Obviously, these four concepts are sin, cos, tg and ctg. This is the basic element of trigonometric function. Unfortunately, many people have studied trigonometric functions for a long time, but they know these four symbols, but they don't really understand their connotation. The so-called trigonometric function is simply the proportional relationship between several sides of a right triangle. Suppose there is a right angle △ ABC, ∠ C = 90, corresponding to the hypotenuse C, ∠ A and ∠ B corresponding to the right angle sides A and B respectively.
Then, Sina = a/c, COSA = b/c, TGA = a/b, and CTGA = B/A. In fact, these four functions are invented to simplify the proportional line segments of right triangle and avoid writing many proportional expressions of line segments every time. SinA stands for the ratio of right angle to hypotenuse, cosA stands for the ratio of adjacent side to hypotenuse, tgA stands for the ratio of opposite side to adjacent side, and ctgA stands for the ratio of adjacent side to opposite side.
Make these simplest concepts clear. Whoever has more foundations in trigonometric functions need not remember them. For example, sin2A+cos2A= 1, tgA ctgA= 1, cosA tgA= sinA, sinA ctgA= cosA. Because these are directly derived from this basic concept, such as cosAtgA= sinA and sinActgA= cosA, these two formulas are upside down, and it is easy to confuse tgA with ctgA, which will be recorded as sinAtgA=cosA or careless.
CosActgA= sinA。 However, as long as we know these four basic concepts, we will know.
Never forget the confusion. Therefore, a truly efficient memory is based on thorough understanding. I have a thorough understanding, and I will never forget it for ten years or eight years, let alone say that class is over. After reading the book, I will forget it after a day.
In senior high school, the biggest change of trigonometric function is not the increase of formulas, but the expansion of basic concepts. That is to say, the value range of trigonometric function has changed from 0 to 90 degrees in junior high school to any angle, that is, from negative infinity to positive infinity. But the four basic concepts of Sina = A/C, COSA = B/C, TGA = A/B and CTGA = B/A have not changed. To learn trigonometric function well in high school, the most fundamental thing is to understand the concept of "unit circle" on the basis of these four basic concepts. When we understand this unit circle, the whole high school formula of trigonometric function can be solved, and no matter how it changes, it can't escape from our palm.
A "standard circle" is a circle with O as the center and a diameter of 1 on the coordinate axis. Make a vertical line from any point on this circle to the X axis, which forms a right triangle with the line connecting the X axis and this point to the center of the circle. As shown in the figure, take any point P(x, y) on the unit circle of the four quadrants in the rectangular coordinate system as PMMO, and then
Here PO= 1, PM=y, then the value of sinO is the length of PM, that is, the ordinate value y of point p ... Similarly,
The only difference between here and junior high school is that the junior high school is 0 to 90 degrees, and all the values are non-negative. There is not only the length of the line segment, but also the vector value, that is, x and y may be negative numbers. In the second quadrant, y is positive and x is negative, so in this quadrant, sinO is positive and cosO is negative; In the third quadrant, X and Y are negative, so sinO and cosO are positive; In the fourth quadrant, y is
Negative, X is positive, so sinO is negative and cosO is positive.
After combing this truth thoroughly, the angle change formula of trigonometric function in high school will not be memorized. What sin(-θ)=-sinθ, cos(-θ)=cosθ? You think the angle is folded in half along the X axis and runs from the first quadrant to the fourth quadrant. Look at the Y corresponding to the fourth quadrant, so sin(-θ)=-sinθ, and the value of X is still positive, so cos(-θ)=cosθ. With this thing, the rest is ever-changing, sin(θ-π/2)=-sin(π/2)=-cosθ, sin (θ-3π/2) =-cos θ, cos (θ+π) =-cos θ ... Anyway, add an angle, that is, PO turns counterclockwise and decreases.
It's negative. We'll know soon enough. In this way, the periodicity of trigonometric functions is fully understood.
Then there are trigonometric functions and difference formulas, which also come from the unit circle and are nothing more than the distance between two points on the unit circle. This deduction is found in all textbooks. It seems that the derivation process is long, but as long as you draw it on the draft paper yourself, the whole process will be clear at a glance. The sum and difference formulas of trigonometric functions are very complicated, not only sin(α+β)=sinαcosβ+ cosαsinβ, sin(α-β)=sinαcosβ-cosαsinβ, cos(α+β)=cosαcosβ-sinαsinβ, cos(α-β)=cosαcosβ+sinαsinβ. These formulas are tossed and turned by rote, which is enough to recite people's mathematical phobia. If you don't use the method of "thoroughly understanding and grasping the law" to remember, you will never learn trigonometric functions well.
In fact, we just need to remember the formula sin(α+β)=sinαcosβ+ cosαsinβ, and the rest can be calculated according to our basic concepts. Because we have a standard circle in our hearts, no matter how the angle changes, as long as there seems to be an alarm clock in our brain: add an angle and the pointer will rotate counterclockwise; Subtract an angle and the pointer will rotate clockwise. With this thing, you won't be confused about how to change.
So, SIN (α-β) = SIN [α+(-β)] = SIN α COS (-β)+COS α SIN (-β), there is one more symbol, which is negative, so turn the pointer clockwise to the fourth quadrant, where Y is negative, X is positive, SIN value becomes negative, and COS value is still positive, so
sin(α-β)= sin[α+(-β)]
Similarly, COS (α+β) =-SIN (α+β+π/2) =-SIN α COS (β+π/2)-COS α SIN (β+π/2), where π/2 is added, the pointer should turn counterclockwise and SIN should become COS. According to our unit circle, we can get.
Cos( α+β) formula. Similarly, cos (α-β) = cos [α+(-β)], we can easily know.
Cos( α-β) formula. As for tg( α+β), tg(α-β), ctg(α+β), ctg(α-β),
We only need to know four basic concepts: Sina = A/C, COSA = B/C, TGA = A/B, CTGA = B/A, which is enough.
tg(α+β)= sin(α+β)/ cos(α+β)，tg(α-β)= sin(α-β)/ cos(α-β)……
By analogy, the seemingly complicated sum and difference formulas of two angles are clearly arranged in your mind. After a long time, you will not remember a symbol or a sequence wrong. Is this memory effect comparable to any opportunistic method? !
As for the double angle formula of trigonometric function, it is relatively simple. Since we know that sin(α+β)=sinαcosβ+ cosαsinβ, then sin2α = sin (α+α) = sin α cos α+cos α sin α = 2sinα cos α. The following formulas of cos2α, tg2α and ctg2α can be easily worked out according to the concept of unit circle and these four basic concepts, and there is no need to memorize them deliberately. So the trigonometric function of junior high school and senior high school is so complicated. In fact, just remember two points: first, Sina = A/C, COSA = B/C, TGA = A/B, CTGA = B/A; Second, the graphic changes of the unit circle.
Who doesn't remember? Everyone can remember these two things, but why do so many people regard trigonometry in junior and senior high schools as a fearful road? Many people are just confused in complicated formulas and forget the most basic relationship between the most basic concepts and knowledge. Therefore, if we have a headache when learning a seemingly complicated knowledge and feel that we have forgotten some seemingly complicated formulas when memorizing it, please go back to the most basic place immediately to understand and find the rules. This is the only way to remember effectively.
"The correct learning method can turn ordinary people into geniuses; The wrong learning method will turn a genius into an idiot. " Mark my words.