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

Description

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

Part One: Learning Objectives

Understand the concept of periodic functions

Understand the periodicity of sine functions and cosine functions, and be able to find the period of functions

Understand the parity and symmetry of trigonometric functions and be able to judge the parity of a given function

PPT on the images and properties of trigonometric functions, part 2: independent learning

Problem guide

Preview textbooks P201-P203 and think about the following questions:

1. What is the definition of periodic function?

2. How to use the definition of periodic function to find the period of positive and cosine functions?

3. What are the odd and even properties of sine and cosine functions?

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 function f(x) is called a periodic function. The non-zero constant T is called the period of this function.

(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).

■Instructions from famous teachers

Two notes on periodic functions

(1) Not every function is a periodic function. If a function is periodic, its period is not necessarily unique.

(2) If T is a period of function f(x), then nT (n∈Z and n≠0) is also a period of f(x).

2. Periodicity and parity of sine function and cosine function

■Instructions from famous teachers

(1) Periodicity of sine and cosine functions

①The periodicity of the sine function and the cosine function is essentially determined by the periodicity of angles with the same terminal sides;

② From the induction formula sin(x+2kπ)=sin x(k∈Z), cos(x+2kπ)=

cos x(k∈Z) can also explain their periodicity.

③The period T = 2πω of the functions y = Asin (ωx + φ) and y = Acos (ωx + φ) (where A, ω, and φ are constants, and A≠0, ω>0).

(2) About the parity and evenness of sine and cosine functions

① The sine function is an odd function, and the cosine function is an even function. As reflected in the image, the sine curve is symmetrical about the origin O, and the cosine curve is symmetrical about the y-axis;

②Sine curves and cosine curves are both centrally symmetrical figures and axis-symmetrical figures.

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) If sinπ4+π2=sinπ4, then π2 is a period of the sinusoidal function y=sin x. ()

(2) The function y=sin x, x∈(-π, π] is an odd function. ()

(3) Because sin(2x+2π)=sin 2x, the minimum positive period of the function y=sin 2x is 2π.()

(4) If T is the period of function f(x), then kT, k∈N* is also the period of function f(x). ()

Among the following functions, the one with a minimum positive period of 4π is ()

A． y＝sin x　B． y＝cosx

C． y＝sinx2 D. y＝cos 2x

The function y=2sin2x+π2 is ()

A． An odd function with period π B. Even function with period π

C． An odd function with period 2π D. Even function with period 2π

Graphics and properties of trigonometric functions PPT, the third part: interactive lecture and practice

Periodic problem of sine and cosine functions

Find the minimum positive period T of the following trigonometric functions:

(1)f(x)=sinx+π3;

(2)f(x)=12cos(2x+π3);

(3)f(x)＝|sin x|.

regular method

How to find the period of a function

(1) Definition method: closely follow the definition of periodic function, and seek a non-zero constant T that satisfies f(x+T)=f(x) for any real number x. This method is mainly suitable for abstract functions.

(2) Formula method: For functions of the form y = Asin (ωx + φ) and y = Acos (ωx + φ) (where A, ω, φ are constants, and A≠0, ω>0), T = 2πω can be used to beg.

(3) Image method: The image of the function can be drawn, and the period of the function can be judged with the help of the image. This method is generally used especially for functions containing absolute values.

Track training

1. Assume function f(x)=sin12x-π3, then the minimum positive period of f(x) is ()

A． π2B． π

C． 2π D. 4π

2. Assume a>0, if the minimum positive period of the function y=sin(ax+π) is π, then a=________.

Parity problem of sine and cosine functions

Determine the parity of the following functions.

(1)f(x)=cos2x+5π2;

(2)f(x)=sin(cos x).

regular method

Three steps to determine parity of a function using definitions

[Note]For parity issues related to trigonometric functions, it is often necessary to use induced formulas to simplify them first, and then judge the parity of the function.

Graphics and Properties of Trigonometric Functions PPT, Part 4: Feedback on Compliance

1. Assume function f(x)=sin(2x-π3), then the minimum positive period of f(x) is ()

A． π2B． π

C． 2π D. 4π

2. It is known that a∈R, function f(x)=sin x-|a|, x∈R is an odd function, then a is equal to ________.

3. The graph of function f(x)=2cos 2x+1 is symmetrical about ________ (fill in the "origin" or "y-axis").

4. Determine the parity of the following functions:

(1)f(x)=sin3x4+3π2;

(2)f(x)=sin |x|;

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Update Time: 2024-10-03

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

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