Trigonometric function lines solve inequalities through three solution methods.
sine line,
cosine,
tangent line,
The main core is the direction of the directed line segment with the trigonometric function value and the positive and negative length of the trigonometric function value, as well as the absolute value.
After carefully reading the definition and content of the line solution of trigonometric functions to inequalities, Yu Hua held a pencil and drew a standard rectangular coordinate system consisting of the Y-axis and the
A circle is drawn at a distance of 0, and an extension line is drawn from the center point 0 to the first quadrant, passing through the circle.
The angle between the extended line and the center point is denoted as α, the intersection point of the extended line and the circle is denoted as A, a perpendicular to the X-axis is drawn through point A, and the perpendicular point is denoted as B.
"So, the sine line is a directed line segment → BA, the cosine line is a directed line segment → OB, the tangent line is a directed line segment → CD, the second quadrant should be drawn like this..." Yu Hua watched with great interest. He learned the limit last night and it was difficult to understand.
The trigonometric function line knowledge points are simple and easy. I feel that my whole body is full of strength again. The pencil redraws a rectangular coordinate system and circle on the scratch paper. Based on the knowledge points, the trigonometric functions of the second, third and fourth quadrants are drawn.
Wire.
Four different quadrants of trigonometric function lines are drawn. Next is a test question about solving inequalities with trigonometric function lines, which originated from Professor Hardy of Cambridge University.
What is the variation interval of X that makes sin x ≤ cos x hold?
"According to the trigonometric function line, sinx=BA, cosx=0B, in order to make sinx≤cosx true, the change interval should be -3π/4≤x≤π/4. It is very simple. Just remember the formula and apply it directly
Just go up and it's done." Yu Hua calculated quickly, quickly drew the rectangular coordinate system and the circle on the scratch paper, and drew an extension line from the center point 0 to the first quadrant through the circle, marking the angles and intersections respectively, and then divided by three, five, and two
Solve the test questions.
This question only needs to find the corresponding trigonometric function line. As long as the line is found, it is easy to solve. All you need to do is calculate the numerical range of
Simple and easy.
Looking further down, Yu Hua is delighted. There is a large wave of test questions, far more than analytic geometry. There are more basic test questions and modified test questions about trigonometric functions and line solutions to inequalities. Basically, they are all written by Professor Hardy of the University of Cambridge. They are layered in difficulty.
The purpose of rising is to improve students' proficiency and increase their experience.
Of course, in the eyes of countless students, Professor Hardy's good intentions turned into meticulous torture.
"Go ahead, go ahead..." Yu Hua rubbed his hands excitedly, his heart full of fighting spirit, and spit out a mouthful of white mist. Others were afraid of this wave of experience, but he was very happy with it.
Nowadays, Yu Hua has basically mastered about 80% of the basic knowledge points in high school arithmetic. The remaining 20% are all difficult points that require a lot of time and energy to overcome. Trigonometric function lines are one of them.
The more test questions you have and the more experience you have, the closer you will get to your goal of getting into Tsinghua University.
Go, go, go!
With a clear mind and a sharp mind, Yu Hua solved several questions in one go. He became more and more proficient in solving inequalities with trigonometric functions. Soon, he came to the last variable question.
Using trigonometric function lines, write the set of angles α that satisfy the following conditions.
(1)sinα≥√2￣/2;
(2)cosα≤1/2;
(3)｜cosα｜＞｜sinα｜.
It is indeed the finale question, the comprehensive form of trigonometric function lines + inequalities + sets.
Yu Hua was startled, feeling a hint of difficulty and feeling challenged. He drew the standard coordinate system and the unit circle on the scratch paper, and then drew the extension lines of the first and second quadrants to complete the drawing.
(1)∵ Within [0,2π), sinπ/4=sin3π/4=√2￣/2, 0A, 0B are the terminal sides of π/4 and 3π/4 respectively. It can be seen from the sine line that satisfies sinα≥
The terminal edge of the angle √2￣/2 is within the minor arc AB,
The solution set of ∴sinα≥√2￣/2 is,
{α|π/4+2kπ≤α≤3π/4+2kπ,k∈Z};
(2), ∵ within [0,2π), cosπ/3=cos5π/3=1/2……
While deducing and calculating, he solved the problem according to the standards and wrote down the steps to solve the problem. After writing eloquently for ten minutes, Yu Hua finally put down the pencil, with a sense of accomplishment on his face. The final problem given by Professor Hardy was solved.
The scratch paper is full of dazzling mathematical symbols and characters. Behind these characters, it is shown that the key points of solving inequalities with trigonometric functions have been completely mastered.
Feeling quite proud in his heart, Yu Hua came back to his senses and suddenly noticed a person standing next to him. He looked up and saw a boy in a black tunic suit standing by the long table. He was thin and wearing round glasses and exuded a radiant air.
He had a strong scholarly aura, his face was blank, and his eyes were filled with incredulity.
Yu Hua felt the other person's gaze and was a little confused: "Uh... what's wrong with you?"
This person is also a student in Class 1 of Science. I forgot his name, Yu Hua. His grades don't seem to be very good on weekdays, and he seems to be studying hard again.
"Gulu."
The thin boy swallowed his throat, looked at Yu Hua, his eyes were full of shock, and quickly bowed and asked: "Yu Hua, can you do this trigonometric problem?"
Trigonometry!
This is a recognized super difficult point in the Science Class 1. Whether they are the scumbags in the class or the class leaders, every time they are faced with the question of trigonometry, they all cry out.
However, what does he see now?
Classmate Yu Hua solved the final question of trigonometry, which is about solving inequalities on lines of trigonometric functions.
This question that Yu Hua solved has been seen by a thin boy. It is recognized as a classic problem in the third-year science class. It comes from Professor Hardy of Cambridge University. He went back this winter vacation to study this question specifically. He was puzzled by the solution. He looked at it for a while each time.
You must be drowsy, and even the symptoms of insomnia have been relieved a lot.
"Well, I understand a little bit." Feeling the gaze from the thin boy, Yu Hua was startled and said modestly.
modesty.
A true academic master will not be arrogant and arrogant. Humility is the last word.
Hearing Yu Hua's answer, the thin boy looked embarrassed, with a hint of hope in his eyes: "Yu Hua, can you tell me how to solve the problem?"
"No problem, you should understand the definition of trigonometric function lines, right?" Yu Hua nodded in agreement upon hearing this. He was just explaining the topic to clear up doubts. This kind of trivial matter is not worth mentioning, but he is still very willing to help others.
"A bit." The thin boy nodded, with joy in his heart. He found a round stool and sat next to Yu Hua, as if listening.
"This question looks difficult, but it is actually very simple. As long as you understand the definition of the trigonometric function line and master the induction formula for conversion, look at the coordinate system. My idea of solving the problem is as follows. First determine the trigonometric function line within the unit circle.
, the first answer is the first and second quadrants. From [0,2π), sinπ/4=sin3π/4=√2￣/2, combined with the sinusoidal line, we can get sinα≥√2￣/2
The solution set of .”
Seeing the attitude of his classmates, Yu Hua was very pleased. He opened the draft paper he had just finished writing, pointed to the first angle α question in the finale, and talked while talking.