Common Formula in Junior High School Mathematics Competition (Urgent)

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)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

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 = 2px y 2 =-2px x 2 = 2py x 2 =-2py.

Lateral area of a straight prism S=c*h lateral area of an oblique prism s = c' * h.

Lateral area of a regular pyramid S= 1/2c*h' lateral area of a regular prism S= 1/2(c+c')h'

The lateral area of the frustum of a cone S = 1/2(c+c')l = pi(R+R)l The surface area of the ball S=4pi*r2.

Lateral area of cylinder S=c*h=2pi*h lateral area of cone s =1/2 * c * l = pi * r * l.

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

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

Borrow the first floor below. Ha! !

1. inductive formula

sin(-a)=-sin(a)

cos(-a)=cos(a)

sin(π2-a)=cos(a)

cos(π2-a)=sin(a)

sin(π2+a)=cos(a)

cos(π2+a)=-sin(a)

sin(π-a)=sin(a)

cos(π-a)=-cos(a)

sin(π+a)=-sin(a)

cos(π+a)=-cos(a)

2. The trigonometric function of the sum and difference of two angles

sin(a+b)= sin(a)cos(b)+cos(α)sin(b)

cos(a+b)= cos(a)cos(b)-sin(a)sin(b)

sin(a-b)= sin(a)cos(b)-cos(a)sin(b)

cos(a-b)= cos(a)cos(b)+sin(a)sin(b)

tan(a+b)= tan(a)+tan(b) 1-tan(a)tan(b)

tan(a-b)= tan(a)-tan(b) 1+tan(a)tan(b)

3. Sum-difference product formula

sin(a)+sin(b)= 2s in(a+B2)cos(a-B2)

Crime (1)? sin(b)=2cos(a+b2)

cos(a)+cos(b)= 2cos(a+B2)cos(a-B2)

cos(a)-cos(b)=-2s in(a+B2)sin(a-B2)

4. Double angle formula

sin(2a)=2sin(a)cos(b)

cos(2a)= cos 2(a)-sin 2(a)= 2cos 2(a)- 1 = 1-2 sin 2(a)

5. Half-angle formula

sin2(a2)= 1-cos(a)2

cos2(a2)= 1+cos(a)2

tan(a2)= 1-cos(a)sin(a)= Sina 1+cos(a)

6. General formula

sin(a)=2tan(a2) 1+tan2(a2)

cos(a)= 1-tan 2(a2) 1+tan 2(a2)

tan(a)=2tan(a2) 1-tan2(a2)

7. Other formulas (derived)

Answer? Sin (a)+b? Cos(a)=a2+b2sin(a+c) where tan(c)=ba.

Answer? Sin (a)+b? Cos(a)=a2+b2cos(a-c) where tan(c)=ab.

1+sin(a)=(sin(a2)+cos(a2))2

1-sin(a)=(sin(a2)-cos(a2))2

Respondent: Muyun 2006- gatekeeper level 311-2416:13.

Common formulas of college entrance examination mathematics

1. De Morgan formula.

4. Three forms of secondary resolution function ① General formula; ② Vertex type; ③ Zero type.

Step 5: Set up

The world is increasing its functions;

The upper limit is a decreasing function.

Let the function be differentiable in a certain interval, and if it is, it is increasing function; If there is, it is the subtraction function.

6. Symmetry of function image: ① The function image is symmetrical about a straight line; ② The image of the function is symmetrical about the straight line.

7. Symmetry of two function images: ① Function and function images are symmetrical about a straight line (i.e. axis symmetry); ② Function and image of function are symmetrical about a straight line; ③ The image of function sum is symmetrical about the straight line y = x. 。

8. The power of the fractional index (,and).

(,and).

9.。

10. The formula for changing the base of logarithm. Inference.

1 1. (the sum of the first n items in the sequence is).

12. General formula of arithmetic progression;

Its first n terms and formulas.

13. General formula of geometric series;

Sum formula of sum of the first n terms.

14. The general formula of equal ratio difference series: is

;

The summation formula of the first n terms is.

15. Repay in installments (mortgage loan) in RMB yuan each time (loan yuan, paid off in installments, interest rate RMB yuan).

16. Basic relation of equiangular trigonometric function, =,.

17. Inductive formulas of sine and cosine

18. Formula of sum angle and difference angle

;

(sine square formula);

The quadrant of the auxiliary angle is determined by the quadrant of the point.

19. Double angle formula.

. .

20. The periodic formula function of trigonometric function, the sum function of x∈R, and the period of x∈R (a, ω is constant, and A≠0, ω > 0); The period of the function, (a, ω, constant, A≠0, ω > 0).

2 1. sine theorem.

22. Cosine theorem; ; .

23. area theorem (1) (representing the heights on the sides of a, b and c respectively).

(2) .

(3) .

24. The triangle interior angle sum theorem in △ABC, there are

25. The formula of the distance between two points on the plane

= (A，B)。

26. Let a=, b= and b 0 be parallel and perpendicular to the vector, then

a b b=λa。

a b(a 0) a？ b=0。

27. The formula for dividing the line segment is set to,, which is the equinox of the line segment, which is a real number, and then

( ).

28. The coordinate formula of the triangle center of gravity The coordinates of the three vertices of Delta ABC are,,, so the coordinate of the center of gravity of Delta ABC is.

29. Point translation formula (after translation, the corresponding point of any point P(x, y) on Figure F is, and the coordinates are).

30. Commonly used inequalities:

(1) (take "=" if and only if A = B).

(2) (Take "=" if and only if A = B).

(3)

(4) Cauchy inequality

(5)

3 1. Extreme value theorem is known to be positive, so there is.

(1) If the product is a constant value, then the sum has a minimum value;

(2) If the sum is a fixed value, then the time product has a maximum value.

32. One-dimensional quadratic inequality, if the sign is the same, its solution set is outside the two roots; If its sign is different, its solution is somewhere in between. In short, it exists between two identical symbols and two different symbols.

;

33. Inequality with absolute value when a >; At 0, there is

Or ...

34. Unreasonable inequality (1).

(2) .

(3) .

35. Exponential inequality and logarithmic inequality (1) when,

; .

(2) When,

;

36. Slope formula (,).

37. Four equations of a straight line

(1) Point inclination (straight line passes through a point with a slope of).

(2) Oblique intercept (b is the intercept of a straight line on the Y axis).

(3) Two-point formula () (,()).

(4) General formula (where a and b are not both 0).

38. If two straight lines are parallel and vertical (1),

① ; ② .

(2) If and A 1, A2, B 1 and B2 are not zero,

① ; ② ;

39. Angle formula. (,,)

( , , ).

When the straight line is a straight line, the angle between the straight line l 1 and l2 is.

40. Distance from point to straight line (point, straight line:).

Four Equations of 4 1. Circle

Standard equation of (1) circle.

(2) General equation of circle (> 0).

(3) The parametric equation of the circle.

(4) Equation of the diameter of a circle (the end point of the diameter of a circle is,).

42. The parameter equation of ellipse is.

43. Elliptic focal radius formula.

44. The focal radius formula of hyperbola

, .

45. The fixed point on the parabola can be set to p or p, where.

46. The image of quadratic function is parabola: (1) vertex coordinates are; (2) The coordinate of the focus is: (3) The equation of the alignment is.

47. The chord length formula of the intersection of straight line and conic curve

The chord endpoint a is obtained by eliminating y from the equation, which is the inclination angle and slope of the straight line.

48. Two kinds of symmetry problems of conic curves;

(1) The curve whose center is symmetrical about a point is.

(2) The curve about line symmetry is

49. "Four lines" An equation For the general quadratic curve, the equation is obtained by substitution, substitution, substitution, substitution and substitution.

The tangent, tangent chord, midpoint chord and midpoint equation of a curve are all derived from this equation.

50. The * * line vector theorem has a real number λ for any two vectors A and b(b≠0) and a‖b in space, so a = λ b. 。

5 1. For any point o in the space and three points A, B and C of the non-* * line,

Then the four points P, A, B and C are * * * planes.

52. The included angle formula of two vectors in space is COS < a, b > = (a =, b =).

53. The angle between a straight line and a plane (the normal vector of the plane).

54. The plane angle of dihedral angle or (,is the normal vector of plane).

55. Let AC be α, BC ⊥ any straight line in AC, the vertical foot be C, the angle formed by AO and AB be, AB and AC be, and AO and AC be.

56. If the angle formed by the line segment sandwiched between dihedral angles and the two half planes of dihedral angles is, and the angle formed by the sides of dihedral angles is θ, then there is;

(If and only if the equal sign holds).

57. If the distance formula between two points in space is A and B, then

= .

58. The distance from a point to a straight line (the point is on a straight line, the direction vector a= and the vector b= a straight line).

59. The distance between straight lines on different planes (the vertical vectors of two straight lines on different planes are arbitrary points, which is the distance between them).

60. The distance from a point to a plane (the normal vector of a plane is a diagonal line through the plane,).

6 1. formula for the distance between two points on a straight line in different planes

(The angle formed by two straight lines A and B in different planes is θ, and the length of the common vertical line is H. Take two points E, F,,, and on the straight lines A and B respectively).

62.

(The projective length and included angle of a segment on three mutually perpendicular straight lines are respectively) (The formula that the diagonal length of a cuboid is in several squares is a special case).

63. Area projection theorem

The areas of planar polygons and their projections are respectively, and the sharp dihedral angle formed by their planes is.

64. euler theorem (Euler formula) (number of vertices V, number of edges E and number of faces F of a simple polyhedron)

65. If the radius of a ball is r, then its volume is and its surface area is.