1, sum of interior angles of polygons: sum of interior angles of n polygons is equal to (n-2) 180? (n≥3, n is a positive integer), and the sum of external angles is equal to 360?
2. Proportional theorem of parallel lines;
(1) Proportional theorem of parallel lines: three parallel lines cut two straight lines, and the corresponding line segments are proportional.
As shown in the figure: a‖b‖c, straight line l 1 and l2 intersect with straight lines A, B and C respectively, and intersect with points A, B and C..
D, e, f, with: (Figure 1)
(2) Inference: A straight line parallel to one side of a triangle cuts the other two sides (or extension lines on both sides), and the corresponding line segments obtained are proportional.
As shown in the figure: △ABC, where DE‖BC, DE intersects AB and AC, and points D and E have: (Figure 2) (Figure 3).
* 3. Projective theorem in right triangle: as shown in figure: Rt△ABC, ∠ ACB = 90o, CD⊥AB in D, then there is.
4, the relevant properties of the circle:
(1) vertical diameter theorem: If a straight line has one of the following five properties? Any two attributes:
(1) passes through the center of the circle; ② Vertical chord; ③ bisect the chord; (4) bisecting the lower arc of the chord; ? (5) bisecting the optimal arc of the chord, then this straight line has three other properties. Note: When ① and ③ are satisfied, the chord cannot be the diameter.
(2) The arcs sandwiched by two parallel chords are equal.
(3) What is the degree of the central angle? A number is equal to the degree of the arc it subtends.
(4) The angle of the arc is equal to half of its central angle.
(5) circumference? An angle is equal to half the degree of the arc it subtends.
(6) Same arc or equal? The angles of the arcs are equal.
(7) In the same circle or in the same circle, the circular arcs with equal circumferential angles are equal.
(8)90? The angle of the circle? The right chord is the diameter, whereas the circumferential angle of the diameter is 90? The diameter is the longest chord.
(9) Diagonal complementation of quadrilateral inscribed in a circle.
5. Inside and outside center of triangle: The center of inscribed circle of triangle is called the inner center of triangle. The heart of a triangle is the intersection of three internal bisectors. The center of the circumscribed circle of an angle is called the outer center of a triangle.
Common conclusion: (1)Rt△ABC has three sides (A, B and C), so what is its inscribed circle radius? (fig. 6);
(2) If the circumference of △ ABC is (Figure 7-0), the area is S, and the radius of its inscribed circle is R, then (Figure 7);
* 6, chord tangent angle theorem and its inference:
(1) Chord angle: the angle whose vertex is on the circle, one side intersects the circle and the other side is tangent to the circle is called chord angle. As shown in the figure: ∠PAC is the chord tangent angle.
(2) Chord angle theorem: the degree of chord angle is equal to half of the degree of arc it encloses.
If AC is the chord of ⊙O, PA is the tangent of ⊙O, and A is the tangent point, then (Figure 8)
Inference: The chord tangent angle is equal to the circumferential angle of the clamped arc (the function proves that the angles are equal).
If AC is the chord of ⊙O, PA is the tangent of ⊙O, and A is the tangent point, then (Figure 9) (Figure 10).
* 7, intersecting chord theorem, secant theorem, secant theorem:
Chord intersection theorem: two chords intersect in a circle, and the product of the length of two straight lines divided by the intersection point is equal. As shown in Figure ①, namely: PA PB = PC PD.
Secant theorem: two secant lines of a circle are drawn from a point outside the circle, and the length of each secant line is equal to the product of the intersection point of the circle.
As shown in Figure ②, namely: PA PB = PC PD.
Secant theorem: the tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the middle term in the length ratio of the two lines where this point intersects the secant. As shown in Figure ③, namely: PC2 = PA PB.
(Figure 1 1)
8. Area formula:
(1) s positive δ =? (figure 12)? × (side length) 2.
②S parallelogram = base × height.
③S diamond = bottom × height =? (figure 13)? × (diagonal product), (Figure 14)?
④S circle = π R2.
⑤l circumference = 2π r.
6 arc length l =? (figure 15)? .
⑦ (Figure 16)
⑧S cylindrical edge = bottom perimeter × height = 2π RH, and s total area = S edge +S bottom = 2π RH+2π R2.
Pet-name ruby S cone edge = x base perimeter X bus = π Rb, S total area = S edge +S base = π Rb+π R2.
mathematical formula
1, integers (including: positive integer, 0, negative integer) and fractions (including: finite decimal and infinite cyclic decimal) are rational numbers. For example: -3,? (figure 17)? ,0.23 1,0.737373…,? (figure 18)? ,? (figure 19)? . ? The decimal of infinite cycle is called irrational number. Such as: π,-(Figure 20)? 0. 1 0 1 001... (there are1zeros between two1). Rational numbers and irrational numbers are collectively called real numbers.
2、? Absolute value: a≥0? (Figure 2 1)? 丨 a 丨 = a；; ? A≤0 (Figure 2 1) Buy A Buy =-A. For example, buy-? (fig. 22)? What =? (fig. 22)? ; 3. 14-π = π-3. 14.
3. An approximation, starting with the number on the left that is not 0 and ending with the last number. All the numbers are called this? Significant number of approximation. For example, 0.05972 is accurate to 0.00 1 to get 0.060, and two significant figures 6,0 are obtained.
4. Write a number as a× 10n? In the form of (where 1 ≤ A < 10, n is an integer), this notation is called scientific notation. For example, -40700 =-4.07× 105, 0.000043 =? 4.3× 10-5.
5. Multiplication formula (factorization formula in turn): ① (a+b) (a-b) = A2-B2. ② (a b) 2 = A2 2ab+B2。 ③? (a+b)(a2-ab+b2)=a3+b3。 ④(a-b)(a2+a b+B2)= a3-B3； a2+b2=(a+b)2-2ab，(a-b)2=(a+b)2-4ab。
6. The essence of power operation: ①? am×an = am+n②am÷an = am-n③(am)n = am。 ④ (ab) n = anbn。 ((Figure 23))? )n=？ n？ .
⑥ A-N = (Figure 24), especially: (? (fig. 23)? )-n=(？ (fig. 25)? )n？ ⑦? A0 = 1 (a ≠ 0)。 For example: A3× A2 = A5, A6 ÷ A2 = A4, (A3) 2 = A6, (3a3? )3=27a9，(-3)- 1=-？ (fig. 26)? ,5-2=? (fig. 27)? =? (fig. 28)? ,? ((fig. 29)? )-2=(? (fig. 30)? )2=? (Figure 3 1)? ,(-3. 14)? = 1,? ((Figure 22)- (Figure 18)? )0= 1.
7. Quadratic radical: ①? (? (fig. 32)? )2=a？ (a≥0)，②？ (fig. 34)? = 丨 a 丨, ③? (Figure 35-0)? =? (fig. 32)? ×? (fig. 33)? ,④? (fig. 35)? =? (fig. 36)? (a>0，b≥0)？ Such as: ①? (3? (fig. 20)? )2=45.②? (fig. 37)? = 6.③ When a < 0,? (fig. 38)? =-a (Figure 33). ④? (fig. 39)? The square root of 4 = 2. (Concepts of square root, cube root and arithmetic square root)
8. Unary quadratic equation: For equation: AX2+BX+C = 0:
① The formula for finding the root is x =? (fig. 40)? , among them? △ = B2-4ac is called root? The discriminant of.
When △ > 0, the equation has two unequal real roots;
When △ = 0, the equation has two equal real roots;
What time? When △ < 0, the equation has no real root. Note: When △≥0, the equation has real roots.
② If the equation has two real roots x 1 and x2, and the quadratic trinomial AX2+BX+C can be decomposed into a (X-X 1) (X-X2).
(3) One with roots A and B? What is a quadratic equation? x2-(a+b)x+ab=0。
9. The image of the linear function y = kx+b (k ≠ 0) is a straight line (b is the ordinate of the intersection of the straight line and the Y axis, that is, the intercept of the linear function on the Y axis). When k > 0, y? It increases with the increase of x (the straight line rises from left to right); When k < 0, y decreases with the increase of x (the straight line decreases from left to right). Especially when b = 0 and y = kx? (k≠0) is also called proportional function (Y is proportional to X), and the image must pass through the origin.
10 and the inverse proportional function y = (k ≠ 0) are called hyperbola. When k > 0, hyperbola is in one or three quadrants (decreasing from left to right in each quadrant); When k < 0, hyperbola is in two or four quadrants (rising from left to right in each quadrant). Therefore, its increase or decrease is contrary to a linear function.
1 1, preliminary statistics: (1) Concept: ① All investigated objects are called the whole, and each investigated object is called the individual. Some individuals extracted from the population are called the sample of the population, and the number of individuals in the sample is called the sample size. ② In a set of data, the number that appears the most (sometimes more than one) is called this set of data.
(2) Formula: How many N numbers are there? x 1，x2，…，xn？ , and then:
① The average value is: (Figure 41);
② Very poor:
The difference between the maximum value and the minimum value of a set of data reflects the range of this set of data. The difference obtained by this method is called extreme range, that is, extreme range = maximum-minimum;
③ Variance:
Data (Figure 44), then = (Figure 42)
Standard deviation: the arithmetic square root of variance.
Data (Figure 45), then = (Figure 43)
The greater the variance of a set of data, the greater the volatility and instability of this set of data.
12, frequency and probability:
(1) frequency =, the sum of each group of frequencies is equal to the total, the sum of each group of frequencies is equal to 1, and the area of each small rectangle in the frequency distribution histogram is each group of frequencies.
(2) Probability
(1) If the probability of event A is expressed by p, then 0 ≤ p (a) ≤1;
P (inevitable event) =1; P (impossible event) = 0;
② Understand the meaning of probability in specific situations, and calculate the probability of simple events by enumeration (including list and tree drawing).
③ The frequency of repeated experiments can be regarded as an estimate of the probability of events;
13, acute trigonometric function:
① Let ∠A be any acute angle of Rt△ABC, then ∠A sine: sinA= =? Cosine of ∠A: cosA= =, Tangent of ∠A: tanA= =? . While sin2a+cos2a = 1.
0 0 Note: The equation has two unequal real roots.
B2-4ac & lt； Note: The equation has no real root, but a complex number of the yoke.
formulas of trigonometric functions
Two-angle sum formula
sin(A+B)= Sina cosb+cosa sinb sin(A-B)= Sina cosb-sinb cosa
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)
Double angle formula
tan2A = 2 tana/( 1-tan2A)ctg2A =(ctg2A- 1)/2c TGA
cos2a = cos2a-sin2a = 2 cos2a- 1 = 1-2 sin2a
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
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 sinasinb = 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)/cosa cosb tanA-tanB = sin(A-B)/cosa cosb
ctgA+ctgBsin(A+B)/Sina sinb-ctgA+ctgBsin(A+B)/Sina sinb
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) 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=2px y2=-2px x2=2py x2=-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
Conical volume formula V= 1/3*S*H Conical 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