my scientific age

Chapter 19
The trigonometric function line solves the inequality through three solutions.

sine line,
cosine line,
Tangent,
The main core is the direction of the directed line segment with the trigonometric value and the positive and negative length of the trigonometric value, and the absolute value.

After carefully reading the definition and content of the trigonometric function line solution inequality, Yu Hua held a pencil and drew a standard rectangular coordinate system consisting of the Y-axis and the X-axis on the scratch paper. The center point was marked with 0, and then the radius was 1. A circle is drawn at the distance of , and an extension line is drawn from the center point 0 to the first quadrant, crossing the circle.

The angle between the extension line and the center point is denoted α, the intersection point of the extension line and the circle is A, the X-axis vertical line passing through point A is denoted as B, and the vertical point is denoted as B.

"So, the sine line is a directed line segment → BA, a cosine line is a directed line segment → OB, and a tangent line is a directed line segment → CD. The second quadrant should be drawn like this..." Yu Hua looked at it with relish, and it was difficult to understand the limit of learning last night. The knowledge points of trigonometric function lines are simple and easy, and I feel that the whole body is full of strength again. The pencil redraws a rectangular coordinate system and a circle on the scratch paper, and draws the trigonometric functions of the second, third and fourth quadrants according to the knowledge points. Wire.

Four different quadrants of trigonometric function lines are drawn, followed by a test question on the solution of inequalities by trigonometric function lines, derived from Hardy, a professor of mathematics at Cambridge University.

What is the variation interval of one of X that makes sin x ≤ cos x.

"According to the trigonometric function line, sinx=BA, cosx=0B, in order to make sinx≤cosx established, the change interval should be -3π/4≤x≤π/4, it is still very simple, just remember the formula and set it directly Go up and you're done." Yu Hua quickly calculated, the scratch paper quickly drew a rectangular coordinate system and a circle, and drew an extension line from the center point 0 to the first quadrant to cross the circle, and marked the corners and intersections. Unravel the test questions.

This problem only needs to find the corresponding trigonometric function line. As long as the line is found, it is easy to solve. It only needs to calculate the value range of X. This is not difficult for Yu Hua, who is a scumbag in a primary school.

Simple and easy.

Looking further down, Yu Huale has a lot of test questions, the number is far more than that of analytic geometry. There are more basic questions and variation questions about trigonometric function line solution inequalities, which are basically written by Professor Hardy of Cambridge University, with different levels of difficulty. The purpose of rising is to improve the proficiency of students and increase their experience.

Of course, in the eyes of countless students, Professor Hardy's good intentions have completely turned into meticulous torture.

"Kai Chong Kai Chong..." Yu Hua rubbed his hands a little excitedly, his heart was full of fighting spirit, and he let out a mouthful of white mist. Others were terrified of this wave of experience, and he was happy with it.

Today, Yu Hua has basically mastered about 80% of the basic knowledge points of high school arithmetic, and the remaining 20% ​​are all difficult points, which require a lot of time and energy to overcome. The trigonometric function line is one of them.

The more test questions and the richer the experience, the closer the primary school scum to the National Tsinghua University goal will be.

Rush!
With a clear mind and a sensitive mind, Yu Hua did several problems in one go, and became more and more proficient in solving inequalities with trigonometric function lines. Soon, he came to the final variable problem.

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 deserves to be the finale question, a complex shape of trigonometric function line + inequality + set.

Yu Hua was stunned for a moment, feeling a hint of difficulty, and a challenge in his heart. He drew the standard coordinate system and unit circle on the scratch paper, and then drew the extension lines of the first and second quadrants to complete the drawing.

(1)∵In [0, 2π), sinπ/4=sin3π/4=√2￣/2, 0A, 0B are the end edges of π/4 and 3π/4, respectively. It can be seen from the sine line that sinα≥ The end edge of the angle of √2￣/2 is within the inferior arc AB,
The solution set of ∴sinα≥√2￣/2 is,

｛α｜π/4+2kπ≤α≤3π/4+2kπ，k∈Z｝;
(2)，∵在［0，2π）内，cosπ/3=cos5π/3=1/2……

While deducing calculations, he followed the standard to solve the problem, wrote down the steps to solve the problem, and wrote eloquently for 10 minutes. Yu Hua finally put down his pencil, with a sense of accomplishment on his face.

The scratch paper is full of a dizzying array of mathematical symbols and characters that indicate that the tricky point of solving inequalities with trigonometric lines has been fully grasped.

Proud in his heart, Yu Hua came back to his senses and suddenly noticed a person standing beside him. Looking up, he saw a boy in a black tunic standing by the long table, with a thin body, round glasses, and an exuding gleam on his body. There is a strong scholar atmosphere, his face is stunned, and there is an incredible color in his eyes.

Yu Hua felt the other party's gaze and was a little puzzled: "Uh... what's the matter with you?"

This person is also a student of the first class of science. I forgot what Yu Hua was called. His grades seemed to be not very good on weekdays, and he seemed to be studying hard again.

"Gollum."

The thin boy swallowed his throat, looked at Yu Hua, his eyes were full of shock, he hurriedly asked, "Yu Hua, can you do this trigonometry problem?"

Trigonometry!
This is a recognized super difficulty in the first class of science. Whether it is a class scumbag or a class leader, every time they face the problem of trigonometry, they all cry.

However, what does he see now?
Yu Hua, a classmate, solved the final problem of solving inequalities of trigonometric function lines in trigonometry.

This problem solved by Yu Hua has been seen by thin boys before. It is a classic problem recognized by the third-year science class. It came from Professor Hardy of Cambridge University. Must be drowsy, and even the symptoms of insomnia have been alleviated a lot.

"Uh, I understand a little bit." Feeling the gaze from the thin boy, Yu Hua froze for a moment and said modestly.

modesty.

A true scholar will not be arrogant and arrogant, and modesty is the last word.

Hearing Yu Hua's answer, the thin boy showed an embarrassed expression, and there was a hint of hope in his eyes: "Yu Hua, can you give me some ideas for explaining the problem?"

"No problem, you should know the definition of the trigonometric function line, right?" Yu Hua nodded in response, explaining the problem and clarifying the confusion. This kind of trivial matter is not worth mentioning, and he is still very helpful.

"A little." The thin boy nodded with joy in his heart. He found a round stool and sat next to Yu Hua, looking like he was listening.

"This problem looks difficult, but it's actually very simple, as long as you understand the definition of the trigonometric function line and master the induction formula for conversion, you look at the coordinate system, my problem-solving idea is this, first determine the trigonometric function line in the unit circle , the first answer is the first and second quadrants. From [0, 2π), sinπ/4=sin3π/4=√2￣/2, and sinα≥√2￣/2 can be obtained by combining the sine line solution."

Seeing the attitude of his classmates, Yu Hua was very relieved. He opened the scratch paper he had just written, pointed to the first corner α of the final question, and talked while talking.

(End of this chapter)