Summary of Important Knowledge Points of Mathematics in Grade Three I. Symmetry of Circle
1, the axis symmetry of the circle
A circle is an axisymmetric figure, and every straight line passing through the center of the circle is its axis of symmetry.
2. The center of the circle is symmetrical
A circle is a central symmetrical figure with the center of the circle as the symmetrical center.
Second, the relation theorem of arc, chord, chord center distance and central angle
1, central angle
The angle of the vertex at the center of the circle is called the central angle.
2, chord center distance
The distance from the center of the circle to the chord is called the chord center distance.
3. Theorem of the relationship between arc, chord, chord center distance and central angle.
In the same circle or in the same circle, the arcs with equal central angles are equal, the chords are equal, and the chord distance is equal.
Inference: In the same circle or equal circle, if one set of quantities in two circles, two arcs, two chords' central angles or two chords' central distances are equal, the corresponding other set of quantities are equal respectively.
Third, the theorem of circle angle and its inference
1, circle angle
The angle whose vertex is on the circle and whose two sides intersect the circle is called the circumferential angle.
2. The theorem of circle angle
An arc subtends a circumferential angle equal to half the central angle it subtends.
Inference 1: the circumferential angles of the same arc or equal arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
Inference 2: the circumferential angle of a semicircle (or diameter) is a right angle; A chord with a circumferential angle of 90 is a diameter.
Inference 3: If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.
Fourthly, the positional relationship between a point and a circle.
Let the radius of ⊙O be r and the distance from point P to the center of O be d, then there are:
D = ⊙ o on point p;
D>r point P is outside ⊙ O.
One lap after three o'clock.
1, a circle passing through three points
Three points that are not on the same straight line determine a circle.
2. The circumscribed circle of a triangle
A circle passing through the three vertices of a triangle is called the circumscribed circle of the triangle.
3. The outer center of the triangle
The center of the circumscribed circle of a triangle is the intersection of the perpendicular lines of the three sides of the triangle, which is called the center of the triangle.
4. Quadrilateral properties of inscribed circle (judging conditions of four-point * * * circle)
Diagonal complementarity of quadrilateral inscribed in a circle.
Five, some basic formulas
Triple angle formula
Sine, cosine and tangent formulas of triple angle
sin3α=3sinα-4sin^3(α)
cos3α=4cos^3(α)-3cosα
tan3α=[3tanα-tan^3(α)]/[ 1-3tan^2(α)]
Derivation of triple angle formula
Additional derivation:
tan3α=sin3α/cos3α
=(sin 2αcosα+cos 2αsinα)/(cos 2αcosα-sin 2αsinα)
=(2sinαcos^2(α)+cos^2(α)sinα-sin^3(α))/(cos^3(α)-cosαsin^2(α)-2sin^2(α)cosα)
Divided by COS 3 (α), we get:
tan3α=(3tanα-tan^3(α))/( 1-3tan^2(α))
sin 3α= sin(2α+α)= sin 2αcosα+cos 2αsinα
=2sinαcos^2(α)+( 1-2sin^2(α))sinα
=2sinα-2sin^3(α)+sinα-2sin^3(α)
=3sinα-4sin^3(α)
cos 3α= cos(2α+α)= cos 2αcosα-sin 2αsinα
=(2cos^2(α)- 1)cosα-2cosαsin^2(α)
=2cos^3(α)-cosα+(2cosα-2cos^3(α))
=4cos^3(α)-3cosα
sin3α=3sinα-4sin^3(α)
cos3α=4cos^3(α)-3cosα
Six, some key knowledge
Remember the definition of trigonometric function skillfully: the trigonometric functions learned in junior high school include sine, cosine, tangent and cotangent, which are actually the ratio of triangle sides. You can separate these two words and remember the definition with the following sentence: An unskilled cook taught his apprentice to kill fish and said the following sentence: Cut the fish phosphorus (neighbor) directly. Positive: sine or tangent, right: opposite positive; Remainder: cosine or cosine, adjacent: adjacent edges indicate that the remainder is adjacent; Tangents are right-angled edges.
Increase and decrease of trigonometric function: positive increase, surplus decrease, special trigonometric function value memory: firstly, remember that the denominators of sine and cosine values of 30 degrees, 45 degrees and 60 degrees are all 2, and the denominators of tangent and cotangent are all 3. Molecules can remember the formula "123,3213927".
Judgment of parallelogram: To prove a parallelogram, two conditions can be met, one is to prove that the opposite sides are equal, the other is to prove that the opposite sides are parallel, and a group of opposite sides can also be proved, and they must be equal and parallel. Slant is a treasure. If we split it, we can't run away. It is also useful if the diagonals are equal. Only "two diagonal lines" can be achieved.
Auxiliary line of trapezoid problem: move the diagonal of trapezoid to make two waists into a line; Move one waist in parallel, with both waists in the "△" position; Extend the waist a little, and there are parallel lines in the "△"; Make two trapezoidal high lines, and the rectangle will be displayed in front of your eyes; Know the center line of the waist, don't forget to make the center line.
Add auxiliary lines Song: Auxiliary lines, how to add them? Finding the pattern is the key. If there is an angle (horizontal) dividing line in the question, it can be vertical on both sides. The middle perpendicular of the line segment leads to the connecting line at both ends, and the connecting line between the two midpoints of the triangle side forms the middle line; A triangle has a midline, and the midline is doubled.
Proportional line segment in the circle: in the case of equal product, change the equal ratio and find similarity vertically and horizontally; Don't be angry, switch to equal lines and ratios, encounter equal ratios, change equal products, quote projective and circular powers, parallel lines, turn proportions, and find the connection between the two ends.
Song: Divide the circle equally, and the value of n must be greater than three, connecting the points in turn and inscribing a regular N-polygon.
Mathematics for senior high school entrance examination 1 important knowledge points: the basic concept of quadratic equation with one variable.
1. The constant term of the unary quadratic equation 3x2+5x-2=0 is -2.
2. The coefficient of the primary term of the unary quadratic equation 3x2+4x-2=0 is 4, and the constant term is -2.
3. The quadratic term coefficient of the unary quadratic equation 3x2-5x-7=0 is 3, and the constant term is -7.
4. Transform the equation 3x(x- 1)-2=-4x into the general formula 3x2-x-2=0.
Knowledge point 2: Cartesian coordinate system and the position of points
1. In the rectangular coordinate system, point A (3 3,0) is on the Y axis.
2. In the rectangular coordinate system, the abscissa of any point on the X axis is 0.
3. In rectangular coordinate system, point A (1, 1) is in the first quadrant.
4. In rectangular coordinate system, point A (-2,3) is in the fourth quadrant.
5. In rectangular coordinate system, point A (-2, 1) is in the second quadrant.
Knowledge point 3: Find the function value of the known independent variable.
1. When x=2, the value of function y= is 1.
2. When x=3, the value of function y= is 1.
3. When x=- 1, the value of function y= is 1.
Knowledge point 4: the concept and nature of basic functions
1. The function y=-8x is a linear function.
2. The function y=4x+ 1 is a proportional function.
3. This function is an inverse proportional function.
4. The opening of parabola y=-3(x-2)2-5 is downward.
5. The symmetry axis of parabola y=4(x-3)2- 10 is x=3.
6. The vertex coordinate of parabola is (1, 2).
7. The image of the inverse proportional function is in the first and third quadrants.
Knowledge point 5: mean, median and mode of data
The average value of 1. data 13, 10,12,8,7 is 10.
2. The pattern of data 3, 4, 2, 4, 4 is 4.
3. The median of data 1, 2,3,4,5 is 3.
Knowledge point 6: Special trigonometric function values
1.cos30 = = root number 3/2.
2.sin260 +cos260 = 1。
3.2sin30 +tan45 =2。
4.tan45 = 1。
5.cos60 +sin30 = 1。
Summary of mathematics learning methods and skills in grade three 1. Prepare carefully before class. The purpose of preview is to listen to the teacher better. Through preview, the mastery level should reach 80%. Listen to the teacher answer these questions with questions that you don't understand in the preview. Preview can also improve the overall efficiency of attending classes. The specific preview method is to finish the topics in the book and draw the knowledge points. The whole process lasted about 65433.
2. Let math class combine with practice. It's no use just listening in math class. When the teacher asks the students to do calculus on the blackboard, they should also practice on the draft paper. Be sure to ask questions you don't understand. Otherwise, if you encounter similar problems in the exam, you may not do it. When you listen to the teacher, you must concentrate on the details, otherwise, you will be destroyed by the ant nest.
Review in time after class. After finishing your homework, sort out what the teacher said that day, and you can do extracurricular problems for about 25 minutes. You can choose the extracurricular books that suit you according to your own needs. The content of the extracurricular problem is probably today's class.
The fourth unit test is to test your recent study. In fact, the score represents your past. The key is to sum up and learn from each exam so that you can do better in the mid-term and final exams. Teachers often take exams without notice and review them in time after class.