Operation of sets:

Set commutative law

A∩B=B∩A

A∪B=B∪A

Set associative law

(A∩B)∩C=A∩(B∩C)

(A∪B)∪C=A∪(B∪C)

Set distribution law

A∩(B∪C)=(A∪B)∩(A∪C)

A ∪( B∪C)=(A∪B)∪( A∪C)

Seth de Morgan's law

Cu(A∩B)=CuA∪CuB

Cu(A∪B)= CuA∪CuB

Set "the principle of inclusion and exclusion"

When we study a set, we will encounter problems about the number of elements in the set. We write the number of elements in finite set A as card(A). For example, A={a, b, c}, then card (A)=3.

Card (A∪B)= Card (A)+ Card (B)- Card (A∪B)

Card (A∪B∪C)= Card (A)+ Card (B)+ Card (C)- Card (A∪B)- Card (C∪A)+ Card (A ∪.

formulas of trigonometric functions

Two-angle sum formula

sin(A+B)= Sina cosb+cosa sinb sin(A-B)= Sina cosb-sinb cosa

cos(A+B)= cosa cosb-Sina sinb cos(A-B)= cosa cosb+Sina sinb

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 = 2 tana/( 1-tan2A)ctg2A =(ctg2A- 1)/2c TGA

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

2 Sina cosb = sin(A+B)+sin(A-B)2 cosa sinb = sin(A+B)-sin(A-B)

2 cosa cosb = cos(A+B)-sin(A-B)-2 sinasinb = 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)/cosa cosb tanA-tanB = sin(A-B)/cosa cosb

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)= 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 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

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 < = > -b≤a≤b

|a-b|≥|a|-|b| -|a|≤a≤|a|

The solution of the unary quadratic equation -b+√(b2-4ac)/2a -b-√(b2-4ac)/2a

The relationship between root and coefficient x1+x2 =-b/ax1* x2 = c/a Note: Vieta theorem.

discriminant

B2-4ac=0 Note: This equation has two equal real roots.

B2-4ac >0 Note: The equation has two unequal real roots.

B2-4ac & lt； Note: The equation has no real root, but a complex number of the yoke.

Reduced power formula

(sin^2)x= 1-cos2x/2

(cos^2)x=i=cos2x/2

General formula of trigonometric function

Let tan(a/2)=t

sina=2t/( 1+t^2)

cosa=( 1-t^2)/( 1+t^2)

tana=2t/( 1-t^2)

When a>0 and a≠ 1, m >；; 0, N>0, then:

( 1)log(a)(MN)= log(a)(M)+log(a)(N)；

(2)log(a)(M/N)= log(a)(M)-log(a)(N)；

(3)log(a)(M^n)=nlog(a)(M)

(4) the formula of bottoming: log (a) m = log (b) m/log (b) a (b >); 0 and b≠ 1)

When a>0 and a≠ 1, a x = n x = ㏒ (a) n

Common abbreviations for logarithmic functions:

( 1)log(a)(b)=log(a)(b)

(2) common logarithm: lg(b)=log( 10)(b)

(3) natural logarithm: ln(b)=log(e)(b)