Chaos in mathematics-angle and radians

Because there is no accurate understanding of the concept of radians, we do not have a good understanding of trigonometric functions (sin, cos), and trigonometric functions are the most frequently used functions in advanced mathematics and analog electronics. As a result, this part of mathematics has been subconsciously rejected by my brain. I am very depressed when I find π and E in various forms that are "inexplicable. Forcing the brain to learn these patterns forces me to vomit like a fly-carrying snack.

In order to save my engineering life as much as possible, I 'd better understand it from the ground up.

When learning ry, the first learning "element" is the line segments (straight lines, edges) and corners. Then there are a variety of more complex graphics, and the nature of these graphics is also basically through the first line and angle. When measuring a line segment, we use "length": a line segment with a length of 1, a line segment with a length of 2, or even, there are also line segments with a length of '\ SQRT {2}' (in other words, the irrational number is discovered by measuring the diagonal length of a square with a side length of 1 ). Therefore, for a line segment, we use a real number to measure it, or distinguish a line segment from another line segment (the same length of line estimation, through translation and rotation can be re-synthesized "one ", the other one disappears because there is no difference ~). Line Segments correspond to real numbers in a one-to-one manner. In this way, functions with real numbers as the defined domain can correspond well to functions with "line" as the graph, after the introduction of Cartesian coordinate systems, the powerful force is revealed.

In terms of online measurements, humans naturally choose a more optimized forward direction ". (This also guides the direction for the later angle .)

For the angle, the angle is measured in the first place. 360 °, 180 °, or 90 °. This is very "good ". The brain is always preemptible. From an early age, people can describe it like this and think that "it is the most natural", "no doubt", and "No need to think ". And this result is very "beautiful", because it can be well described for our daily use or frequently encountered "corners", for example, turning around a circle is 360 °, the plane is 180 °, the right angle is 90 °, and the angle between the southeast and the south is 45 °. This is a comfortable integer. (Why are they all easy-to-use integers? Because 360 is divisible by 1, 2, 3, 4, 5, 6, 8, 9, and 10, the smallest integer that can be divisible by such a majority is 360. People choose 360 ° for a week. Of course there is a reason)

From this point of view, the "radians" were defeated.

"How many radians is the right angle ?"

"π/2"

"What is π ?"

"An infinitely non-repeating decimal number, its value is about 3.1415 ......"

For an extremely common angle in daily life, the radians cannot even produce an "exact" value in the daily sense. Isn't that disgusting? I would never really agree to use this ugly thing to represent the angle.

Since then, the angle-related mathematics has gone through my tragedy. If you do not want to learn, you will be confused. If you do not want to learn, you will be confused. When I was young, I did not know how to think and learn. I did not know how to relate what I learned. It was even more tragic.

There are some problems with the angle System: The angle is 60, and people only define the degree, minute, and second, without further precision (for daily life, this precision is enough.) The angle is not continuous. Of course, we have a way to keep it continuous, but it is something that future generations have added to "mathematical research ", these things have no significance for people in daily life and will not be used by everyone. For those who are "doing mathematical research", they should leave the familiar 10-digit system, however, it is not a good choice to continue in the original 60 hexadecimal notation. Think about it. When studying the edge length, we still use the 10 hexadecimal notation, it turns into a 60-to-60 (or a 60/10-to-mixed-to-order). How awkward is it, and how much mental burden is it. Concise and consistent is a virtue ~!

Therefore, when angle is used to study angle and trigonometric function, it is a static and special case study. In the era of angle, sin is not a "function", but a ratio. It only represents a "opposite edge than a straight angle edge". In order to distinguish it from the "function sin, let's first write the "sin with the length ratio of the edge into uppercase (SIN ". People know that "as long as the angle size remains unchanged, this ratio will not change ". So people want to know the angle (the sin ratio), and the length of one side, they can calculate the length of the other side. This is what people are measuring, sailing, the most common tasks to be solved in astronomy. We can calculate the sin value corresponding to each angle to form a table (sine table ).

The easiest way to make a table is to draw a lot of triangles of various angles, and then determine the length of the right side and the right corner side. After division, the sin ratio of the corresponding angle is obtained. However, the accuracy is of course very poor. It is better to use the "Algebra" method to calculate the sine value of another angle from the sine value of a known angle, so that it can be very accurate. Sin values of some special angles are precisely determined, such as sin (30 °) = 0.5. If we know the relationship between sin (15 °) and sin (30 °, the sin (15 °) value can also be accurately calculated, which can avoid "measurement error" and produce a sine table with very high accuracy. So we study the relationship between sin values of the half-width, double-angle, triple-angle, and so on, and product and difference products, the cosine theorem has become the main content of ancient triangle science. One goal is to obtain several other values (angles and edges) of a triangle ).

Although there are some problems, the angle is the most intuitive way to measure the angle. The most intuitive but not necessarily the most "scientific" or "the best in mathematics ".

This state lasted until the 18th century, and The radian system officially emerged.

First, we will introduce the history of radian generation (from Baidu encyclopedia)

The basic idea of radians is to make the circle radius and circle perimeter have the same measurement unit, and then use the corresponding arc length and circle radius to measure the angle. This idea originated from India. The famous Indian mathematician ariliyvituo (476? -550 ?) The circumference length is 21600 points, and the radius of the circle is 3438 points (that is, the circumference rate is π = 3.142). However, the concept of radian is not explicitly proposed by aliliypeta. Strictly speaking, the concept of radians is drawn by the Swiss mathematician Euler (1707-1783, Euler is a god man !) Introduced in 1748. Euler is different from ariliypeton. If the radius is set to 1 unit, the arc length of the semi-circle is π, And the sine value is 0, which is recorded as sin (π) = 0. Similarly, the arc length of the circumference of 1/4 is π/2, and the sine is 1, which is recorded as sin (π/2) = 1. Thus, the center angle of the semi-circular and the 1/4 arc is represented by π and π/2 respectively. Other angles can also be pushed based on this type. (Previously, people used the length of a line segment to define trigonometric functions. In his epoch book "Introduction to infinitely small analysis" published in 1748, Euler proposed that the trigonometric function is the ratio of the corresponding trigonometric function line to the circle radius, and set the circle radius to 1, this greatly simplifies the study of trigonometric functions, which is one of Euler's important achievements in the history of mathematics. In addition, Euler puts forward the concept of radians in chapter 8 of the above work. He believes that if the radius is taken as a unit length, then the length of the semi-circle is π, And the sine of the center angle is 0, that is, sin (π) = 0. Similarly, the arc length of the semi-circle is π/2, and the sine of the center angle is 1, which can be recorded as sin (π/2) = 1)

The proposal of radians and trigonometric curve makes the research of triangles enter the "function" stage. Developed a modern triangle. The radian is "the arc length of the angle is longer than the upper radius". My question is: "The radius depends on the side of the angle and you can naturally think of it. Why is the arc rather than the other, to define radians?" If no answer is found, you can guess:

Perhaps the most intuitive and simple idea is that the side to which the angle is compared with the upper "radius", but in this case, the angle increases constantly. Once the angle exceeds 180 °, the length of this side will decrease, which does not meet the expectation of increasing the angle. You can only take another path. People have thoroughly studied and "simple" shapes and circles. The circle is harmonious! It is the most "symmetric" Image in nature. Therefore, if an arc is used to define an angle, the angle is then equals, then equals, and the description is defined, there are also corresponding "Images" that will not change at all. But for the same angle, different radius will obviously be different, which is not good, so we should define it with "arc-length ratio radius.

With the radians, We will associate the angle with the "length. In addition to the fact that the radian system does not have the characteristics of "The angle system is so integer", all other aspects are "not the angle System", and the advantage of this "integer" is of little significance in mathematics research. Radians do not have the advantages of angle. The definition naturally communicates the "angle, arc length, radius" and defines itself as a real number with the same length.

Such a definition has a profound influence on "Triangle learning. In this case, sin (x) has so many wonderful and concise properties and expressions. Note that sin (X) is no longer a static ratio, but a "function". This function is a "unit circle, starting from the X axis to the arc length of any point, the relationship between the distance from the point to the X axis ". What changes will happen to the distance from the point to the X axis as the arc length increases. Sin (x) can do all the things sin (x) requires. Therefore, the original ratio (SIN) can also be unified into the calculation of the dependent variable obtained by the sin function given the independent variable. And it has more than that.

Among them, the most "Famous" may be of the following nature:

'\ Lim _ {x \ to 0} \ frac {SiN x} {x} = 1'

The reason for this function is that '\ sin 'X = \ cos x' must be based on this limit, '\ sin 'X = \ cos x' is an important foundation for many triangle-related calculus to be "concise. If this limit is true, X in sin (X) is expressed in radians.

Another very important formula is the legendary Euler's formula:

'E ^ {IX} = Cos x + ishangzhou'

This formula is also an important foundation in advanced mathematics. It can be expressed in a concise way, and must be expressed in radians based on X in sin (X.

Okay, to sum up,

Angle system is a self-centered view of the angle, which is very useful in daily life, but there are many inconveniences in mathematics and engineering. Although radian produces π, which makes me uncomfortable, it brings about a relatively simple form of triangle used in the entire advanced mathematics.

I began to accept it.