Trigonometric function in high school, like a stumbling block, makes many students flinch and fear. The study of trigonometric function in junior high school directly affects the study of trigonometric function in senior high school, because junior high school is the foundation of senior high school. So, what are the knowledge points of trigonometric function in junior high school? What are the junior high schools for trigonometric function formulas? How to recite these formulas? How to learn trigonometric function in junior high school to lay a good foundation for senior high school? Don't worry, here is your answer.
What are the knowledge points of trigonometric function in junior high school and how to learn it?
Steps/methods
Pythagorean Theorem: The sum of squares of two right angles A and B of a right triangle is equal to the square a2+b2=c2 of hypotenuse C.
2. As shown in the figure below, in Rt△ABC, ∠C is a right angle, then ∠A's acute trigonometric function is (∠A can be replaced by ∠B):
3. The sine value of any acute angle is equal to the cosine value of other angles; The cosine of any acute angle is equal to the sine of the remaining angles.
4. The tangent of any acute angle is equal to the cotangent of its complementary angle; The cotangent of any acute angle is equal to the tangent of its complementary angle.
Trigonometric function values of special angles of 5, 0, 30, 45, 60 and 90 (important)
6. Sine and cosine increase or decrease:
When 0 ≤ α≤ 90, sinα increases with α and cosα decreases with α.
7. Increase or decrease of tangent and cotangent: When 0
What are the knowledge points of trigonometric function in junior high school and how to learn it?
What are the knowledge points of trigonometric function in junior high school and how to learn it?
What are the knowledge points of trigonometric function in junior high school and how to learn it?
What are the knowledge points of trigonometric function in junior high school and how to learn it?
Next, you should be familiar with the junior high school formula of trigonometric function.
Constant deformation formula of trigonometric function;
Trigonometric function of sum and difference of two angles in junior middle school;
cos(α+β)=cosα cosβ-sinα sinβ
cos(α-β)=cosα cosβ+sinα sinβ
sin(α β)=sinα cosβ cosα sinβ
tan(α+β)=(tanα+tanβ)/( 1-tanαtanβ)
tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)
Double angle formula of trigonometric function in junior middle school;
sin(2α)=2sinα cosα
cos(2α)=cos^2(α)-sin^2(α)=2cos^2(α)- 1= 1-2sin^2(α)
tan(2α)=2tanα/[ 1-tan^2(α)]
Triple angle formula of trigonometric function in junior middle school;
sin3α=3sinα-4sin^3(α)
cos3α=4cos^3(α)-3cosα
Half-angle formula of trigonometric function in junior middle school:
sin^2(α/2)=( 1-cosα)/2
cos^2(α/2)=( 1+cosα)/2
tan^2(α/2)=( 1-cosα)/( 1+cosα)
tan(α/2)= sinα/( 1+cosα)=( 1-cosα)/sinα
General formula of trigonometric function in junior middle school:
sinα=2tan(α/2)/[ 1+tan^2(α/2)]
cosα=[ 1-tan^2(α/2)]/[ 1+tan^2(α/2)]
tanα=2tan(α/2)/[ 1-tan^2(α/2)]
Formula of product sum and difference of trigonometric function in junior middle school;
sinαcosβ=( 1/2)[sin(α+β)+sin(α-β)]
cosαsinβ=( 1/2)[sin(α+β)-sin(α-β)]
cosαcosβ=( 1/2)[cos(α+β)+cos(α-β)]
sinαsinβ=-( 1/2)[cos(α+β)-cos(α-β)]
Junior high school trigonometric function and differential product formula;
sinα+sinβ= 2 sin[(α+β)/2]cos[(α-β)/2]
sinα-sinβ= 2cos[(α+β)/2]sin[(α-β)/2]
cosα+cosβ= 2cos[(α+β)/2]cos[(α-β)/2]
cosα-cosβ=-2 sin[(α+β)/2]sin[(α-β)/2]
What are the knowledge points of trigonometric function in junior high school and how to learn it?
What are the knowledge points of trigonometric function in junior high school and how to learn it?
Finally, how to learn trigonometric function in junior high school to master it well and lay a solid foundation for trigonometric function in senior high school?
Because junior high school trigonometric function is actually the basis of senior high school trigonometric function, so I will give an example of senior high school:
I remember one year, a senior one student came to me and said that he had studied mathematics in senior one, hoping I could give him some advice. He came to my office with a set of documents, with a question on it:
y=sinx23sinxcosx4cosx2
Find the maximum of this function.
I think a senior one student can't even solve this problem, which shows that his level is too average. I can explain this problem to him in a few words, but I can't just tell him this problem. I have to consider what the child's problem is, otherwise he won't do the same problem.
I asked him, "Will the power reduction formula?"
He said he didn't know.
I thought I met a "master" today. I continued to ask, "Can you know the double angle formula of trigonometric function?"
He thought for a moment: "I don't remember."
I continued to push back: "Can you know the trigonometric function of the sum and difference of two angles?"
He thought about it: "sin(αβ) seems to be equal to sinαsinβcosαcosβ."
I want to jump. A freshman can't remember the trigonometric function of the sum and difference of two angles. What else can I say? But I'm also stubborn. I usually help students, no matter how bad they are, I will help them to the end. I want to throw caution to the wind today. I have to dig out the root of his inability to go back and ask him, "Do you know the trigonometric function theorem at any angle?"
He said he didn't know.
Later, go back to the next day's content: "Do you know the theorem of acute trigonometric function?"
He said, "Teacher, can you be more specific?"
I said, "What is sinα equal to in a right triangle?"
His eyes lit up: "sinα is equal to the ratio of the opposite side of the hypotenuse."
I said, "That's it." Ask again: "What is cosα equal to?"
"Cosine α is equal to the ratio of the adjacent side to the hypotenuse."
"What about tanα?"
"It means that the opposite side is greater than the adjacent side."
I finally breathed a sigh of relief and said, "Son, you are so amazing that you even remember this thing. Start with it."
In order to solve this student's problem, I have been giving him back the contents of the second grade, starting from the second grade.
I said, "According to my thinking, we should translate this right triangle to the bottom of the rectangular coordinate system. You see that the hypotenuse in the rectangular coordinate system becomes the terminal edge of an angle, so how do you say that sinα equals? What is cosα equal to? "
As soon as he thought about it, the definition of trigonometric function at any angle appeared, and then with trigonometric function at any angle, I led him to deduce the basic relationship, square relationship, quotient relationship and reciprocal relationship between trigonometric functions at the same angle, all of which were deduced by himself. I continue to guide the students forward. As a result, under my guidance, the student spent two hours, starting from the definition of acute trigonometric function, and deduced all the formulas of trigonometric functions he learned in high school. I watched, his nose was sweating and he was very devoted.
I said, "Let's call it a day. I think you have solved all trigonometric functions in these two hours. "
He was taken aback and asked, "Teacher, how long has it been? Did it really take two hours? "
I said, "Look at your watch. We leave at eight o'clock. Look, it's past ten now. "
He said, "Teacher, learning is really interesting! I have studied mathematics for so many years, and I have never found such a feeling once. How can I calculate all trigonometric functions in these two hours? "
I smiled and asked, "Do you still need to remember the formula of trigonometric function now?"
He said, "I don't need to remember. I'm sure I can remember now. Because I can deduce it, am I still afraid? "
On the basis of understanding, it is a good way to remember. When you encounter a formula that you can't remember, you can deduce it yourself. Even if I can't remember it at the moment during the exam, it's too late to push it now. And you have deduced it several times, and that formula will gradually become your eternal memory.
It can be seen that we should remember on the basis of understanding. In fact, many questions, you understand, just remember; If you don't understand, hard memory may not be remembered for a long time. Even if you remember it, you will soon forget it.
Many theorems in mathematics are difficult for you to write down, but if you prove this theorem again, it will be vividly displayed in front of you, and you don't have to remember this theorem.
What are the knowledge points of trigonometric function in junior high school and how to learn it?
What are the knowledge points of trigonometric function in junior high school and how to learn it?
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Matters needing attention
After junior high school understands trigonometric functions, you can draw inferences from one another, and in this way, there are many formulas. If we recite these formulas, the burden is very heavy. But my students basically don't need to remember the formula of trigonometric function, and they can master it well. I asked the students to deduce these formulas in detail to see how they were obtained and push them down step by step along the source. After the students pushed it once, they felt that the formula was just like their own invention. It's easy to remember the formula again, even if they forget it, just push it again from the beginning.