The content of mathematics compulsory two in Senior High School

The content of mathematics compulsory two in Senior High School

Two knowledge points of compulsory high school mathematics
1、 Straight line and equation
(1) The inclination angle of a straight line
Definition: the angle between the positive direction of the x-axis and the upward direction of the straight line is called the inclination angle of the straight line. In particular, when the straight line is parallel to or coincident with the x-axis, we specify that its inclination angle is 0 degree. Therefore, the value range of the inclination angle is 0 °≤ α＜ 180 °
(2) The slope of a straight line
① Definition: the tangent of a straight line whose inclination angle is not 90 ° is called the slope of the straight line. The slope of the straight line is usually expressed by K. that is, the slope reflects the inclination of the straight line and the axis
At that time,; at that time,; at that time, there was no existence
② The slope formula of a straight line passing through two points:
Pay attention to the following four points: (1) at that time, the right side of the formula is meaningless, the slope of the straight line does not exist, and the inclination angle is 90 °;
(2) K has nothing to do with the order of P1 and P2; (3) the later slope can be obtained directly from the coordinates of two points on the straight line instead of the inclination angle;
(4) The inclination angle of a straight line can be obtained by calculating the slope of the coordinates of two points on the line
(3) Linear equation
① Point oblique: the slope of the straight line is k, and it passes through the point
Note: when the slope of the line is 0 °, k = 0, and the equation of the line is y = Y1
When the slope of a straight line is 90 degrees, the slope of the straight line does not exist, and its equation can not be expressed in the form of oblique point. But because the abscissa of every point on L is equal to x1, its equation is x = x1
② Oblique section: the slope of the line is k, and the intercept of the line on the Y axis is B
③ Two point formula: () straight line two points,
④ Cut moment formula:
Where the line intersects the axis at a point, and intersects the axis at a point
⑤ General formula: (a, B are not all zero)
Note: the scope of application of various equations is special, such as:
A line parallel to the x-axis: (B is a constant); a line parallel to the y-axis: (a is a constant);
(5) Linear system equation: that is, a line with a common property
（1） Parallel line system
A system of lines parallel to a known line (a constant that is not all zero): (C is a constant)
（2） Vertical line system
A system of lines perpendicular to a known line (a constant that is not all zero): (C is a constant)
（3） Straight line system passing through fixed point
(I) the system of lines with slope k: the line passes through a fixed point;
(II) the equation of the line system passing through the intersection of two lines is
(is a parameter), where the line is not in the line system
(6) Two lines parallel and perpendicular
When, when,
；
Note: when using the slope to judge whether a straight line is parallel or vertical, pay attention to the existence of the slope
(7) The intersection of two lines
intersect
The point of intersection coordinates is a set of solutions of the equations
The system of equations has no solution; the system of equations has innumerable solutions and coincidence
(8) The formula of distance between two points: let two points be in the plane rectangular coordinate system,
be
(9) Formula of distance from point to line: distance from point to line
(10) Distance formula of two parallel straight lines
Take any point on any line, and then convert it into the distance from point to line
2、 The equation of circle
1. The definition of circle: the set of points whose distance to a certain point in the plane is equal to the fixed length is called circle. The fixed point is the center of the circle and the fixed length is the radius of the circle
2. The equation of circle
(1) Standard equation, center of circle, radius R;
(2) General equation
At that time, the equation represents a circle. At this time, the center of the circle is and the radius is
At that time, it represents a point; at that time, the equation does not represent any graph
(3) The method of solving circular equation is as follows
Generally, the undetermined coefficient method is used: set first and then find out. Three independent conditions are needed to determine a circle,
We need to get a, B, R; if we use the general equation, we need to get D, e, f;
In addition, we should pay more attention to the geometric properties of the circle: for example, the vertical line of the string must pass through the origin, so as to determine the position of the center of the circle
3. Position relationship between line and circle:
There are three kinds of position relationship between line and circle: separation, tangency and intersection
(1) Let a line, a circle, and the distance from the center of the circle to l be;;
(2) Tangent line passing through a point outside the circle: ① K does not exist, verify whether it is true; ② K exists, set a point oblique equation, use the distance from the center of the circle to the straight line = radius, solve K, and get the equation [certain two solutions]
(3) The tangent equation passing through a point on a circle: circle (x-a) 2 + (y-b) 2 = R2, the tangent equation passing through a point on a circle is (x0, Y0), then (x0-a) (x-a) + (y0-b) (y-b) = R2
4. The position relation between circle and circle is determined by comparing the sum (difference) of the radii of two circles with the center distance (d)
Let's set a circle,
The position relation of two circles is usually determined by comparing the sum (difference) of the radii of two circles with the center distance (d)
At that time, the two circles were separated from each other. At this time, there were four common tangent lines;
At that time, the two circles were circumscribed, and the central line passed through the tangent point. There were two external tangent lines and one internal tangent line;
At that time, the two circles intersect, the common chord is bisected vertically by the central line, and there are two external tangent lines;
At that time, two circles were inscribed, and the connecting line passed through the tangent point, and there was only one common tangent line;
At that time, the two circles contained; at that time, they were concentric circles
Note: if two points on a circle are known, the center of the circle must be on the vertical line; if two circles are known to be tangent, the center of the two circles and the tangent point are collinear
The auxiliary line of a circle is usually connected with the center of the circle and tangent line or the center of the circle and the midpoint of the chord
3、 A preliminary study of solid geometry
1. Structural characteristics of column, cone, table and ball
(1) Prisms:
Geometric features: two bottom surfaces are congruent polygons parallel to corresponding sides; side and diagonal surfaces are parallelogram; side edges are parallel and equal; section parallel to bottom surface is congruent polygon with bottom surface
(2) Pyramid
Geometric features: the side and diagonal planes are triangles; the section parallel to the bottom is similar to the bottom, and its similarity ratio is equal to the square of the ratio of the distance and height from the vertex to the section
(3) Prisms:
Geometric features: ① the upper and lower bottom surfaces are similar parallel polygons; ② the side surfaces are trapezoids; ③ the side edges intersect the vertices of the original pyramid
(4) Cylinder: definition: it is formed by the rotation of the line on one side of the rectangle and the rotation of the other three sides
Geometric features: ① the bottom is a congruent circle; ② the generatrix is parallel to the axis; ③ the axis is perpendicular to the radius of the bottom circle; ④ the side expansion is a rectangle
(5) Cone: definition: a right triangle with a right side as the axis of rotation, rotating a circle
Geometric features: ① the bottom is a circle; ② the generatrix intersects the apex of the cone; ③ the side expansion is a sector
(6) Round platform: definition: it is formed by one circle of rotation with the vertical of right angled trapezoid and the waist at the bottom as the rotation axis
Geometric features: ① the upper and lower bottom surfaces are two circles; ② the side generatrix intersects the vertex of the original cone; ③ the side expanded view is an arch
(7) Sphere: definition: a geometric body formed by a semicircle surface rotating one circle with the straight line of the semicircle diameter as the rotation axis
Geometric features: ① the cross section of the sphere is a circle; ② the distance from any point on the sphere to the center of the sphere is equal to the radius
2. Three views of space geometry
Three views are defined: front view (the light is projected from the front of the geometry to the back); side view (from left to right); and
Top view (top down)
Note: the front view reflects the height and length of the object; the top view reflects the length and width of the object; the side view reflects the height and width of the object
3. The visual drawing of space geometry
The characteristics of oblique two survey drawing method are as follows: 1. The line segment which was originally parallel to X axis is still parallel to X and the length is not changed;
② The original line segment parallel to y axis is still parallel to y, and its length is half of the original
4. Surface area and volume of cylinder, cone and platform
(1) The surface area of a geometry is the sum of the areas of its faces
(2) Surface area formula of special geometry (C is the perimeter of the bottom, h is the height, oblique height, l is the generatrix)
(3) Volume formula of cylinder, cone and platform
(4) The surface area and volume formula of sphere: v =; s=
4. The position relation of space point, line and plane
Axiom 1: if two points of a line are in a plane, then all points of the line are in the plane
Application: judge whether the line is in the plane
Axioms are expressed in symbolic language
Axiom 2: if two non coincident planes have a common point, then they have and only have one common line passing through the point
Symbol: plane α and β intersect, the intersection line is a, denoted as α ∩ β = a
Symbol language:
The function of axiom 2:
① It is a method to determine the intersection of two planes
② It shows the relationship between the intersection line of two planes and the common point of two planes: the intersection line must pass through the common point
③ It can judge that a point is on a straight line, which is an important basis to prove that several points are collinear
Axiom 3: there is only one plane passing through three points which are not on the same line
Inference: a line and a point outside the line determine a plane; two intersecting lines determine a plane; two parallel lines determine a plane
Axiom 3 and its inferential function: ① it is the basis for determining the plane in space; ② it is the basis for proving the coincidence of planes
Axiom 4: two lines parallel to the same line are parallel to each other
The position relation between spatial straight line and straight line
① Definition of out of plane line: two different lines in any plane
② Properties of out of plane lines: neither parallel nor intersecting
③ Out of plane straight line judgment: the straight line passing through a point outside the plane and a point in the plane is in the plane, but the straight line of the store is out of plane straight line
④ Angle formed by straight lines of different planes: make parallel, make two lines intersect, get acute angle or right angle, that is, the angle formed. The range of angle formed by two straight lines of different planes is (0 ° 90 °). If the angle formed by two straight lines of different planes is right angle, we say that the two straight lines of different planes are perpendicular to each other
The steps of finding the angle of a straight line on a different plane are as follows:
A. Using the definition to construct the angle, one can be fixed, the other can be translated, or two can be translated to a special position at the same time, and the vertex is selected at a special position. B. prove that the angle is the angle C. use the triangle to find the angle
(7) Isometric theorem: if two sides of an angle are parallel to each other, then the two angles are equal or complementary
(8) The positional relationship between spatial line and plane
The line is in the plane - there are countless common points
Symbolic representation of three kinds of positional relations: a α a ∩ α = a a a ∩ α
(9) The positional relationship between two planes: parallel - no common point; α‖ β
Intersection - there is a common line. α ∩ β = B
5. Parallel problems in space
(1) The judgment and properties of the parallel of straight line and plane
Judging theorem of line plane parallel: if a straight line outside the plane is parallel to a straight line in the plane, the straight line is parallel to the plane
Line parallel line plane parallel
If a line is parallel to a plane, the plane passing through the line intersects the plane,
So this line is parallel to the intersection line. The line plane is parallel to the line
(2) The judgment and properties of parallel plane and plane
The judgement theorem of two parallel planes
(1) If two intersecting lines in one plane are parallel to the other plane, then the two planes are parallel
(line plane parallel → plane plane parallel),
(2) If two groups of intersecting lines are parallel in two planes, the two planes are parallel
(line parallel → plane parallel),
(3) Two planes perpendicular to the same line are parallel,
The property theorem of two parallel planes
(1) If two planes are parallel, the line in one plane is parallel to the other plane. (plane parallel → line plane parallel)
(2) If two parallel planes intersect the third plane, their intersection lines are parallel
7. Vertical problems in space
(1) The definition of line, surface and line surface perpendicularity
① The perpendicularity of two straight lines in different planes