a^2-b^2=(a+b)(a-b)
a^3+b^3=(a+b)(a^2+ab+b^2)
a^3-b^3=(a-b)(a^2+ab+b^2)
Triangle inequality
|a+b|≤|a|+|b|
|a-b|≤|a|+|b|
| a |≤b & lt； = & gt-b≤a≤b
|a-b|≥|a|-|b|
-|a|≤a≤|a|
The solution of the unary quadratic equation -b+√ (b 2-4ac)/2a-b-√ (b 2-4ac)/2a
The relationship between root and coefficient x1+x2 =-b/ax1* x2 = c/a Note: Vieta theorem.
discriminant
δ = b 2-4ac = 0 Note: The equation has two equal real roots.
δ=b^2-4ac>； 0 Note: The equation has two unequal real roots.
δ=b^2-4ac<； Note: The equation has no real root, but a complex number of the yoke.
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)/Sina sinb-ctgA+ctgBsin(A+B)/Sina sinb
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)=n^2
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 theorem a/sinA=b/sinB=c/sinC=2R (note: where r represents the radius of the circumscribed circle of a triangle).
Cosine Theorem B 2 = A 2+C 2-2 ACCOSB (Note: Angle B is the angle between side A and side C)
The standard equation of a circle is (x-a)2+(y-b)2=r2 (note: (a, b) is the center coordinate).
The general equation x2+y2+Dx+Ey+F=0 (note: D2+E2-4f >; 0)
Parabolic standard equation y 2 = 2px y 2 =-2px x 2 = 2py x 2 =-2py.
Side area of right-angle prism S=c*h
Side area of oblique prism S=c'*h
The side area of a regular pyramid is S= 1/2c*h'
The side area of the prism is S = 1/2(c+c')h'
The lateral area of the frustum of a cone is s =1/2 (c+c') l = pi (r+r) l.
The surface area of the ball S=4pi*r2.
The lateral area of the cylinder is s = c * h = 2π * h
The lateral area of the cone is s =1/2 * c * l = π * r * l.
The arc length formula l=a*r a is the radian number r > of the central angle; 0
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 straight cross-sectional area and l is the side length.
Cylinder volume formula V=s*h
Cylinder v = π * r 2h