From the teacher's point of view, this is a less difficult content in high school mathematics, which is usually arranged in the first semester of senior one.
From the students' point of view, many people will encounter the first setback in high school in trigonometric function: there are too many formulas to remember; And some formulas are easy to be confused.
This paper aims to help high school students overcome the difficulty of "too many trigonometric formulas to remember". The basic idea is: remember the core formula, and then deduce the rest formulas from the core formula.
Trigonometric function has been introduced into junior high school mathematics. Introducing arbitrary angle and arc system into senior high school mathematics, redefining trigonometric function;
As shown in the above figure, the trigonometric function of any angle is defined as follows:
"Definition 1"
"Definition 2"
"Definition 3"
Please note that the following identities can be obtained immediately:
Formula 1
This formula can be regarded as an inference of Pythagorean theorem. In this paper, it is written as "formula 1". Note that there are many variations of this formula, such as:
This formula is very concise and important. This formula will be used to derive the following formula; It is also an ordinary test center. So add a box here to emphasize.
Formula 2
Formula 3
From the function point of view, the minimum positive period of sine and cosine function is.
Ancient Greek philosophers thought that the circle was the most perfect geometric figure in the world. The reason is that a circle has infinite symmetry axes; All straight lines passing through the center of the circle are its symmetry axes.
As shown in the above figure, it is the intersection of the terminal edge of sum and the unit circle.
According to symmetry, if the coordinate of a point is, the coordinate of the point is:. Therefore, the following formula is obtained:
Formula 4
Formula 5
These two formulas show that cosine function is even function; The sine function is odd function.
The terminal edge of and is the same straight line, and the intersection point with the unit circle is, so:
Equation 6
Formula 7
Formulas related to complementary angles can be derived from two pairs of formulas in the previous section:
Equation 8
Formula 9
The cosine formula of the difference between two angles is as follows:
Formula 10
Note: Cosine function is an even function, so the cosine value of sum is equal.
The above formula can be expressed in words as follows:
The cosine of the difference between two angles is equal to the product of the cosine of these two angles plus the product of the sine of these two angles.
There are many ways to derive this formula. A simpler method is to use the distance formula in cartesian coordinate system. Because the theme of this paper is "how to remember", only a schematic diagram is given here, and the derivation process of this formula will be introduced later.
Note: This formula is the core of the core formula, so it must be kept in mind. Other formulas about the sum and difference of two angles can be derived from this formula.
In this paper, it is written as "formula 10".
The cosine formula of the sum of two angles can be easily derived from the cosine formula of the difference between two angles:
Formula 1 1'
The sine formula of the sum and difference of two angles can be summarized as follows:
Formula 12
This set of formulas related to complementary angle is called "induced formula" in textbooks. Traditionally, we usually talk about inductive formulas first, and then talk about the cosine and sine of the sum and difference of two angles. However, from the students' point of view, the most puzzling thing is the inductive formula.
In fact, the inductive formula can also be derived from the sine and cosine formula of the sum and difference of two angles.
We have previously derived the following formula about the sum and difference of two angles:
On the other hand, the sine and cosine values of right angles are also known:
Substituting the sine and cosine values of right angles into the cosine formula of the sum and difference of two angles, we can get:
Formula 13
Formula 14
Formula 15
Substituting the sine and cosine values of a right angle into the sine formula of the sum and difference of two angles, we can get:
Formula 16
Formula 17
Formula 18
In the process of deriving "formula 10", our basis is Pythagorean theorem and triangle congruence, so there is nothing wrong with this and there is no "circular argument".
From the point of view of examination, it is better to remember these six formulas related to complementary angles in this way of thinking.
From the formula 10, the cosine formula of double angles can be derived:
Formula 19
Combined with the previous formula 1, there are two other forms of this formula:
Formula 20
Equation 2 1
Applying the square difference formula, another form can be obtained:
Equation 22
There are many variants of the double-angle cosine formula. In the process of solving problems, we sometimes use the following inferences:
Equation 23
Equation 24
From "formula 12", the sine formula of double angle can be deduced:
Equation 25
Combining the definition of tangent function and the properties of sine and cosine function, the following formula can be derived:
Equation 26
Equation 27
Equation 28
Equation 29
Formula 30
Equation 3 1
Equation 32
Equation 33
Equation 34
Cosine formula of sum and difference of two angles:
The above two formulas can be obtained by addition and subtraction:
Equation 35
Equation 36
Sine formula of sum and difference from two angles:
You can get the sum of the two formulas:
Equation 37
Pay attention to the following relations:
Combining the previous three formulas, the following formula can be derived:
Equation 38
Equation 39
Equation 40
In the textbooks of the last century, there was an independent chapter devoted to introducing the product and difference, and the product formula of difference. There is no special chapter in the new curriculum standard textbook of People's Education Press, but these formulas are introduced in a group of exercises. See: Exercise of Math 3.2-Compulsory 4. Textbooks are arranged in this way, probably to cultivate students' habit of exploring by themselves.
From the point of view of examination, if you are familiar with this formula, you may have more ideas when answering some questions, or improve the efficiency of answering questions. So we must attach great importance to the exercises in the textbook.
There are many ways to deduce the triple angle formula. Below we use the method of sum-difference product to deduce.
Equation 4 1
Equation 42
Equation 43
Equation 44
It should be noted that the triangle basically does not appear alone in the examination questions. Formula 4 1 and formula 4 1 may have more chances to play than the other two formulas. No matter what the form, mastering the basic methods and ideas is the right way to solve the problem, and rote memorization can't solve the problem.
Learning mathematics is learning deduction. When discussing the experience of learning mathematics, we will hear experts say from time to time, "I never recite formulas, but I always push them when I use them."
Trigonometric functions are characterized by many formulas. However, after being familiar with the derivation process, we will find that the status of these formulas is not equal. For example, the relationship between bamboo roots, branches and leaves. If we master the bamboo roots, the branches and leaves of bamboo can grow naturally.
Remember the formula, know how to use the formula, and you can score. About how to use the formula, I will introduce it with concrete examples in the subsequent articles.
Interested students can watch this first:
20 13 two questions of science mathematics in the national volume 17