The classical example of trigonometric function "the image and property of sine function and cosine function" in senior one mathematics The best is moderate difficulty, comprehensive, wide range of knowledge points

The classical example of trigonometric function "the image and property of sine function and cosine function" in senior one mathematics The best is moderate difficulty, comprehensive, wide range of knowledge points

The best way is to have moderate difficulty and strong comprehensiveness,

The maximum and minimum of y = 2sinx + 3cosx?

Y = square of 2 + square of 3 times sin (x + α)
Here α is a constant value, which does not need to be calculated
The maximum value of sin (x + α) is 1 and the minimum value is - 1
So the maximum value of Y is 13 under the radical
The minimum value is 13 under the root sign
Such problems can be solved by this method
The maximum value of y = xsinx + ycosx is the square of X + the square of Y

Given the function f (x) = sin (((K π x) / 5) + π / 3) (k > 0), when the independent variable x changes between any two integers (including the integer itself), it contains at least one period, and the value range of K is obtained

2*5/k=10

What is the maximum value of sine function, cosine function and tangent function,

Sine function: maximum value 1, minimum value - 1,
Cosine function: Max 1, min - 1
Tangent function: no maximum value

The properties of sine cosine function image

As shown in the figure, its common form is y = SiNx
The period is 2 π and it is an odd function
① Maximum value: when x = 2K π + (π / 2), K ∈ Z, y (max) = 1. ② minimum value: when x = 2K π + (3 π / 2), K ∈ Z, y (min) = - 1 zero point: (K π, 0), K ∈ Z
It is a monotone increasing function on [- π / 2 + 2K π, π / 2 + 2K π], K ∈ Z and a monotone decreasing function on [π / 2 + 2K π, 3 π / 2 + 2K π], K ∈ Z
The maximum value of y = asin (ω x + φ) + Ba > 0 is y = a + B,
If the minimum value is y = B-A, the period of the function can be calculated by T = 2 π / w (the tangent function is t = π / W)
Cosine function cosx is obtained by shifting sine function SiNx to the left by π / 2 unit. It is easy to get that cosine function is even function with the same range as SiNx. Its properties can be deduced by referring to the above sine function

The image and properties of sine function

Definition and theorem
Definition: for any real number x, it corresponds to the unique angle (equal to this real number in radian system), and this angle corresponds to the unique sine value sin X. in this way, for any real number x, there is a unique definite value SiN x corresponding to it. According to this correspondence rule, the function established is expressed as f (x) = SiN x, which is called sine function
The theorem of sine function: in a triangle, the ratio of each side to its diagonal sine is equal, that is, a / sin a = B / sin B = C / sin C
In the right triangle ABC, ∠ C = 90 °, y is a right angle side, R is an oblique side, X is another right angle side (in the coordinate system, this is the base), then sin a = Y / R, r = √ (x ^ 2 + y ^ 2)
Define domain
Real number set R
range
[- 1,1] (embodiment of boundedness of sine function)
Maximum and zero
① When x = 2K π + (π / 2), K ∈ Z, y (max) = 1
② When x = 2K π + (3 π / 2), K ∈ Z, y (min) = - 1
Zero point: (K π, 0), K ∈ Z
Symmetry
It is both axisymmetric and centrosymmetric
1) Symmetry axis: symmetry about the straight line x = (π / 2) + K π, K ∈ Z
2) Centrosymmetry: with respect to points (K π, 0), K ∈ Z symmetry
Periodicity
Minimum positive period: y = asin (ω x + φ) t = 2 π / | ω|
Parity
Odd functions (whose images are symmetric about the origin)
Monotonicity
It is monotonically increasing on [- π / 2 + 2K π, π / 2 + 2K π], K ∈ Z
It is monotone decreasing on [π / 2 + 2K π, 3 π / 2 + 2K π], K ∈ Z

How to transform sine function and cosine function 、、、、、、、、、、、、、、、、

sinx=cos(π/2-x)

The relationship between cosine function and sine function

(cosx)^2+(sinx)^2=1

Sine plus cosine What is the result of asinx + bcosx

√(a^2+b^2)*sin[x+arctan(b/a)]
Is the tangent value of arc

Sine function cosine function problem The sine function obtains the maximum value of 1 if and only if x = ---, and the minimum value - 1 if and only if x = ---

Sine function obtains maximum value 1 if and only if x = Π / 2 + 2K Π, and minimum value-1 (K ∈ z) if and only if x = - Π / 2 + 2K Π