/(ctg b-ctgA)
Double angle formula
tan2A=2tanA/( 1-tan2A)
ctg2A=(ctg2A- 1)/2ctga
cos2a = cos2a-sin2a = 2 cos2a- 1 = 1-2 sin2a
half-angle formula
sin(A/2)=√(( 1-cosA)/2)
sin(A/2)=-√(( 1-cosA)/2)
cos(A/2)=√(( 1+cosA)/2)
cos(A/2)=-√(( 1+cosA)/2)
tan(A/2)=√(( 1-cosA)/(( 1+cosA))
tan(A/2)=-√(( 1-cosA)/(( 1+cosA))
ctg(A/2)=√(( 1+cosA)/(( 1-cosA))
ctg(A/2)=-√(( 1+cosA)/(( 1-cosA))
Sum difference product
2sinAcosB=sin(A+B)+sin(A-B)
2cosAsinB=sin(A+B)-sin(A-B)
2cosAcosB=cos(A+B)-sin(A-B)
-2sinAsinB=cos(A+B)-cos(A-B)
sinA+sinB = 2 sin((A+B)/2)cos((A-B)/2
cosA+cosB = 2cos((A+B)/2)sin((A-B)/2)
tanA+tanB=sin(A+B)/cosAcosB
tanA-tanB=sin(A-B)/cosAcosB
ctgA+ctgBsin(A+B)/sinAsinB
-ctgA+ctgBsin(A+B)/sinAsinB
The sum of the first n terms of some series
1+2+3+4+5+6+7+8+9+…+n = n(n+ 1)/2
1+3+5+7+9+ 1 1+ 13+ 15+…+(2n- 1)= N2
2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1)
12+22+32+42+52+62+72+82+…+N2 = n(n+ 1)(2n+ 1)/6
13+23+33+43+53+63+…n3 = N2(n+ 1)2/4
1 * 2+2 * 3+3 * 4+4 * 5+5 * 6+6 * 7+…+n(n+ 1)= n(n+ 1)(n+2)/3
sine law
a/sinA=b/sinB=c/sinC=2R
note:
In ...
rare
Represents the radius of the circumscribed circle of a triangle.
cosine theorem
B2 = a2+C2-2 acco b
Note: Angle B is the included angle between side A and side C..
the standard equation of the circle
(x-a)2+(y-b)2=r2
Note: (a, b) is the central coordinate.
Circular general equation
x2+y2+Dx+Ey+F=0
Note: D2+E2-4f > 0
Parabolic standard equation
y2=2px
y2=-2px
x2=2py
x2=-2py
Transverse area of right prism
S=c*h
Oblique prism side area
S=c'*h
Side area of regular pyramid
S= 1/2c*h '
Transverse area of regular prism
S= 1/2(c+c')h '
Yuantai lateral area
s = 1/2(c+c’)l = pi(R+R)l
Surface area of ball
S=4pi*r2
Cylindrical side area
S=c*h=2pi*h
Cone lateral area
S= 1/2*c*l=pi*r*l
Arc length formula
l=a*r
A is the radian number r of the central angle
& gt0
Sector area formula
s= 1/2*l*r
Cone volume formula
V= 1/3*S*H
Cone volume formula
V= 1/3*pi*r2h
Oblique prism volume
V=S'L
Note: where s' is the area of the straight line,
L is the length of the side.
Cylinder volume formula
V=s*h
cylinder
V=pi*r2h