Multiplication and factorization:
a2-b2=(a+b)(a-b)
a3+b3=(a+b)(a2-ab+b2)
a3-b3=(a-b)(a2+ab+b2)
Trigonometric 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-b+√(B2-4ac)/2a
Relationship between root and coefficient
X1+x2 =-b/ax1* x2 = c/a note: Vieta's theorem.
Discriminant b2-4a=0 Note: The equation has two equal real roots.
B2-4ac >0 Note: The equation has real roots.
B2-4ac & lt； 0 Note: The equation has multiple yokes.
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 = 2 cos((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 theorem: a/sinA=b/sinB=c/sinC=2R Note: where r represents the radius of the circumscribed circle of a triangle.
Cosine theorem: b2=a2+c2-2accosB Note: Angle B is the included angle between side A and side C..
The standard equation of a circle: (x-a)2+(y-b)2=r2 Note: (a, b) is the center coordinate.
General equation of circle: x2+y2+Dx+Ey+F=0 Note: D2+E2-4f > 0
Parabolic standard equation: y2=2px
y2=-2px
x2=2py
x2=-2py
Side area of right-angle prism: S=c*h
Side area of oblique prism: S = c'* h
Side area of a regular pyramid: S= 1/2c*h'
Side area of prism: S = 1/2(c+c')h'
Area of frustum side: s =1/2 (c+c') l = pi (r+r) l.
Surface area of the ball: S=4pi*r2.
Area of cylinder side: s = c * h = 2pi * h
The lateral area of the cone: s =1/2 * c * l = pi * r * l.
Arc length formula: l=a*r, where a is the radian number of the central angle r >; 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=pi*r2h