Mathematics website daquan

(2) .

(3) .

Note: if a > 0 and p is an irrational number, then ap stands for a definite real number. The above operational properties of rational exponential power are applicable to the exponential power of irrational numbers.

33. Reciprocal formula of exponential and logarithmic expressions

34. Change the formula of logarithm base.

(,and, and,).

Inference (,and, and,).

35. Four operations of logarithm

If a > 0, a≠ 1, m > 0, n > 0, then

( 1) ;

(2) ;

(3) .

36. Set a function and remember it. If the domain of is, then, and; If the range of is, and. For the case of, it needs to be tested separately.

37. Logarithmic inversion inequality and its generalization

If,,, then the function

(1) when, and is increasing function.

(2) If is the decreasing function of sum.

Inference: Suppose,,, and, then

( 1) .

(2) .

38. The question of average growth rate.

If the base of the original output value is n and the average growth rate is 0, then the total output value of time is.

39. The relationship between the same series formula and the sum of the first n items

The sum of the first n items in the sequence is.

40. arithmetic progression's general formula

;

The first n terms and formulas are as follows

General formula of 4 1. geometric series

;

The summation formula of the first n terms is

Or ...

42. Equal ratio difference series: the general formula is

;

The first n terms and formulas are as follows

43. Installment payment (mortgage loan)

The repayment of each installment is RMB (the loan is RMB, which is paid off in installments with the interest rate of RMB per installment).

44. Common triangular inequalities

(1) If, then.

(2) If, then.

(3) .

45. The basic relationship of trigonometric functions with the same angle

, = , .

46. Sine and cosine induction formula

47. Sum angle and difference angle formula

;

(sine square formula);

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

48. Double angle formula

49. Triple angle formula

. .

50. Periodic formula of trigonometric function

Function, x∈R and the period of the function, x∈R(A, ω is a constant, and A≠0, ω > 0); The period of the function, (a, ω, constant, A≠0, ω > 0).

5 1. sine theorem

52. Cosine theorem

;

53. Area theorem

(1) (representing the heights of sides A, B and C respectively).

(2) .

(3) .

54. The theorem of triangle interior angle sum

In △ABC, there are

55. General solution of simple trigonometric equation

Specifically, there are

56. The simplest triangular inequality and its solution set

57. Algorithm of product of real number and vector

Let λ and μ be real numbers, then

(1) Binding Law: λ (μ a) = (λ μ a);

(2) The first distribution law: (λ+μ) a = λ a+μ a;

(3) The second distribution law: λ (a+b) = λ a+λ b 。

58. The number of vector product algorithm:

( 1) a？ b= b？ A (commutative law);

⑵(a)？ b= (a？ b)= a？ b= a？ (b) and:

(3)(a+b)？ c= a？ c +b？ c.

59. The basic theorem of plane vector

If e 1 and e 2 are two non-linear vectors on the same plane, then there are only a pair of real numbers λ 1 and λ2 for any vector on this plane, so that a = λ 1e 1+λ 2e2.

The vectors e 1 and e2 of non-* * lines are called a set of bases representing all vectors in the plane.

60. Vector parallel coordinate representation

Let a=, b=, and b 0, then a b(b 0).

53. the quantitative product (or inner product) of a and b.

Answer? b=|a||b|cosθ。

6 1. A? Geometric meaning of b

Quantity product a? B is equal to the product of the length of a |a| and the projection of b in the direction of a |b|cosθ.

62. Coordinate operation of plane vector

(1) Let A = and B =, then a+b=.

(2) Let A = and B =, then a-b=.

(3) Let A and B, then.

(4) Let a=, then a=.

(5) Let A = and B =, then A? b=。

63. The included angle formula of two vectors

(a=，b=)。

64. The distance formula between two points on the plane

(A, B).

65. Parallelism and verticality of vectors

Let a=, b=, b 0, then

A||b b=λa。

a b(a 0) a？ b=0。

66. Fixed fraction formula of line segment

Let,, be the equinox of a line segment, be a real number, and then

( ).

67. Coordinate formula of triangle center of gravity

The coordinates of the three vertices of △ABC are, respectively, and the coordinates of the center of gravity of △ABC are.

68. Point translation formula

Note: Any point P(x, y) on Figure F is translated to point and coordinate.

69. Several conclusions of "vector translation"

(1) points are translated by vector a= to get points.

(2) The image of the function is translated with vector a= to obtain an image, and the resolution function is.

(3) The image is translated by vector a=, and the resolution function of is.

(4) Curve: If an image is obtained after translation with vector a=, the equation is.

(5) Vector m= The vector obtained after translation according to vector a= is still m=.

70. Necessary and Sufficient Conditions of Five "Heart" Vector Forms of Triangle

Let it be a point on the plane and the opposite sides of the angle are respectively, then

(1) is the outer center.

(2) as the center of gravity.

(3) I care.

(4) for the heart.

(5) seek benefits for others.

7 1. Common inequalities:

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

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

(3)

(4) Cauchy inequality

(5) .

72. Extreme value theorem

As we all know, all are positive numbers, so there is

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

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

Publicity is well known, there are

(1) If the product is a constant value, it is maximum when it is maximum;

When it is the smallest, it is the smallest.

(2) If the sum is a fixed value, it is the maximum value and the minimum value;

When it is the smallest, it is the largest.

73. A quadratic inequality has two solution sets if it has the same symbol; If its sign is different, its solution is somewhere in between. In short, it exists between two identical symbols and two different symbols.

;

74. Absolute inequality

When a> is at 0, there is

Or ...

75. unreasonable inequality

( 1) .

(2) .

(3) .

76. Exponential inequality and logarithmic inequality

(1) When,

;

(2) When,

;

77. Slope formula

( 、 ).

78. Five 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) Intercept type (horizontal intercept and vertical intercept of straight line respectively)

(5) General formula (where a and b are not 0 at the same time).

79. Parallelism and verticality of two straight lines

(1) If,

① ;

② .

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

① ;

② ;

80. Angle formula

( 1) .

( , , )

(2) .

( , , ).

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

8 1. Angle formula to

( 1) .

( , , )

(2) .

( , , ).

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

82. Four common linear equations

(1) Fixed-point linear equations: The linear equations passing through the fixed point are (except straight lines), where the coefficients are undetermined; The equation of the straight line system passing through the fixed point is, where is the undetermined coefficient.

(2)*** point linear equations: the linear equations passing through the intersection of two straight lines are (division), where λ is the coefficient to be found.

(3) Parallel linear equations: When the slope k is constant and b changes along a straight line, it means parallel linear equations. The linear system equation parallel to the straight line is (), and λ is a parameter variable.

(4) Vertical linear system equation: The linear system equation perpendicular to the straight line (A≠0, B≠0) is, and λ is the parameter variable.

83. Distance from point to straight line

(Point, straight line:).

84. Or expressed plane area.

If a straight line is set, the plane area represented by or is:

If, when the same symbol is used, it means the area above the straight line; When the symbol is different from the symbol, it indicates the area below the straight line. In short, the sign is above and the sign is below.

If, when the same symbol is used, it represents the area to the right of the straight line; When the sign is different from the sign, it indicates the left area of the straight line. In short, the sign is on the right and the sign is on the left.

85. Or expressed plane area.

Set the curve (), and then

Or the plane area is:

The upper and lower parts of the plane area represented;

The upper and lower parts of the plane area represented.

86. Four kinds of equations of 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,).

87. Equation of circle system

(1) The equation of the circle system passing through the point is

Where is the equation of a straight line and λ is the coefficient to be found.

(2) Line: The equation of the circle system intersecting with circle: is, and λ is the undetermined coefficient.

(3) Passing the circle: the equation of the circle system intersecting the circle: is, and λ is the undetermined coefficient.

88. The positional relationship between a point and a circle

There are three positional relationships between a point and a circle.

If, then

The point is outside the circle; A point on a circle; The point is in the circle.

89. The positional relationship between a straight line and a circle

There are three positional relationships between a straight line and a circle:

;

One of them is.

90. How to determine the position relationship between two circles

Let the center of two circles be O 1, O2, and the radius be r 1, r2.

;

Tangent equation of 9 1. circle

(1) known cycles.

(1) If the tangent point is known to be on the circle, there is only one tangent, and its equation is

When outside the circle, it represents the tangent equation passing through two tangent points.

② The tangent equation of a point outside the circle can be set as, and then k can be obtained by using the tangent condition. There must be two tangents at this time. Be careful not to miss the tangent parallel to the Y axis.

③ The tangent equation with a slope of k can be set as, and then using the tangent condition to find b, there must be two tangents.

(2) Known cycles.

The tangent equation of this point on the (1) circle is:

② The tangent equation of a circle with slope is.

92. The parameter equation of ellipse is.

93. Elliptic focal radius formula

, .

94. Inside and outside the ellipse

The (1) point is inside the ellipse.

(2) The point is outside the ellipse.

95. The tangent equation of ellipse

The tangent equation of a point on (1) ellipse is.