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Commonly used formulas in senior high school mathematics
1 trigonometric inequality
| a+b |≤| a |+| b | | a-b |≤| a |+| b | | a |≤b & lt； = & gt-b≤a≤b
|a-b|≥|a|-|b|-|a|≤a≤|a|
Solutions of quadratic equation in one variable -b+√ (B2-4ac)/2a, -b-√ (B2-4ac)/2a
The relationship between root and coefficient x1+x2 =-b/ax1x2 = c/a Note: Vieta 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 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
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))
Sum-difference product of 4
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)/cosAcosBtanA-tanB = sin(A-B)/cosAcosB
ctgA+ctgBsin(A+B)/Sina sinb-ctgA+ctgBsin(A+B)/Sina sinb
The sum of the first n terms in some sequences is1+2+3+4+5+6+7+8+9+…+n = n (n+1)/21+3+5+7+9+/kloc-0.
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 = 2 pxy 2 =-2 pxy 2 = 2 pxy 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 = 2pi h.
The lateral area of the cone is s = 1/2 c l = pi r l.
The arc length formula l = a ra 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 = pi r2h
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 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 x 2+y 2+dx+ey+f = 0 note: d 2+e 2-4f > 0.
Parabolic standard equation y 2 = 2pxy 2 =-2pxx 2 = 2pyx 2 =-2py.
Side area of straight prism S = C H side area of oblique prism s = c 'h.
The side area of a regular pyramid s = 1/2c h' The side area of a regular prism S= 1/2(c+c')h'
The lateral area of the frustum S = 1/2(c+c')l = pi(R+R)l The surface area of the ball s = 4pi R2.
Cylindrical lateral area s = c h = 2pi h conic lateral area s = 1/2 c l = pi r l.
The arc length formula l = a ra 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
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
Double angle formula
tan2A=2tanA/[ 1-(tanA)^2]
cos2a=(cosa)^2-(sina)^2=2(cosa)^2- 1= 1-2(sina)^2
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))
cot(A/2)=√(( 1+cosA)/(( 1-cosA))cot(A/2)=-√(( 1+cosA)/(( 1-cosA))
Sum-difference product of 5
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
The sum of the first n terms of some series
1+2+3+4+5+6+7+8+9+…+n = n(n+ 1)/2
2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1)5
1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2+…+n^2=n(n+ 1)(2n+ 1)/6
1^3+2^3+3^3+4^3+5^3+6^3+…n^3=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
Seven Common Derivative Formulas
1, y=c(c is a constant) y'=0.
2、y=x^ny'=nx^(n- 1)
3、y=a^xy'=a^xlna
4、y=e^xy'=e^x
5、y=logaxy'=logae/x
6、y=lnxy'= 1/x
7、y=sinxy'=cosx
8、y=cosxy'=-sinx
9、y=tanxy'= 1/cos^2x
10、y=cotxy'=- 1/sin^2x
1 1、y=arcsinxy'= 1/√ 1-x^2
12、y=arccosxy'=- 1/√ 1-x^2
13、y=arctanxy'= 1/ 1+x^2
14、y=arccotxy'=- 1/ 1+x^2
How to learn high school mathematics well
First, basic knowledge points are the beginning of solving mathematical problems.
Remember all the definitions and formulas in the book. Students who don't want to recite can turn over the books and read them several times every day, which is also helpful to remember the knowledge points of mathematics.
Or find a small notebook to copy formulas or definitions for easy reading and carrying. It's okay. Just read and write more. This is really a trick to learn mathematics efficiently.
Second, pay attention to the steps to solve mathematical problems
The process of solving big math problems is graded according to the steps, and you must not fool around at will. This time, the students may not care, but when the results come out, they realize that it is too late to regret anything.
So you must write down the standardized answers that the teacher told you in class, so that you can use them as standard steps when you solve problems in the future, so that you can try not to lose points. This is an efficient learning method, and you can score more points without losing points.
Third, the collection and binding of classic questions
In general, the last two big questions in mathematics account for half of one side of the roll paper, so after each lecture, the teacher will write the standard answers in the blank, and if possible, cut off this half page.
The same is true for future papers, which are bound together, like a wrong book or exercise book, and can be used for review if nothing happens. On the back of the paper, you can copy some similar big deformation questions or classic questions that have appeared in the college entrance examination over the years.
You can also do it again by yourself to strengthen your understanding of this math problem. Long-term accumulation will significantly improve your math scores.
The Solution of Math Skills in Senior High School
1, applicable condition: [straight line passing through the focus], which must have ecosA=(x- 1)/(x+ 1), where a is the included angle between the straight line and the focus axis, which is an acute angle. X is the separation ratio and must be greater than 1. Note: The above formula is applicable to all conic curves. If the focus is internally divided (meaning that the focus is on the cutting line segment), use this formula; If it is divided (focusing on the extension line of the section), the right side is (x+ 1)/(x- 1), and the rest remains unchanged.
2. the periodicity of the function (note 3): 1, if f(x)=-f(x+k), then T = 2k.
3. If f(x)=m/(x+k)(m is not 0), then T = 2k3. If f(x)=f(x+k)+f(x-k), then T=6k. Note: A. Periodic function, the period must be infinite B. Periodic function can have no minimum period, such as constant function. C. periodic function plus periodic function is not necessarily a periodic function, for example, y=sinxy=sin pie x is not a periodic function.
4. The problem of symmetry (a problem that countless people can't understand) can be summarized as follows: 1, if it is satisfied on r (the same below): f(a+x)=f(b-x) is a constant, and the symmetry axis is x = (a+b)/2; 2. The images of functions y=f(a+x) and y=f(b-x) are symmetric about x=(b-a)/2; 3. If f(a+x)+f(a-x)=2b, the image of f(x) is symmetrical about the center of (a, b).
5, the parity of the function is 1, and for odd function belonging to R, there is f (0) = 0; 2. For parametric functions, odd function has no even power term, and even functions have no odd power term. 3. Parity has little influence and is generally used to fill in the blanks.
6. The law of explosive power of series: 1, in arithmetic progression: s odd =na medium, for example, s13 =13a7 (13 and 7 are lower angles); 2 In arithmetic progression, S(n), S(2n)-S(n) and S(3n)-S(2n) are equal. In geometric series, when the common ratio is not negative, the above two terms are equal. When q=- 1, 4, the explosion intensity formula of geometric series is not necessarily true: s (n+m).
7, the ultimate weapon of the sequence, the characteristic root equation. Forget it if you don't understand. Firstly, the formula is introduced: for an+ 1=pan+q(n+ 1 is the lower corner, and n is the lower corner), if a 1 is known, then the characteristic root x=q/( 1-p), then the general term formula of the series is an = (A6550). The second order is a bit troublesome and not commonly used. So I won't go into details. I hope the students will remember the above formula. Of course, this type of sequence can be constructed (both sides are added at the same time)
Tips for improving mathematics in senior three.
First, master the theorem of mathematical formula.
Basically, it is to do examples in textbooks. The ideas of the examples in the textbook are relatively simple. After reading these examples, you can do an example corresponding to the knowledge points yourself. The process of doing problems is the best process of memorizing mathematical formulas and theorems. This step cannot be saved. Don't try to memorize mathematical formulas and theorems. Only by remembering while using can we really understand and use it.
The examples in the textbook are finished, so do the exercises after class. Some of the exercises after class are comprehensive questions. Combining the new knowledge points with the previously learned knowledge points is helpful for further study and consolidation.
Second, carry out training on special topics and difficult problems.
Don't be afraid of difficult problems when doing problems. Some students put it down when they see it, and keep practicing the questions they can do, which is difficult to improve. You can try to do some more difficult problems, don't be afraid. If you don't solve the problem all the time, your exam results will definitely fail.
First of all, when you see a difficult problem, you should do it boldly, be active in thinking and think more about knowledge points. This method won't work. It doesn't matter. Analyze it again, review the questions and find other ways. If you can't do it all the time, you can refer to the answer, see how the answer is answered, what is the idea of solving the problem, and whether the solution is impossible or unexpected, and then summarize and reflect on yourself.