"Images and Properties of Trigonometric Functions" Trigonometric Functions PPT Courseware (Lesson 2: Periodicity and Parity of Sine and Cosine Functions)

Description

"Images and Properties of Trigonometric Functions" Trigonometric Functions PPT Courseware (Lesson 2: Periodicity and Parity of Sine and Cosine Functions)

Part One: Learning Objectives

1. Understand the definitions of periodic functions, periods, and minimum positive periods.

2. Can find the periods of functions y=Asin(ωx+φ) and y=Acos(ωx+φ). (emphasis)

3. Master the parity of functions y=sin x, y=cos x, and be able to judge the parity of simple trigonometric functions. (Key point, easy to confuse)

core competencies

1. Cultivate logical reasoning literacy through periodic research.

2. Use parity and the relationship between images to improve your intuitive imagination.

PPT on the images and properties of trigonometric functions, part 2: independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. periodicity of function

(1) Periodic function: For function f(x), if there is a _________ such that when x takes every value in the domain, there is _________, then the period of this function is _____.

(2) Minimum positive period: If there is a minimum _________ among all periods of the periodic function f(x), then this minimum _________ is called the _________ of f(x).

2. Periodicity and parity of sine and cosine functions

First try

1. The function y=2sin2x+π2 is ()

A． Odd functions with period π

B. Even function with period π

C． An odd function with period 2π

D. Even function with period 2π

2. The parity of function f(x)=2sin 2x is ()

A． odd function

B. even function

C． both odd and even functions

D. non-odd and non-even functions

3. Function f(x)=3sinπx2-π4, the minimum positive period of x∈R is ________.

4. If the function y=f(x) is a function with a period of 2, and f(5)=6, then f(1)=________.

The images and properties of trigonometric functions PPT, the third part: cooperative exploration to improve literacy

Periodic problems and simple applications of trigonometric functions

[Example 1] Find the period of the following function:

(1)y=sin2x+π4;

(2)y＝|sin x|.

[Ideas] (1) Method 1: Find a non-zero constant T so that f(x+T)=f(x) is always true.

Method 2: Use the periodic formula of y=Asin(ωx+φ) to calculate.

(2) Draw the graph of the function and observe the period.

[Solution] (1) Method 1: (Definition method) y=sin2x+π4

＝sin2x＋π4＋2π＝sin2x＋π＋π4,

So the period is π.

Method 2: (Formula method) y=sin2x+π4 where ω=2, T=2πω=2π2=π.

(2) The drawing is as follows:

Observing the image, we can see that the period is π.

regular method

How to find the period of a trigonometric function:

(1) Definition method: that is, using the definition of periodic function to solve.

(2) Formula method: For functions of the form y=Asin(ωx+φ) or y=Acos(ωx+φ) (A, ω, φ are constants, A≠0, ω≠0), T=2π|ω&# 124;.

(3) Image method: that is, finding the period of a function by observing its image.

Reminder: The minimum positive period of y=|Asin(ωx+φ)|(A≠0,ω≠0) is T=π|ω|.

Track training

1. Use the definition of periodic functions to find the period of the following functions.

(1)y=cos 2x, x∈R;

(2)y=sin13x-π4, x∈R.

Judgment of parity of trigonometric functions

[Example 2] Determine the parity of the following functions:

(1)f(x)=sin-12x+π2;

(2)f(x)=lg(1-sin x)-lg(1+sin x);

(3)f(x)＝1＋sin x－cos2x1+sin x.

regular method

1. There are two aspects that should be grasped when judging the parity of a function:

First, check whether the domain of the function is symmetric about the origin;

Second, look at the relationship between f(x) and f(-x).

2. For the judgment of the parity of a trigonometric function, sometimes the functional formula can be simplified first according to the induction formula and then judged.

Reminder: Studying function properties should follow the principle of "domain first".

Class summary

1. "f(x+T)=f(x)" is an identity within the domain of definition, that is, it is true for every value in the domain of definition. T is a non-zero constant. The period T is the increasing value of the independent variable x that causes the function value to appear repeatedly. The period The graph of the function repeats every other period.

2. "f(x+T)=f(x)" in the definition of periodic function is for every x value in the definition domain. Only individual x values ​​satisfy f(x+T)=f(x). It cannot be said that T is The period of y=f(x).

3. On the number axis, the domain is symmetrical about the origin, which is a necessary condition for the function to have parity. Therefore, to determine the parity of a function, we must first check whether its domain is symmetrical about the origin. If so, then judge the relationship between f(-x) and f(x); if not, then the function is neither an odd function nor an even function.

Graphics and properties of trigonometric functions PPT, part 4: Complying with standards and solidifying double bases in class

1. Thinking and analysis

(1) If sin2π3+π6=sinπ6, then 2π3 is a period of the function y=sin x. ()

(2) All periodic functions have a minimum positive period. ()

(3) The function y=sin x is an odd function. ()

[Tips] (1)×. Because for any x, sin2π3+x and sin x are not necessarily equal.

(2)×. Not all functions have a minimum positive period. For example, if the function f(x)=5 is a periodic function, there is no minimum positive period.

(3) ×. The domain of the function y=sin x is {x|2kπ≤x≤2kπ+π, k∈Z}, which is not symmetrical about the origin, so it is neither odd nor even.

2. As shown in the figure are the graphs of four functions defined on R, among which the graph that is not a periodic function is ()

3. If the function y=f(x) is an odd function with period 3 defined on R and f(1)=3, then f(5)=________.

4. Determine the parity of the following functions:

(1)f(x)=-2cos 3x;

(2)f(x)=xsin(x+π).

Keywords: Free download of PPT courseware for high school People's Education A version of Mathematics compulsory course I, PPT download of images and properties of trigonometric functions, PPT download of trigonometric functions, PPT download of periodicity and parity of sine and cosine functions, .PPT format;

For more information about the PPT courseware "Graphics and Properties of Trigonometric Functions, Periodicity and Parity of Sine and Cosine Functions", please click on the ppt tag of Trigonometric Functions ppt Graphics and Properties of Trigonometric Functions ppt Periodicity and Parity of Sine and Cosine Functions .

"End of Chapter Review Lesson" Trigonometric Functions PPT:

"End of Chapter Review Course" Trigonometric Functions PPT Basic relations and induction formulas for congruent angle trigonometric functions [Example 1] (1) It is known that sin(-+)+2cos(3-)=0, then sin +cos sin -cos =________ . (2) It is known that f()=sin2-cos2-tan-+sin..

"End of Chapter Review Improvement Course" Trigonometric Functions PPT:

"End of Chapter Review and Improvement Course" Trigonometric Functions PPT comprehensively improves the basic relational expressions and induced formulas of trigonometric functions with the same angle. It is known that cos(+)=-12, and the angle is in the fourth quadrant, calculate: (1) sin(2-); (2)sin[＋(2n＋1)]+sin(＋)sin(-)cos..

"Applications of Trigonometric Functions" Trigonometric Functions PPT download:

"Applications of Trigonometric Functions" Trigonometric Function PPT Download Part One: Learning Objectives 1. Understand that trigonometric functions are important function models that describe periodic changing phenomena, and be able to use trigonometric function models to solve some simple practical problems. (Key points) 2. Practical questions...

File Info

Update Time: 2024-11-15

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

Preview