What to learn after completing the fourth compulsory course of high school mathematics

Compulsory 51Linear programming.

Truth be told, there is no formula for linear programming.

It's just a series of inequalities.

And the formula of the sequence.

is equal difference: an=a1+(n-1)d

sn=[(a1+an)*n]/2

a1*n+n*(n-1)d/2

Ratio: an=a1*q (n-1).

sn=[a1（1-q^n)]/(1-q)

a1-an*q）/（1-q）

General term (find any term): an=(a1+an) d(tolerance)-1

n(number of terms)The formula for finding the number of terms is n=(an-a1) d+1

Here are some of the applications

1+2+3+..n=n(n+1)/2

2。 1^2+2^2+3^2+..n^2=n(n+1)(2n+1)/6

3。 1^3+2^3+3^3+..n^3=( 1+2+3+..n)^2=n^2*(n+1)^2/4

4。 1*2+2*3+3*4+..n(n+1)=n(n+1)(n+2)/3

5。 1*2*3+2*3*4+3*4*5+..n(n+1)(n+2)=n(n+1)(n+2)(n+3)/4

1+(1+2)+(1+2+3)+(1+2+3+4)+.1+2+3+..n)

1*2+2*3+3*4+..n(n+1)]/2

n(n+1)(n+2)/6

1+(1+1)+(1+1+2)+(1+1+2+3)+.1+1+2+3+..n)

n+1)*1+[1*2+2*3+3*4+..n(n+1)]/2

n+1)+n(n+1)(n+2)/6

1-1/(n+1)=n/(n+1)

2/2*3+2/3*4+2/4*5+..2/n(n+1)=(n-1)/(n+1)

2*3*4*..n-1)/2*3*4*..n

n+1)^2 (2n^2+2n-1) /12–1

Compulsory 5 Regardless of the order of 14523 or 12345, learn 5 after compulsory 4

For example, in terms of the nature of trigonometric function images, the compulsory test questions account for 5 points every year, and in the use of sine and cosine to understand triangles, it accounts for 12 points for major questions, and one point in the college entrance examination falls to thousands of people, and trigonometric functions cannot be ignored!

Let's give you an analysis, what are the 10 points of high school mathematics compulsory four, which are the contents that we need to master, and what is the effect of doing the questions after mastering them.

1. Arbitrary angle and radian system.

The main content includes the generalization of angles and the definition of the radian system, which is the basic content of the trigonometric function module.

2. Trigonometric functions of arbitrary angles.

The main content includes the trigonometric line problem in the unit circle and the basic relationship of the trigonometric function at the same angle.

3. Induction formula for trigonometric functions.

The main content includes all the induction formulas in the trigonometric function part, and there are many formulas that need to be memorized.

4. Images and properties of trigonometric functions.

The main content includes images of sine, cosine, and tangent functions.

The properties of sine function, cosine function and tangent function are basic knowledge in trigonometric functions and even in future studies, and students need to pay attention to them.

5. Image of the function y=asin(x+).

The main content includes the function y=asin(x+) image and image transformation.

6. Basic concepts and linear operations of plane vectors.

The main content includes the basic concepts of vectors, as well as the addition and subtraction operations of vectors and the multiplication of numbers and their geometric meanings.

7. The product of the quantity of the plane vector.

The main contents include the relevant concepts of vector quantity products, the geometric significance and properties of vector quantity products, as well as the application of vectors in geometry and the application of vectors in physics.

8. Trigonometric functions of the sum and difference of two angles.

The main contents include the sine, cosine, and tangent formulas of the sum and difference of two angles, as well as the transformation of the tangent formula of the sum and difference of two angles, and how to form a trigonometric function of asin +bcos into a trigonometric function of an angle.

9. Sine, cosine, and tangent formulas for double angles.

It is mainly the deformation formula of the double angle formula and the double angle formula, which is the more important knowledge content in the trigonometric function module.

10. Simple trigonometric identity transformation.

The focus is on the trigonometric basic formulations commonly used for trigonometric identity transformations, as well as the application of the function asin + BCOS. In exams, evaluation, proof of trigonometric identities, and the ultimate value of trigonometric functions are usually examined.

Watching online classes and listening to the teacher's explanations can be very good for learning high school mathematics compulsory 4. It's actually simple.

Do more exercises, only through a lot of practice, can you consolidate the knowledge points you understand in class.

You should lay a good foundation in mathematics, have the most basic mathematical thinking ability and the most basic way of thinking, and learn the compulsory four high school mathematics courses through careful listening and after-class summary and review.

There are a lot of induction formulas in the textbook, and it's hard to remember.

I'll teach you how to memorize the induction formula, is it good? If you can learn to be broad, you don't need to memorize so many formulas in the textbook, because it is a combination of the formulas classified in the textbook.

Convert the angle to the form k 2 or k 90°, and then memorize the mantra "odd and even do not change, sign in quadrant".

"Odd and even unchanged" means to say:

If k is an even number, then the preceding sign of the three-call difference angle function does not change

If k is an odd number, then the preceding trigonometric sign is to be changed, and the principle of change is: sin cos; cos→sin;tan→cot,cot→tan.

"Symbol to see quadrant" means to determine the final symbol according to the quadrant where the angle is located

I'll give you an example:

sin1730°＝sin(19×90°＋20°)

Step 1: Here k 19 is an odd number, so change sin to cos;

Step 2: Make sure that the terminal edge of 1730° is in the fourth quadrant, then you will know that the sign for sin1730° is " ".

Therefore, sin1730° sin(19 90° 20°) cos20°

As for how to judge the symbols of various trigonometric functions in the four quadrants, you can also remember the mantra "one is perfect; Second, it happens that the leather strings; the third is the cut; Four Cosine".

The meaning of this twelve-word mantra is to say:

The four trigonometric values for any one of the angles in quadrant 1 are " ".

In quadrant 2, only the sine is " " and the rest are " ".

The inscribed function in quadrant 3 is " and the chord function is " ".

In quadrant 4, only the cosine is " " and all the rest are " ".

If you can grasp the meaning of this text, then there is actually only one induction formula I never ask my students to memorize the induction formula in the textbook in my teaching, and ask them to understand the induction formula according to the above paragraph, and the effect is very good, you can try it too

Compulsory 4 is not difficult, it is not difficult, the vector must be grasped"Triangular swift matching or parallelogram refers to the law of the finger shape", the trigonometric function part should be familiar with the formula, how to turn the 2x angle into a once.

The first chapter of Compulsory 2 is Solid Geometry.

Chapter 2 is the positional relationship between points, lines, and planes.

Chapter 3 is Straight Lines and Equations.

Chapter 4 is Circles and Equations.

Compulsory 3: Chapter 1 Preliminary Algorithms.

Chapter 2 Statistics.

Chapter 3 Probability.

Compulsory 4: Chapter 1 Trigonometry.

Chapter 2 Planar Vectors.

Chapter 3 Trigonometric Identity Transformations.

Compulsory 5: Chapter 1 Solving Triangles.

Chapter 2 Series.

Chapter 3 Inequality.

f(sin2y) = root number (1-sin2y) = root number [(siny) 2-2sinycosy + (cosy) 2] = |siny-cosy|=siny-cosy (because y belongs to (3/4, in this case siny>0, cosy<0, and |siny|<|cosy|）

In the same way, f(-sin2y)=|siny+cosy|=-siny-cosy

Thus f(sin2y) + f(-sin2y) = -2cosy

The root number 3sin k/x has the property of a trigonometric function, and the minimum positive period of this function is 2 ( k)=2k

The adjacent maximum and minimum points are (k2, root 3) and (k 2, -root 3), respectively

On the circle x 2 + y 2 = k, substituting the coordinates of one of the points into this equation gives k 2 4 + 3 = k

k 2-4k + 3 = 0

The solution yields k = 3 or 1

Therefore, the minimum positive period for f(x) is 6 or 2

A+75 in the third or fourth quadrant.

sin(a+75)<0

sin²+cos²=1

So sin(a+75)=-2 2 3

cos（105º-a）+sin（a-105º）=-cos[180-(105-a)]-sin(105-a)]=-cos(75+a)-sin(a+75)=(2√2-1)/3

Classification of formulas.

The basic relationship of conformal trigonometric functions.

tan α=sin α/cos α

Two formulas that are commonly used for different conditions.

sin2 +cos2 =1 the adjacent angle of tan tan = 1

Acute angle trigonometric formula.

Sine: sin = opposite side of hypotenuse cosine: cos = of adjacent edge of hypotenuse tangent: tan = of opposite edge of adjacent edge of cot of opposite edge of cot.

Double angle formula.

sin2a=2sina•cosa cos2a=cos^2 a-sin^2 a=1-2sin^2 a=2cos^2 a-1 tan2a=(2tana)/(1-tan^2 a)

Triple angle formula.

sin3 =4sin ·sin( 3+ )sin( 3- )cos3 =4cos ·cos( 3+ )cos( 3- )tan3a = tan a · tan( 3+a)· tan( 3-a) triple angle formula derivation sin3a =sin(2a+a) =sin2acosa+cos2asina =2sina(1-sin 2a)+(1-2sin 2a)sina =3sina-4sin 3a cos3a =cos(2a+a) =cos2acosa-sin2asina =(2cos 2a-1)cosa-2(1-cos a)cosa =4cos 3a-3cosa sin3a=3sina-4sin 3a =4sina(3 4-sin 2a) =4sina[( 3 2) 2-sin 2a] =4sina(sin 260°-sin 2a) =4sina(sin60°+sina)(sin60°-sina) =4sina*2sin[(60+a) 2]cos[(60°-a) 2]*2sin[(60°-a) 2]cos[(60°-a) 2] =4sinasin(60°+a)sin(60°-a) cos3a=4cos 3a-3cosa =4cosa(cos 2a-3 4) =4cosa[cos 2a-( 3 2) 2] =4cosa( cos 2a-cos 230°) =4cosa(cosa+cos30°)(cosa-cos30°) =4cosa*2cos[(a+30°) 2]cos[(a-30°) 2]* =-4cosasin(a+30°)sin(a-30°) =