1. Senior two mathematics volume two compulsory three knowledge points induction
The trigonometric function of acute angle defines sine (sin), cosine (cos), tangent (tan), cotangent (cot) and secant (sec) of acute angle A, and cotangent (csc) is called the trigonometric function of acute angle A..
Sine is equal to the hypotenuse of the opposite side; Sina = account
Cosine (cos) is equal to the ratio of adjacent side to hypotenuse; cosA=b/c
Tangent (tan) is equal to the opposite side of the adjacent side; tanA=a/b
Cotangent is equal to the comparison of adjacent edges; cotA=b/a
Secant is equal to the hypotenuse than the adjacent edge; secA=c/b
Cotangent (csc) is equal to the ratio of hypotenuse to edge. cscA=c/a
The relationship between trigonometric functions with complementary angles
sin(90 -α)=cosα，cos(90 -α)=sinα，
tan(90 -α)=cotα，cot(90 -α)=tanα。
Square relation:
sin^2(α)+cos^2(α)= 1
tan^2(α)+ 1=sec^2(α)
cot^2(α)+ 1=csc^2(α)
Product relationship:
sinα=tanα cosα
cosα=cotα sinα
tanα=sinα secα
cotα=cosα cscα
secα=tanα cscα
cscα=secα cotα
Reciprocal relationship:
tanα cotα= 1
sinα cscα= 1
cosα secα= 1
Acute angle formula of trigonometric function
Trigonometric function of sum and difference of two angles;
sin(A+B)=sinAcosB+cosAsinB
sin(A-B)=sinAcosB-cosAsinB？
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)
cot(A+B)=(cotA cotB- 1)/(cot B+cotA)
cot(A-B)=(cotA cotB+ 1)/(cot b-cotA)
Trigonometric function of trigonometric sum:
sin(α+β+γ)= sinαcosβcosγ+cosαsinβcosγ+cosαcosβsinγ-sinαsinβsinγ
cos(α+β+γ)= cosαcosβcosγ-cosαsinβsinγ-sinαcosβsinγ-sinαsinαsinβcosγ-sinαsinβcosγ
tan(α+β+γ)=(tanα+tanβ+tanγ-tanαtanβtanγ)/( 1-tanαtanβ-tanβtanγ-tanγtanα)
Auxiliary angle formula:
Asinα+bcosα = (A2+B2) (1/2) sin (α+t), where
sint=B/(A^2+B^2)^( 1/2)
cost=A/(A^2+B^2)^( 1/2)
tant=B/A
asinα+bcosα=(a^2+b^2)^( 1/2)cos(α-t),tant=a/b
Double angle formula:
sin(2α)=2sinα cosα=2/(tanα+cotα)
cos(2α)=cos^2(α)-sin^2(α)=2cos^2(α)- 1= 1-2sin^2(α)
tan(2α)=2tanα/[ 1-tan^2(α)]
Triple angle formula:
sin(3α)=3sinα-4sin^3(α)
cos(3α)=4cos^3(α)-3cosα
Half-angle formula:
sin(α/2)= √(( 1-cosα)/2)
cos(α/2)= √(( 1+cosα)/2)
tan(α/2)=√(( 1-cosα)/( 1+cosα))= sinα/( 1+cosα)=( 1-cosα)/sinα
Reduced power formula
sin^2(α)=( 1-cos(2α))/2=versin(2α)/2
cos^2(α)=( 1+cos(2α))/2=covers(2α)/2
tan^2(α)=( 1-cos(2α))/( 1+cos(2α))
General formula:
sinα=2tan(α/2)/[ 1+tan^2(α/2)]
cosα=[ 1-tan^2(α/2)]/[ 1+tan^2(α/2)]
tanα=2tan(α/2)/[ 1-tan^2(α/2)]
Product sum and difference formula:
sinαcosβ=( 1/2)[sin(α+β)+sin(α-β)]
cosαsinβ=( 1/2)[sin(α+β)-sin(α-β)]
cosαcosβ=( 1/2)[cos(α+β)+cos(α-β)]
sinαsinβ=-( 1/2)[cos(α+β)-cos(α-β)]
Sum-difference product formula:
sinα+sinβ= 2 sin[(α+β)/2]cos[(α-β)/2]
sinα-sinβ= 2cos[(α+β)/2]sin[(α-β)/2]
cosα+cosβ= 2cos[(α+β)/2]cos[(α-β)/2]
cosα-cosβ=-2 sin[(α+β)/2]sin[(α-β)/2]
Derived formula:
tanα+cotα=2/sin2α
tanα-cotα=-2cot2α
1+cos2α=2cos^2α
1-cos2α=2sin^2α
1+sinα=(sinα/2+cosα/2)^2
2. The second volume of high school mathematics requires summarizing three knowledge points.
The parity of 1. function (1) If f(x) is an even function, then f (x) = f (-x);
(2) If f(x) is odd function and 0 is in its domain, then f(0)=0 (which can be used to find parameters);
(3) The parity of the judgment function can be defined in equivalent form: f (x) f (-x) = 0 or (f (x) ≠ 0);
(4) If the analytic formula of a given function is complex, it should be simplified first, and then its parity should be judged;
(5) odd function has the same monotonicity in the symmetric monotone interval; Even functions have opposite monotonicity in symmetric monotone interval;
2. Some questions about compound function.
Solution of the domain of (1) composite function: If the domain is known as [a, b], the domain of the composite function f[g(x)] can be solved by the inequality a≤g(x)≤b; If the domain of f[g(x)] is known as [a, b], find the domain of f(x), which is equivalent to x∈[a, b], and find the domain of g(x) (that is, the domain of f(x)); When learning functions, we must pay attention to the principle of domain priority.
(2) The monotonicity of the composite function is determined by "the same increase but different decrease";
3. Function image (or symmetry of equation curve)
(1) Prove the symmetry of the function image, that is, prove that the symmetry point of any point on the image about the symmetry center (symmetry axis) is still on the image;
(2) Prove the symmetry of the image C 1 and C2, that is, prove that the symmetry point of any point on C 1 about the symmetry center (symmetry axis) is still on C2, and vice versa;
(3) curve C 1: f (x, y) = 0, and the equation of symmetry curve C2 about y=x+a(y=-x+a) is f(y-a, x+a)=0 (or f(-y+a,-x+a) =
(4) Curve C 1:f(x, y)=0 The C2 equation of the symmetrical curve about point (a, b) is: f(2a-x, 2b-y) = 0;
(5) If the function y=f(x) is constant to x∈R, and f(a+x)=f(a-x), then the image y=f(x) is symmetrical about the straight line x=a;
(6) The images of functions y=f(x-a) and y=f(b-x) are symmetrical about the straight line x=;
4. The periodicity of the function
(1)y=f(x) for x∈R, f(x+a)=f(x-a) or f (x-2a) = f (x) (a >: 0) is a constant, then y=f(x) is a period of 2a.
(2) If y=f(x) is an even function, and its image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 2 ~ a;
(3) If y=f(x) odd function, whose image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 4 ~ a;
(4) If y=f(x) is symmetric about points (a, 0) and (b, 0), then f(x) is a periodic function with a period of 2;
(5) If the image of y=f(x) is symmetrical (a ≠ b) about straight lines x = a and x = b, then the function y = f (x) is a periodic function with a period of 2;
(6) When y=f(x) equals x∈R, f(x+a)=-f(x) (or f(x+a)=, then y = f (x) is a periodic function with a period of 2;
5. Equation k=f(x) has a solution k∈D(D is the range of f(x));
3. Senior two mathematics volume two compulsory three knowledge points induction
1. Commutation division is a method to find the common divisor. This algorithm was first proposed by Euclid around 500 BC, so it is also called Euclid algorithm. 2. The so-called exchange method is to divide a larger number by a smaller number for a given two numbers. If the remainder is not zero, the smaller number and the remainder form a new pair of numbers, and the above division is continued until the larger number is divided by a decimal. At this time,
3. Multiphase subtraction is a method to find the common divisor of two numbers. Its basic process is: for a given two numbers, subtract the smaller number from the larger number, then compare the difference with the smaller number, subtract the number from the larger number, and continue this operation until the obtained numbers are equal, then this number is the common divisor.
4. Qin algorithm is a method to calculate the value of univariate quadratic polynomial.
5. The commonly used sorting methods are direct insertion sorting and bubble sorting.
6. The carry system is an agreed counting system for the convenience of counting and operation. "All in one" is a K-base system, and the base of the base system is K.
7. The method of converting decimal number into decimal number is: first, write decimal number as the sum of the product of the number on each bit and the power of k, and then calculate the result according to the operation rules of decimal number.
8. The method of converting decimal number into decimal number is: divide by k, and the remainder. That is to say, k is used to continuously divide the decimal number or quotient until the quotient is zero, and then the remainder obtained each time is arranged as an inverse number, which is the corresponding decimal number.
4. The second volume of high school mathematics requires summarizing three knowledge points.
Population and sample ① In statistics, the whole research object is called population.
② Call each research object an individual.
③ The total number of individuals in the population is called the total capacity.
④ In order to study the related properties of the population, we generally randomly select a part from the population: x 1, x2, ..., _ research, which we call samples. The number of individuals is called sample size.
simplerandom sampling
Also known as pure random sampling. That is, on the whole, there is no grouping, classification, queuing, etc. , completely follow.
Machine-based measurement unit extraction. The characteristics are: the probability of each sample unit being extracted is the same (the probability is equal), and each unit of the sample is completely independent, and there is no certain correlation and exclusion between them. Simple random sampling is the basis of other sampling forms. This method is usually only used when the difference between the whole units is small and the number is small.
Common methods of simple random sampling
(1) draw lots
② Random number table method
③ Computer simulation method
④ Direct extraction with statistical software.
In the sample size design of simple random sampling, we mainly consider:
① General variation;
② Allowable error range;
③ Degree of probability assurance.
draw lots
(1) Number each object in the measurement group;
(2) Prepare the lottery tool and implement it;
③ Measure or investigate each individual in the sample.
5. The second volume of high school mathematics requires summarizing three knowledge points.
Sum formula of equal ratio sequence
(1) geometric series: a(n+ 1)/an=q(n∈N).
(2) General formula: an = a1× q (n-1); Generalization: an = am× q (n-m);
(3) summation formula: sn = n× a1(q =1) sn = a1(1-q n)/(1-q) = (a1)
(4) nature:
(1) if m, n, p, q∈N, m+n=p+q, then am×an = AP×AQ；;
(2) In geometric series, every k term is added in turn and still becomes a geometric series.
③ If m, n, q∈N and m+n=2q, then am× an = AQ 2.
(5) "G is the equal ratio mean of A and B" and "G 2 = AB (G ≠ 0)".
(6) In geometric series, the first term a 1 and the common ratio q are not zero. Note: an in the above formula represents the nth term of geometric series.
Derivation of summation formula of proportional series: Sn=a 1+a2+a3+...+an (common ratio q) q * sn = a1* q+a2 * q+a3 * q+...+an * q = a2+a3+a4+. sn=a 1-a 1*q^nsn=(a 1-a 1*q^n)/( 1-q)sn=(a 1-an*q)/( 1-q)sn=a 1( 1-q^n)/( 1-q)sn=k*( 1-q^n)~y=k*( 1-a^x)。