Trigonometric function is a kind of transcendental function in elementary function in mathematics. Their essence is the mapping between the set of arbitrary angles and a set of ratio variables. The usual trigonometric function is defined in the plane rectangular coordinate system, and its domain is the whole real number domain. The other is defined in a right triangle, but it is incomplete. Modern mathematics describes them as the limit of infinite sequence and the solution of differential equation, and extends their definitions to complex system.
Because of the periodicity of trigonometric function, it does not have the inverse function in the sense of single-valued function.
Trigonometric functions have important applications in complex numbers. Trigonometric function is also a common tool in physics.
It has six basic functions:
Function name sine cosine tangent cotangent secant cotangent
Symbol sin cos tan cot sec csc
Sine function sin(A)=a/h
Cosine function cos(A)=b/h
Tangent function tan(A)=a/b
Cotangent function cot(A)=b/a
In a certain change process, the two variables X and Y, for each value of X within a certain range, Y has a certain value corresponding to it, and Y is a function of X. This relationship is generally expressed by y=f(x).
The development history of function concept
1. Early concept of function-function under geometric concept
/kloc-Galileo in the 0/7th century (Italy, 1564- 1642), in his book "Two New Sciences", almost all contained the concept of the relationship between functions or variables, and expressed the relationship between functions in the language of words and proportions. Descartes (France, 1596- 1650) noticed the dependence of one variable on another around his analytic geometry 1673. However, because he didn't realize that the concept of function needed to be refined at that time, no one had defined the function until Newton and Leibniz established calculus in the late17th century.
1673, Leibniz first used "function" to express "power". Later, he used this word to represent the geometric quantities of each point on the curve, such as abscissa, ordinate, tangent length and so on. At the same time, Newton used "flow" to express the relationship between variables in the discussion of calculus.
/kloc-Function Concept in the 0/8th Century ── Function under Algebraic Concept.
John? Bernoulli John (Rui, 1667- 1748) defined the concept of function on the basis of the concept of Leibniz function: "a quantity consisting of any variable and any form of constant." He means that any formula composed of variable X and constant is called a function of X, and he emphasizes that functions should be expressed by formulas.
1755, Euler (L. Euler, Switzerland, 1707- 1783) defines a function as "if some variables depend on other variables in some way, that is, when the latter variable changes, the former variable also changes, and we call the former variable a function of the latter variable."
Euler (L. Euler, Switzerland, 1707- 1783) gave a definition: "The function of a variable is an analytical expression composed of this variable and some numbers or constants in any way." He put John? The definition of function given by Bernoulli is called analytic function, which is further divided into algebraic function and transcendental function, and "arbitrary function" is also considered. It is not difficult to see that Euler's definition of function is better than John's? Bernoulli's definition is more general and has a wider meaning.
13.19th century function concept-function under correspondence.
182 1 year, Cauchy (France, 1789- 1857) gave a definition from the definition of variables: "Some variables have certain relationships. When the value of one variable is given, the values of other variables can be determined accordingly, so the initial variable is called independent variable. The word independent variable appeared for the first time in Cauchy's definition, and pointed out that functions don't need analytic expressions. However, he still believes that functional relationships can be expressed by multiple analytical expressions, which is a great limitation.
1822, Fourier (France,1768-1830) found that some functions have also been expressed by curves, or they can be expressed by one formula, or they can be expressed by multiple formulas, thus ending the debate on whether the concept of functions is expressed by only one formula and pushing the understanding of functions to a new level.
In 1837, Dirichlet (Germany, 1805- 1859) broke through this limitation and thought that it was irrelevant how to establish the relationship between x and y. He broadened the concept of function and pointed out: "For every definite value of X in a certain interval, Y has one or more definite values. This definition avoids the description of dependence in function definition and is accepted by all mathematicians in a clear way. This is what people often call the classic function definition.
After the set theory founded by Cantor (German, 1845- 19 18) played an important role in mathematics, veblen (American, veblen, 1880- 1960) used "set" and ".
The Concept of Modern Function —— Function under Set Theory
F. Hausdorff defined the function in 19 14 with the fuzzy concept of "ordered couple" in the outline of set theory, avoiding the two fuzzy concepts of "variable" and "correspondence". In 192 1, Kuratowski defined "ordered pair" with the concept of set, which made Hausdorff's definition very strict.
In 1930, the new modern function is defined as "If there is always an element Y determined by set N corresponding to any element X of set M, then a function is defined on set M, and it is denoted as y=f(x). Element x is called an independent variable and element y is called a dependent variable. "
The terms function, mapping, correspondence and transformation usually have the same meaning.
But the function only represents the correspondence between numbers, and the mapping can also represent the correspondence between points and between graphs. It can be said that the mapping contains functions.
Positive proportional function:
The image of the proportional function y=kx(k is constant, k≠0) is a straight line passing through the origin. When x >; 0, the image passes through three or one quadrant and rises from left to right, that is, y increases with the increase of x; When k < 0, the image passes through two or four quadrants and decreases from left to right, that is, y decreases with the increase of x.
It is precisely because the image of the proportional function y=kx(k is constant, k≠0) is a straight line that we can call it a straight line y=kx.
(another: the origin of the Chinese name "function"
In the book Algebra translated by China mathematician Li (181-1882), the word "function" was translated into Chinese for the first time, and this translation is still in use today. As for why the concept is translated in this way, the book explains that "whoever believes in this variable is a function of that variable"; "Faith" here means tolerance. )
An in-depth study of a function
Xu ruohan
When learning a function, according to the requirements of middle school, we should further study its practical application and how to change the position of the image.
Piecewise function in practical problems
[Example 1] (Wuhan, 2005) Xiaoming rides his bike from home to school in the morning, going uphill first and then downhill. The itinerary is as shown. How long will it take Xiao Ming to ride home from school if the speed of going up and down the hill remains the same when he returns?
Analysis: the speed of uphill and downhill is different, so the problem should be studied in two sections.
According to the information provided by the function image, we can know that when Xiaoming goes to school from home, the uphill distance is 3600 meters and the downhill distance is 9600-3600 = 6000 meters.
∴ uphill speed is 3600÷ 18=200 (m/min).
Downhill speed is 6000 ÷ (30- 18) = 500 (m/min).
When Xiao Ming came home, the uphill journey was 6000 meters, and the downhill journey was 3600 meters. It took 6000÷200+3600÷500 = 37.2 minutes.
Application in physics discipline
[Example 2] (Huanggang City, 2004) When a class of students explored the relationship between spring length and external force, the corresponding data recorded in the experiment were as follows:
Find the resolution function of y about x and the range of independent variables.
Analysis: According to the knowledge of physics, the spring is deformed (elongated) under the action of external force (gravity of hanging heavy objects), and the relationship between external force and pointer position can be expressed by a linear function; But the external force on each spring is limited, so we must find the range of independent variables.
According to the known data, it is found that during the spring stretching process,
Let y=7.5 and get x=275.
The function of ∴ is
Note that the dividing point between two paragraphs is x=275, not x=300.
Application of linear translation
[Example 3] In the rectangular coordinate system (Heilongjiang Province in 2005), points A (-9,0), P (0 0,3) and C (0, 12) are known. Q: Is there a point Q on the X-axis, so that a quadrilateral with points A, C, P and Q as its vertices is a trapezoid? If it exists, find the analytical formula of straight line PQ; If it does not exist, please explain why.
Analysis: Which two sides are parallel in the studied trapezoid? There are two possibilities: if, that is, the straight line CA is translated, the analytical formula of the straight line CA can be easily obtained through point P as follows.
The analytical formula of the straight line obtained after translation is
if
Translation line PA: through point C.
Get a straight line:
The line intersects the X axis at the point (-36,0).
The analytical formula of straight line is
How to understand the concept of function
Cao Yang
Function is an extremely important basic concept in mathematics. In middle school mathematics, functions and their related contents are very rich and occupy a great weight. Mastering the concept of function is very useful for future study. Looking back on the development history of the concept of function, Leibniz first adopted "function" as a mathematical term. He first put forward the concept of function in his paper 1692, but its meaning is quite different from the current understanding of function. In modern junior high school mathematics curriculum, the definition of function is "variable theory". Namely:
In a certain change process, there are two variables X and Y. If there is a unique definite value corresponding to each definite value of X within a certain range, then Y is called a function of X, X is called an independent variable, and Y is called a dependent variable.
It clearly points out that the independent variable X can take any value within a given range, and the dependent variable Y takes a unique and certain value every time according to certain laws. But junior high school does not require mastering the range of independent variables (just look at several functions to be learned in junior high school, and you will know that this definition is completely sufficient and easy to understand for junior high school students).
The concept of function is very abstract, which is difficult for students to understand. To understand the concept of function, we must clarify two points: First, we must clarify the relationship between independent variables and dependent variables. In a certain change process, there are two variables X and Y. If Y changes with X, then X is called independent variable and Y is called dependent variable. If x changes with y, then y is called independent variable and x is called dependent variable. Second, the core of function definition is "one-to-one correspondence", that is, given the value of an independent variable X, there is a uniquely determined value of a dependent variable Y corresponding to it. Such correspondence can be "one independent variable corresponds to one dependent variable" (abbreviated as "one-to-one") or "several independent variables correspond to one dependent variable" (abbreviated as "many-to-one"), but it cannot be "one independent variable corresponds to multiple dependent variables".
One-to-one, many-to-one and one-to-many.
It's a function, a function, not a function
Figure 1
Here are four examples to help you understand the concept of function:
Example 1 The length of the spring is 10 cm. When the spring is pulled by F(F is within a certain range), the length of the spring is expressed by Y, and the measured data are shown in Table 1:
Table 1
Tension f (kg)
1
2
three
four
…
Spring length y(c)
…
Is the length y of the spring a function of the tension f?
Analysis: Information can be read from the table. When the tensile forces are 1kg, 2kg, 3kg and 4kg respectively, they all correspond to the length y of a spring, which satisfies the definition of the function, so the length y of the spring is a function of the tensile force f ... Usually, the first line of the function given in tabular form is the value of the independent variable, and the second line is the value of the dependent variable.
Example 2 Figure 2 shows the highest and lowest temperatures in a certain area every month of the year.
Figure 2
Figure 2 describes the relationship between which variables? Can you regard one of the variables as a function of the other?
Analysis: Three variables are given in the figure, namely the highest temperature, the lowest temperature and the month. As can be seen from the figure, the maximum and minimum temperatures change with the change of the month, and the maximum and minimum temperatures of each month are unique, so the maximum (or minimum) temperature is a function of the month. We can also find that the highest temperature in July and August is the same, which means that two independent variables correspond to the same dependent variable. Generally speaking, the horizontal axis represents the independent variable and the vertical axis represents the dependent variable for functions given in the form of images.
Example 3 Is the relationship between the following variables a functional relationship? Explain why.
(1) the relationship between the area s and the radius r of a circle;
(2) When the car is traveling at a speed of 70km/h, the relationship between the distance S (km) traveled by the car and the time T (h) used;
(3) The area of an isosceles triangle is the relationship between its base length y (cm) and its base height x (cm).
Analysis: (1) The relationship between the area s and the radius r of a circle is that when the radius is determined, the area s of the circle is also uniquely determined, so the relationship between the area s and the radius r of the circle is a functional relationship.
⑵ The relationship between the distance S (km) and the time t (hours) used is that when the time t is determined, the distance s is also uniquely determined, so the relationship between the distance S (km) and the time t (hours) used is a functional relationship.
(3) The relationship between the base length ycm and the base height xcm is that when the base height X is determined, the base length Y is also uniquely determined, so the relationship between the base length ycm and the base height xcm is a functional relationship.
Generally speaking, the function given in the form of relation has dependent variables on the left of the equal sign and unknowns on the right of the equal sign as independent variables.
Example 4 In the following images, the one that cannot express the functional relationship is ().
Analysis: In the above four pictures, A, C and D can all represent functional relationships, because any given value of independent variable X has a unique Y value corresponding to it, but in Figure B, any given value of independent variable X has two different Y values corresponding to it, so this question should choose B.
[Question 2.9] Let M be a four-digit number less than 2006, and it is known that there is a positive integer n, so that M-n is a prime number and mn is a complete square number, and find all four-digit M that meet the conditions.