(1) multiplication and factorization
a2-B2 =(a+b)(a-b)； a3+B3 =(a+b)(a2-a b+B2)； a3-b3=(a-b)(a2+ab+b2).
(2) Trigonometric inequality
| a+b |≤| a |+| b |； | a-b |≤| a |+| b |； | a |≤b-b≤a≤b； |a-b|≥|a|-|b|-|a|≤a≤|a| .
(3) The solution of the unary quadratic equation:-b+√ (B2-4ac)/2a-b-b+√ (B2-4ac)/2a.
(4) Relationship between root and coefficient: x1+x2 =-b/ax1* x2 = c/a, note: Vieta's theorem.
(5) discriminant formula
1)b2-4a=0。 Note: The equation has two equal real roots.
2)B2-4ac & gt； 0, note: the equation has real roots.
3)b2-4ac
2. formulas of trigonometric functions
The formula of (1) sum of two angles
sin(A+B)= Sina cosb+cosa sinb； sin(A-B)= Sina cosb-sinBcosA； 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).
(2) Double angle formula
tan2A = 2 tana/( 1-tan2A)； ctg2A =(ctg2A- 1)/2c TGA； cos2a = cos2a-sin2a = 2 cos2a- 1 = 1-2 sin2a .
(3) 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))。
(4) Sum-difference product formula
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 sinas inb = 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)/cosa cosb； tanA-tanB = sin(A-B)/cosa cosb； ctgA+ctgBsin(A+B)/Sina sinb； -ctgA+ctgBsin(A+B)/sinAsinB
(5) the first n terms and formulas 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 .
(6) Sine theorem: a/sinA=b/sinB=c/sinC=2R, note: where r represents the radius of the circumscribed circle of a triangle.
(7) Cosine theorem: b2=a2+c2-2accosB. Note: Angle B is the included angle between side A and side C. ..
3. Memorize the knowledge points of liberal arts mathematics in senior high school.
(1) collection
The concept of 1) set is not defined, but the properties are the same. There is a sub-intersection and a complementary set, and the operation result is a set.
2) Three characteristics of set elements: difference and disorder; The elements of the set are the same, and the two sets are equal.
3) Symbolization of writing norms, namely enumeration and description; In the description, the curly braces and the object xy must be clearly seen.
4) Pay attention to the point set of the number set, which is a pair of real numbers; Sets of elements belong to each other, and sets talk about inclusion.
5)0 and the empty set are different, and the correct distinction is successful; If there is any difficulty in operation, Wayne's number axis will help.
(2) Common logical terms
1) True or false is a proposition, and the conditional conclusion is clear; Proposition has four forms, divided into two pairs, the same truth and the same false.
2) If P is a true proposition of Q, sufficient conditions for P and Q; Q is a necessary condition for P, and both the original and inverse are true and necessary.
3) There are three ways to judge conditions, citing the definition of counterexample; ; Set method from small to large, inverse proposition equivalence method.
4) Logical conjunctions are either not or true when the proposition is true; The proposition is false, and the non-proposition is opposite.
5) Negative type of proposition, negative or proposition; Negative form, negation and proposition of or proposition.
6) Generally, there are two quantifiers, both of which are quantifiers; There is one existential quantifier, whose full name is two propositions.
6) full name proposition negation, especially proposition affirmation; It contains the negative form of quantifiers and rewrites the negative conclusion of quantifiers.
(3) the concept of function
1) function structure, and the domain is defined by the law of range; There are three forms of functions, list image analysis.
2) There are three kinds of special functions, subsection combination and compound; There are many requirements for the domain, and the denominator of the score is not 0.
3) Even roots must be non-negative and the power of 0 must be positive; If the cardinality is not 1, it is a positive number, and zero and negative numbers have no logarithm.
4) The tangent function is not straight, and the serial number is a positive integer; Must meet the practical significance of the intersection of multiple functions.
5) Solving the function value domain and defining the formula image; Part of the whole observation method became monotonous in Yuan Dynasty.
6) the inequality method of separating the discriminant formula of constant from the mean value theorem; How to find the analytical formula, the topic is often androgynous.
7) Abstract resolution function, which is substituted into substitution collocation method and equation thought elimination method; Specify the type of analysis formula,
8) Use the undetermined coefficient method. The nature of parity is monotonous, and the observation image is the most beautiful; To prove this point in detail,
We must master this definition. Combinatorial functions are monotonous, judging that they are regular, and increasing is equal to increasing,
10) increase or decrease equals increase, decrease equals decrease, and decrease equals decrease. Monotonicity of composite function,
1 1) the same increase but different decrease. The parity of compound function, even plus or minus even equals even, odd plus or minus odd equals odd.
12) Even addition and subtraction is odd and odd, even multiplication and division is even, odd multiplication and division is odd, and odd multiplication and division is odd.
13) periodic symmetry, and the observation structure is the most feasible; Automorphism means periodicity, while introversion means symmetry.
14) center symmetry, function period; Function zero equation root, abscissa of image intersection point;
15) There are several zeros in the function. Draw a picture to see the intersection. The two endpoints are replaced and multiplied by zero.
4. Summarize the necessary knowledge points of liberal arts mathematics.
(1) Set related concepts
Three characteristics of elements in 1) set:
2) element determinism: mutual difference and disorder.
3) Representation methods of sets: enumeration and description.
4) Note: Common number sets and their notation: non-negative integer sets (that is, natural number set) are recorded as: n positive integer sets, N* or N+ integer sets Z rational number sets Q real number sets R.
(2) the basic relationship between sets
1) "inclusion" relation-subset, note: BA has two possibilities. A is a part of b; A and b are the same set. On the other hand, set A is not contained in set B, or set B does not contain set A. ..
2) A set without any elements is called an empty set and denoted as φ; It is stipulated that an empty set is a subset of any set and an empty set is a proper subset of any non-empty set. A set of n elements, including 2n subsets and 2n- 1 proper subset.