sin(a+b)=sinacosb+cosasinb
sin(a-b)=sinacosb-cosasinb
Look at this formula, you can write it as:
Sum is addition, difference is subtraction, before and after, before positive remainder and after JUNG WOO.
After understanding, it can be summed up as: harmony but different, former but different, former and positive remainder.
cos(a+b)=cosacosb-sinasinb
cos(a-b)=cosacosb+sinasinb
Look at this formula, you can write it as:
Sum is subtraction, difference is addition, the same as before and after, and the rest are before and after.
After understanding, it can be summed up as follows: harmony but difference are opposites, the same before and after, and the rest are before.
There is also a formula for trigonometric functions:
Parity remains the same, the symbol looks at the quadrant!
Knowledge carding of trigonometric function at the end of senior one.
1. 1 arbitrary angle and arc system
2. quadrant angle: in rectangular coordinate system, the vertex of the angle coincides with the origin, the starting edge of the angle coincides with the non-negative semi-axis of the shaft, and the ending edge of the angle is in which quadrant, so the angle is in which quadrant. If the terminal edge of an angle is on the coordinate axis, it is considered that the angle does not belong to any quadrant.
3..① Set of angles with the same terminal edge as (0 ≤ < 360):
(2) Angle group of terminal edge on X axis:
(3) Angle group of terminal edge on Y axis:
(4) Angle set of terminal edge on coordinate axis:
(5) Y = Angle set of terminal edge on X axis:
6. Angle setting of the end edge on the shaft:
⑦ If the angle and the terminal edge of the angle are symmetrical about X, the relationship between the angle and the angle is:
⑧ If the angle and the terminal edge of the angle are symmetrical about the Y axis, then the relationship with the angle:
Pet-name ruby if the angle and the terminal edge of the angle are in a straight line, the relationship with the angle is:
Attending the angle and the terminal edge of the angle are perpendicular to each other, then the relationship with the angle is:
4. Curvature system: the central angle of an arc with a long radius is called radian. 360 degrees =2π radians. If the arc length subtended by the central angle is l, the absolute value of its radian number |, where r is the radius of the circle.
5. Reciprocal formula of radian and angle: 1 rad = () ≈ 57.30 1 =
Note: The radian of positive angle is positive, the radian of negative angle is negative, and the radian of zero angle is zero.
6 .. Angle of the first quadrant:
Acute angle:; Angle less than: (including negative angle and zero angle)
7. Arc length formula: sector area formula:
1.2 trigonometric function at any angle
1. Definition of trigonometric function of any angle: Let it be any angle, P be any point on the terminal edge (different from the origin), and its distance from the origin is,
The value of trigonometric function is only related to the angle, and has nothing to do with the position of point P at the edge of the terminal.
2 .. trigonometric function line
Sine line: MP; Cosine line: om; Tangent: in
3. Symbol of trigonometric function in each quadrant: (one is full of two sines and three are tangent to four cosines)
+ + - + - +
- - - + + -
4. The basic relationship of trigonometric functions with the same angle:
(1) square relation:
(2) Quotient relation: (used to cut strings)
The general square relation is an implicit condition and can be used directly. ※. Pay attention to the substitution of "1"
The inductive formula of 1.3 trigonometric function
1. Inductive formula (write the angle as a table, and use the formula: odd change couple, sign to see quadrant)
Ⅰ) Ⅱ) Ⅲ)
Ⅳ) Ⅴ) Ⅵ)
Images and properties of 1.4 trigonometric function
1. Definition of periodic function: For a function, if there is a non-zero constant that makes every value in the defined domain true, then this function is called a periodic function, and this non-zero constant is called the period of this function. (Not all functions have a minimum positive period)
(1) and the period is.
② or () period.
③
The period of is 2 (as shown).
2. The main properties of three commonly used trigonometric functions
Function y = sinxy = cosxy = tanx
Domain (-∞,+∞) (-∞,+∞)
Range [- 1, 1] [- 1, 1] (-∞, +∞)
Even and odd odd function even function odd function
The minimum positive period is 2π 2π.
Monotonic increase
negative
raise
negative
Continuous increase
symmetrical
Asymmetric axis
3, the shape of the function:
(1) Several physical quantities: a- amplitude; -Frequency (reciprocal of the period); -stage; -Initial stage;
(2) Determination of function expression: A is determined by the maximum value; Determined by the cycle; It is determined by special points on the image. For example, if the image is as shown in the figure, then = _ _ _ _ _ (a:);
(3) Drawing method of function image:
① "five-point method"-set = 0, find the corresponding value, calculate the coordinates of five points, and draw points to get an image; ② Image transformation method: This is a common method for making functional diagrams.
(4) Relationship between function images: ① The vertical coordinates of function images are unchanged, and the horizontal coordinates are left (> 0) or right (
③ The abscissa of the function image remains unchanged, and the ordinate becomes a times of the original one, thus obtaining the image of the function;
④ The abscissa of the function image is unchanged, and the ordinate is up () or down () to get the image.
It is particularly important to note that if the image is obtained through, the translation to the left or right should be done in units.
Example: Take transformation as an example.
Translation unit to the left (left plus right minus)
The abscissa is doubled (the ordinate is unchanged).
The ordinate is quadrupled (the abscissa remains the same).
The abscissa is doubled (the ordinate is unchanged).
Translation unit to the left (left plus right minus)
The ordinate is quadrupled (the abscissa remains the same).
Note: X is always changed in the transformation.
(5) Function properties (the idea of potential substitution): the method of finding symmetry center, symmetry axis and monotone interval (pay special attention to it first).
9. Sine and cosine "three brothers and sisters"-the memory connection of "knowing one and seeking two"