"Graphics of Sine Function and Cosine Function" Trigonometric Function PPT

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"Graphics of Sine Function and Cosine Function" Trigonometric Function PPT

Part One: Explanation of Curriculum Standards

1. According to the definition of sine function, the sinusoidal line of the unit circle can be used to construct the image of the sine function.

2. Master the use of the "five-point method" to construct the graphs of sine functions and cosine functions.

3. Be able to understand the intrinsic relationship between the sine function image and the cosine function image from the perspective of simple image transformation.

Image PPT of sine function and cosine function, part 2: independent preview

1. The image of the sine function

1. (1) As shown in the unit circle in the figure, the terminal side of the angle α intersects the unit circle at B (x0, y0). Can you use the coordinates of point A to express sin α and cos α?

Tip: From the definition of trigonometric functions, we know that sin α=y0, cos α=x0.

(2) In (1), the BM⊥x axis is drawn through point B, and the vertical foot is M. If it is specified that the BM direction is in the same direction as the positive direction of the y-axis, it is positive, and it is negative in the same direction as the negative direction of the y-axis, so that Use the size of BM (including positive and negative) to represent the sine value.

How does the size of sin α change when the angle α is rotated counterclockwise within 0,π/2? What about the angle α at π/2,π?

Tip: When sin α is at α∈ 0,π/2, as α increases, the value of sin α becomes larger, which is an increasing function;

When sin α is at α∈ π/2,π, as α increases, the value of sin α becomes smaller, which is a decreasing function.

Among them, BM can represent the value of sin α, and BM is also called a sine line.

(3) For any real number x, are its sine and cosine values ​​unique? Can sin x and cos x be regarded as functions of the variable x?

Tip: The only one that can.

(4) Analytical expressions and domain of sine and cosine functions

(5) What is the most basic method to draw a function graph? If you use the point drawing method to draw a graph of the sine function y=sin x within [0,2π], what points can be taken?

Tips: The most basic method for plotting function graphs is the plotting point method; use the plotting point method to plot the image of the sine function y=sin x within the range of [0,2π]. It is advisable to use the plotting point method when x=0,π/6, Each point when π/4, π/3, π/2,….

2. Fill in the blanks

Use sine lines to create the graph of sine function

The steps to use sine lines to make the graph of sine function: (1) divide it equally; (2) make sine lines; (3) translate to get points; (4) connect lines.

3. How to get the image of y=sin x when x∈[2π,4π],-2π,0],...?

Tip: According to the induction formula 1, the image of the function y=sin x within [0,2π] can be obtained by shifting left and right.

4. Fill in the blanks

The graph of the sine function y=sin x,x∈R is called a sinusoidal curve.

5. On the graph of function y=sin x,x∈0,2π, what are the key points?

Tip: a highest point, a lowest point, the intersection of the three images and the x-axis.

Image PPT of sine function and cosine function, the third part: exploration and learning

Use the "five-point method" to draw graphs of trigonometric functions

Example 1 Use the "five-point method" to make the graph of the following function:

(1)y=sin x-1,x∈[0,2π];

(2)y=1-1/3cos x,x∈[-2π,2π].

Analysis: (1) First find 5 key points on [0,2π], and then connect them with smooth curves; (2) First use the "five-point method" to make a function on [0, The image on 2π] is then obtained through symmetry or translation to obtain the image on -2π,0].

Solution: (1) List:

Draw points and connect lines, as shown in the picture.

(2) List:

Draw the points and connect the lines to get the image of the function y=1-1/3cos x on [0,2π], and then translate the image 2π units to the left to get the function on [ ;-2π, the image on 2π], as shown in the figure.

Reflection and insights on using the "five-point method" to draw a simple diagram of the function y=Asin x+b(A≠0) (or y=Acos x+b(A≠0)) on [0,2π] A step of:

(1)List:

(2) Drawing points: Draw the following five points in the plane rectangular coordinate system:

(0,y1),(π/2 "," y_2 ),(π,y3),(3π/2 "," y_4 ),(2π,y5).

(3) Connecting lines: Use smooth curves to connect the five points drawn.

Reflection and understanding on the rules of image transformation

1.Translation transformation

(1) The graph of function y=f(x+a) is translated from the graph of function y=f(x) to the left (a>0) or right (a<0) by |a| ;obtained by units;

(2) The graph of function y=f(x)+b is translated upward (b>0) or downward (b<0) by the graph of function y=f(x) by |b| obtained by units.

2. Symmetry transformation

(1) The graph of the function y=|f(x)| is the graph of the function y=f(x) where the part above the x-axis remains unchanged and the part below is symmetrically folded to x above the axis to get;

(2) The graph of the function y=f(|x|) is to keep the part of the graph of the function y=f(x) on the right side of the y-axis stationary and fold it symmetrically to the y-axis. Left side gets;

(3) The graph of function y=-f(x) and the graph of function y=f(x) are symmetrical about the x-axis;

(4) The graph of function y=f(-x) and the graph of function y=f(x) are symmetrical about the y-axis;

(5) The graph of function y=-f(-x) and the graph of function y=f(x) are symmetrical about the origin.

Image PPT of sine function and cosine function, part 4: thinking methods

Use numbers and shapes to combine ideas to solve numerical problems

Typical example: The number of solutions to the equation lg x=sin x is ()

A.0 B.1 C.2 D.3

Question review perspective: This equation cannot be solved with the root formula, and only the number of roots of the equation is required. The functions y=sin x and y=lg x are basic elementary functions, and their images are easy to draw, so the combination of numbers and shapes can be used Method: draw the graphs of two functions in the same plane rectangular coordinate system, observe the number of their intersections, that is, get the number of roots of the equation.

Analysis: Draw the graphs of the functions y=lg x and y=sin x in the same plane rectangular coordinate system. As shown in the figure, when x=5π/2, y=lg5π/2<1, y=sin5π/ 2=1; when x=9π/2, y=lg9π/2>1, y=lg x and y=sin x have no intersection points. As shown in the figure, there are three intersection points, so the equation has three solutions.

Answer:D

The key point of the method is to combine numbers and shapes. The idea of ​​combining numbers and shapes is an important mathematical idea. When studying the roots of equations and the number of roots, if the functions involved in the equations are basic elementary functions, their graphs are easy to draw. In this case, the equations can be The roots are transformed into intersection points of function graphs, and problems are solved through the combination of numbers and shapes, so that abstract algebraic problems can be solved intuitively and vividly.

Image PPT of sine function and cosine function, part 5: practice in class

1. Use the "five-point method" to draw the graph of the function y=2-3sin x. Among the following points, the one that is not one of the five key points is ()

A.(0,2) B.(π/2 "," 1) C.(π,2) D.(3π/2 "," 5)

Analysis: When x=π/2, y=2-3sin x=-1, so (π/2 "," 1) is not a key point.

Answer:B

2. The image of the function y=cos(x+3π) and the image of the cosine function ()

A. Symmetry about the x-axis

B. Symmetry about the origin

C. Symmetrical about the origin and x-axis

D. Symmetrical about the origin and coordinate axis

Analysis: Because y=cos(x+3π)=-cos x, its image and the image of the cosine function y=cos x are symmetrical about the origin and the x-axis.

Answer:C

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Update Time: 2024-10-11

This template belongs to Mathematics courseware People's Education High School Mathematics Edition A Compulsory Course 1 industry PPT template

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