No matter how fertile the soil is, no crops can grow. No matter how hard you struggle, no matter how much creativity you have, no matter how beautiful your youth is. Don't let the pursuit boat anchor in the harbor of fantasy, but raise the sail of struggle and sail for the sea of real life. The senior one channel has compiled a summary of the four required knowledge points in the second volume of senior high school mathematics for you who are struggling. I hope it helps you!

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The first chapter trigonometric function

Positive angle: the angle formed by counterclockwise rotation.

1, negative angle of any angle: the angle formed by clockwise rotation.

Zero angle: an angle formed without any rotation.

2. The vertex of the angle coincides with the origin, the starting edge of the angle coincides with the non-negative semi-axis of the X axis, and the quadrant in which the ending edge falls is called quadrant.

The set of the second quadrant angle is k36090k360 180, k.

The set of the third quadrant angle is K360 180K360270, the set of the fourth quadrant angle is k360270k360, and the set of the angles of the K-end edge on the X-axis is K 180, k.

The angle set of the terminal edge on the y axis is K 18090, and the angle set of the terminal edge on the coordinate axis is K90, k.

The set of the first quadrant angle is k360k36090, k.

3. The set of angles that are the same as the terminal edge of an angle is K360, k.

4. The central angle of an arc with a length equal to the radius is called 1 radian.

5. If the length of the arc opposite to the central angle of a circle with radius r is L, the absolute value of the radian number of the angle is

l.r

180

6. Conversion formula of arc system and angle system: 2360, 1, 157.3. 180.

7. If the central angle of the sector is

For an arc system, if the radius is R, the arc length is L, the circumference is C, and the area is S, then lr, C2rl,

1

1 1

Slrr2。

22

eight

Let an angle of any size be an angle whose distance from the origin is rr, and the coordinates of any point on the terminal side are X and Y, then sin.

0,

yxy

，cos，tanx0.rrx

9. Symbols of trigonometric functions in each quadrant: the first quadrant is all positive, and the sine in the second quadrant is positive.

The tangent of the third quadrant is positive, and the cosine of the fourth quadrant is positive.

10, trigonometric function line: sin, cos, tan ..

2222

1 1, the basic relationship of trigonometric function:1sin2cos21sin1cos, COS 1SIN.

;

2

commit a crime

Tankos

commit a crime

Sintankos, Colorado.

Dark color

12, inductive formula of function:

1sin2ksin，cos2kcos，tan 2k tank . 2 sin，coscos，tantan . 3 sin，coscos，tantan . 4 sin，coscos，tantan。

Formula: The name of the function remains unchanged, and the symbol depends on the quadrant.

5sin

cos，cossin.6sincos，cossin.2222

Formula: Sine and cosine are interchanged, and the sign looks at quadrant.

Translate all points on the images of 13 and ① to the left (right) by one unit length to obtain the image of function ysinx; Then extend (shorten) the abscissa of all points on the function ysinx image to the original position.

1

Multiply (the ordinate is unchanged) to get the image of the function ysinx; then

Extend (shorten) the ordinate of all points on the function ysinx image to the original multiple (the abscissa is unchanged) to get the function.

The image of ysinx.

The abscissa of all points on the ysinx image is extended (shortened) to the original abscissa.

1

Times (ordinate unchanged), get the function.

Image of ysinx; Then all points on the function ysinx image are translated to the left (right).

Unit length, get the function.

Image of ysinx; Then the vertical coordinates of all points on the image of the function ysinx are extended (shortened) to the original time (horizontal

2

Coordinate invariant), the image of function ysinx. 14 and the properties of function ysinx 0,0 are obtained: ① amplitude:; ② cycle:

2

; ③ frequency: f

1

; 4 stages: X; ⑤ Initial stage: .2

Function ysinx, when xx 1, the minimum value is ymin；; When xx2, get the value of ymax, then

1 1

x2x 1x 1x 2 ymaxyminymaximin

22,,2.

yASinx，A0，0，T

2

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2

yACosx，A0，0，T

yASinx，A0，0，T

yACosx，A0，0，T

yASinxb，A0，0，b0，T

2

2

yACosxb，A0，0，b0，T

TyAcotx，A0，0，

yAtanx，A0，0，T

yAcotx，A0，0，T

yAtanx，A0，0，T

three

Chapter II Plane Vector

16, vector: both magnitude and direction. Quantity: only quantity, no direction quantity. Three elements of a directed line segment: starting point, direction and length. Zero vector: A vector with a length of 0. Unit vector: a vector with a length equal to 1 unit. Parallel vector (* * line vector): the direction is the same or opposite.

Equal-length vector: a vector with equal length and the same direction.

17, vector addition operation:

⑵ The characteristics of triangle rule: end to end. ⑵ Characteristics of parallelogram rule: * * starting point.

C

⑶ Triangle inequality: ababab ..

(4) nature of operation: ① reduction of criminal law: abba；;

Abcabc② binding method: ③a00aa.

a

b

abCC

four

5] Coordinate operation: let ax 1, y 1, bx2, y2, then abx 1x2, y 1y2.

18, vector subtraction:

⑴ The characteristics of triangle rule are: * * starting point, even ending point, and direction pointing to reduced vector.

⑵ coordinate operation: let ax 1, y 1, bx2, y2, then abx 1x2, y 1y2.

Let the coordinates of two points be x 1, y 1, x2, y2, then X 1x2, Y 1Y2.

19, vector multiplication:

(1) The product of real number and vector is a vector. This operation is called vector multiplication and is recorded as.

aa；

② At 0, the direction of A is the same as that of A; 0, the direction of a is opposite to that of a; When 0, A0.

⑵ Operation method: ① AA; ②AAA； ③abab。

(3) coordinate operation: let ax, y, then ax, yx, y.

20, vector * * * line theorem: vector aa0 and b*** line, if and only if there are real numbers, make ba.

Let ax 1, y 1, bx2, y2, where b0, then if and only if x 1y2x2y 10, vectors a and bb0*** are connected.

2 1, the basic theorem of plane vectors: If e 1 and e2 are two nonlinear vectors on the same plane, then for any vector A on this plane, there is

And there is only a pair of real numbers 1, 2, so that A 1e 12e2. (The vectors E 1 and E2 of non-* * lines serve as a set of bases for all vectors in this plane) 22. Coordinate formula of the point: the set point is the point on the line segment12,65438.

The coordinates of this point are

x 1x2y 1y2

, is the midpoint formula. When 1,.

1 1

23, the number of plane vector product:

(1) Abakosa 0, B0,0180. The product of zero vector and arbitrary vector is 0.

⑵ Properties: If both A and B are nonzero vectors, then ① abab0. ② When A and B are in the same direction, abab when A and B are reversed.

2

When, abab；; Aaaa or a.③abab

2

⑶ Operation method: ① ABBA; ② Ababa cloth; ③abcacbc。

⑷ Coordinate operation: Let two non-zero vectors ax 1, y 1, bx2, y2, then ABX 1x2Y 1Y2.

222

If ax, y, then axy,

Or let a be ax 1 and y 1, then abxx 12yy 12bx2, y2,

0.

five

extreme

The first chapter trigonometric function

1.

Positive angle: the angle formed by counterclockwise rotation is called positive angle.

Divide the zero-degree angle according to the direction of edge rotation: if a ray does not make any rotation, we say it forms a zero-degree angle. Angle negative angle: the angle formed by clockwise rotation is called negative angle.

The first quadrant angle of {α| k2360 < α < 90+k2360, k ∈ z}

The second quadrant angle {α | 90+K2360 < α < 180+K2360, k ∈ z} belongs to the third quadrant angle {α|180+K2360 < α < 270+K2360. K∈Z} or {α |-90+K2360 < α < K2360, K ∈ z} (inter-image angle): When the last edge of an angle coincides with the coordinate axis, it is called an on-axis angle, which does not belong to any quadrant. 2. Representation of the same angle on the final edge: all angles are the same as the final edge of angle α. 3. Several special position angles:

(1) The angle of the terminal edge on the non-negative semi-axis on the X axis: α = K2360, k ∈ z.

⑵ The illegal terminal angle on the X axis to the semi-axis is α = 180+K2360, K∈Z ⑵ The terminal angle on the X axis is α = K2 180, k ∈ z..

(4) The terminal angle on the Y axis: α = 90+K2 180, and the terminal angle on the coordinate axis: α = K290, k ∈ z..

[6] terminal edge angle on y = x: α = 45+k 2180, k ∈ z.

(7) y =-x: α =-45+k 2180, K ∈ Z or α = 135+K2 180, and k ∈ z ∈ 2 ends in coordinate axis or four quadrant angle.

4. radian: in a circle, the central angle of an arc with a length equal to the radius is called 1 radian angle, which is represented by the symbol rad. 5.6. If the central angle α of a circle with radius r is L, then the related formula of angle α 7. Conversion between angle system and arc system. Unit circle: in rectangular coordinate system, we call a circle with the origin o as the center and the unit length as the radius as the unit circle.

9. Use the unit circle to define the trigonometric function of any angle: let α be any angle, and its terminal edge intersects the unit circle at point P(x, y). Then: (1) Y is called sine of α, and it is recorded as sinα, that is, (2) X is called cosine of α, and it is recorded as cos α (3).

Y is called the tangent of α, and is denoted as tanαx22.

10 . sincos 1sin； cosine

The basic relation α≠kπ+ of trigonometric functions with the same angle

The inductive formula of 1 1. trigonometric function:

πnis(k∈Z):ant2cos

Male sink2sin type cosk 2 cos-tank 2 tank injection kZ

Male crime cos

cosine

Koskoskos

Male sinsin type coscos tetratantan

Male sincos

2

Gong xinke

2

Cosney type

22

various food crops

2

Liutanko

2

Note: the period of ysinx is 2π; The period of y|sinx| is π; The period of y|sinxk| is 2π; Ysin|x| is not a periodic function.

13. Method of obtaining image of function yAsin(x):

y=sin(x+)ysin(x)y①y=sinx

Periodic transformation

Pan left or right || Units

Translation transformation, period transformation and amplitude transformation.

Asin (X)

②y = sinxysinxysin(x)yasin(x) 14。 simple harmonic motion

① Analytical formula: yAsin(x), x[0,+] ② Amplitude: A is the amplitude of this simple harmonic vibration. ③ Period: T4 Frequency: f=

Amplitude transformation

2π

1

T2π

⑤ Phase and initial phase: X is called phase, and the phase when x=0 is called initial phase.

Chapter II Plane Vector

1. vector: In mathematics, we call a quantity with both magnitude and direction a vector. Quantity: We call the quantity with only size and no direction quantity. 2. Directed line segment: A directed line segment is called a directed line segment. Three elements of a directed line segment: starting point, direction and length.

3. Length (modulus) of vector: the size of vector AB, that is, the length (or modulus) of vector AB, is recorded as |AB|.

4. Zero vector: A vector with a length of 0 is called a zero vector, which is recorded as 0, and the direction of the zero vector is arbitrary.

Unit vector: A vector with a length equal to 1 unit is called a unit vector.

5. Parallel vectors: non-zero vectors with the same or opposite directions are called parallel vectors. If vectors A and B are two parallel vectors, they are usually expressed as A ∑ B. ..

Parallel vectors are also called * * * line vectors. We stipulate that the zero vector is parallel to any vector, that is, any vector A has 0 ∑ a..

6. Equal vector: A vector with the same length and direction is called an equal vector. If vectors a and b are two equal vectors, it is usually recorded as a = b.

BC=b，b，7。 As shown in the figure, given a non-zero vector A, take any point A on the plane, let ab=a, then the vector AC is called the sum of A and B, and recorded as ab.

AbABBCAC。

Vector addition: The operation of finding the sum of two vectors is called vector addition. This method of finding vectors is called the triangle rule of vector addition.

8. For zero vector and arbitrary vector A, we specify that a+0 = 0+a = a..

9. Formula and algorithm: ① a1a2+a2a3+...+ana1= 0 ② | a+b | ≤| a |+b |.

(a+b)+ca(b+c)③a+bba④

10. Inverse quantity: ① We stipulate that the vector with the same length and opposite direction as A is called the inverse quantity of A, and it is recorded as -a ... A and-A are opposite.

Quantity.

② We specify whether the inverse of zero vector is zero vector.

③ The sum of any vector and its opposite vector is zero, that is, a+(-a)(=-a)+a=0.

④ If A and B are opposite vectors, then a=-b, b=-a and ab=0.

⑤ We define a-b=a+, that is, subtracting a vector is equal to adding the inverse of this vector. (-b)

1 1. Vector multiplication: Generally speaking, we stipulate that the product of real number λ and vector A is a vector, and this operation is called vector multiplication. Write it down, it is

The length and direction are specified as follows: ①|a|a|② When λ > 0, the direction of A is the same as that of A; When λ < 0, the direction of is the same as that of a.

The opposite direction; When λ=0, a=0.

()a 12。 Algorithm: ①

②()aaa

③(ab)=ab

()a(a)(a)(ab)=ab④⑤

13. Theorem: For vectors a(a≠0) and B, if there is a real number λ that makes b=a, then A and B are * * * lines. On the contrary, vectors a and b are known.

* * * line, a≠0, the length of vector B is μ times the length of vector A, that is |b|=μ|a|, then when A and B are in the same direction, there is B = A；; When a man

Opposite to the direction of B, there is b=a. Then the following theorem is obtained: Vector a(a≠0) and straight line b*** make B = A if and only if there is a real number λ.

14. Basic theorem of plane vectors: If e 1 and e2 are two non-linear vectors in the same plane, then any vector A in this plane has an AND.

There is only one pair of real number 1 and 2, so it is a 1e 12e2. We call the vectors e 1 and e2 of non-* * lines as a set of bases representing all vectors on this plane.

Bottom.

15. Angle between vectors A and B: Two non-zero vectors A and B are known. For OAa and OBb, call AOB (0 ≤θ≤ 180).

When the included angle θ between vectors A and B is 0, the directions of A and B are the same; When θ = 180, A and B are opposite. If the angle between A and B is 90, we say that A is perpendicular to B, and we call it ab.

16. Supplementary conclusion: It is known that vectors A and B are two vectors that are not * * * lines, m, n∈R, and m=n=0 if manb0.

17. Orthogonal decomposition: decomposing a vector into two mutually perpendicular vectors is called orthogonal decomposition of vectors.

18. The coordinates of the sum (difference) of two vectors are respectively equal to the sum (difference) of the corresponding coordinates of these two vectors. That is, if a(x 1, y 1) and b(x2, y2), then

ab(x 1x2，y 1y2)，ab(x 1x2，y 1y2)

19. The coordinates of the product of a real number and a vector are equal to the corresponding coordinates of the original vector multiplied by this real number. That is, if a(x 1, y 1), then a(x 1, y 1).

20. if and only if x 1y2-x2y 1=0, the line of vectors a and b (b≠0)**.

x 1x2y 1y2

2 1. Fixed-point coordinate formula: P 1PPP2, the coordinate of point p is (,).

1 1

(1) when the point p is on the line segment P 1P2, it is called the inner bifurcation point of the line segment P 1P2, and when the point p is on the extension line of the line segment P 1P2, it is called the outer bifurcation point of the line segment P 1P2, λ.

B

Then OCOAOB, where λ+μ= 1

23. scalar product (inner product): given two non-zero vectors a and b, we weigh |a||b|cos as the scalar product (or inner product) of a and b, and record it as a2b, that is, a2b=|a||b|cos. Where θ is the included angle between a and b,

|a|cos(|b|cos) is called the projection of vector a in direction b (b in direction a). We specify the number of zero vectors and arbitrary vectors.

The product is 0.

24. Geometric meaning of A2B: The product of quantity A2B is equal to the product of the length of A |a| and the projection of B in the direction of A |b|cos.

25. product of quantity algorithm: ① a2b = b2a2 (λ a) 2b = λ (a2b) = a2 (λ b) ③ (a+b) 2c = a2c+b2c222222224 (ab) a2abb ⑤ (ab) a2abb ⑤.

26. The product of two vectors is equal to the sum of the products of their corresponding coordinates. That is abx 1x2y 1y2. Then:

22

2

① If a(x, y), then |a|xy, or |a|. If the coordinates of the starting point and the midpoint of the directed line segment representing vector A are (x2x 1, y2y 1) respectively.

(x 1, y 1)(x2, y2), then a, |a|

(x 1, y 1)(x2, y2)② let a and b, then abx 1x2y 1y20ab0.

(x 1，y 1)(x2，y2)27。 Let A and B be nonzero vectors, and A, B and θ be the included angle between A and B. According to cross product's definition and coordinate table,

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Display available: cos

|a||b|

Chapter III Triangular Identity Transformation

Cs 1。 The concise cosine formula C (α+β) of the sum of two angles: OOS2. Concise cosine formula C (α-β) for the difference between two angles: C.

csocsnisniso

Koskosnis

3. The formula features of the sum (difference) cosine formula of two angles: ① left plus sign and right minus sign. ② Sum and difference of products of homonym functions. ③ α and β are called single angle, α β.

The cosine of sum (difference), called complex angle, is obtained from the sine and cosine of a single angle. ④ "use", "opposition" and "change"

Is4。 The abbreviation of sine formula of sum of two angles is S (α+β): NIS5. Abbreviation of sine formula of two angular differences S (α-β): n

Isocosine curve

Niszk

6. The formula characteristics and uses of the sum (difference) sine formula of two angles: ① The left and right operation symbols are the same. ② On the right is the sum and difference of the products of different nominal functions, and the sine value.

Tisso

The first part is trigonometric function and trigonometric identity transformation.

The representation of the corner of the test center 1. Representation of the same angle of the terminal edge:

All angles that are the same as the terminal edge of the angle, plus the angle, can form a set: {β | β = K2360+α, k ∈ z} 2. Representation method of quadrant angle: the set of the first quadrant angle is {α, the set of the second quadrant angle is {α, and the set of the third quadrant angle is {α.

|k2360