"Images and Properties of Trigonometric Functions" Trigonometric Functions PPT (Properties and Images of Tangent Functions in Lesson 4)

Description

"Images and Properties of Trigonometric Functions" Trigonometric Functions PPT (Properties and Images of Tangent Functions in Lesson 4)

Part One: Learning Objectives

Master the definition domain and value range of tangent function

Be able to use the graph of the tangent function to study its monotonicity and use monotonicity to solve its corresponding problems

Master the periodicity and odd-evenness of the tangent function

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

Problem guide

Preview textbooks P209-P212 and think about the following questions:

1. How to draw the graph of the tangent function with the help of the unit circle?

2. How are the properties of the tangent function different from those of the sine function?

3. Is the tangent function a monotonic function within its domain?

A preliminary exploration of new knowledge

Graph and properties of function y=tan x

■Instructions from famous teachers

(1) The tangent function is an increasing function in every open interval -π2+kπ, π2+kπ (k∈Z). It cannot be said that a function is a monotonically increasing function within its domain and has no monotonically decreasing interval.

(2) The three-point and two-line method is commonly used to draw the tangent function image: "three points" refers to (-π4,-1), (0,0), (π4,1), and "two lines" refers to x=- π2 and x=π2, roughly draw the diagram of the tangent function on (-π2, π2) and then expand it to the left and right to get the tangent curve.

self-test

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

(1) The domain and range of the tangent function are both R.()

(2) The tangent function is an increasing function in the entire domain. ()

(3) The tangent function has no maximum and minimum values ​​in the definition domain. ()

(4) There exists a certain interval where the tangent function is a decreasing function. ()

The domain of function f(x)=tanx+π6 is ()

A.x|x∈R，x≠kπ－π2，k∈Z

B.{x|x∈R,x≠kπ,k∈Z}

C.x|x∈R, x≠kπ+π6, k∈Z

D.x|x∈R, x≠kπ+π3, k∈Z

The minimum positive period of the function y=tan2x+π4 is ()

A． π2B． π

C． 2πD. 3π

The minimum value of function f(x)=tan x on [-π3, π4] is ________.

The monotonically increasing interval of the function y=tanx-π4 is ________.

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

Domain and value range of tangent function

(1) The domain of the function y=tan(2x-π4) is ________.

(2) The value range of function y=tan2x+4tan x-1 is ________.

regular method

Method to find the domain of tangent function

(1) ① When finding the domain of a function related to the tangent function, in addition to the general requirements for finding the domain of the function, it is also necessary to ensure that the tangent function y=tan x is meaningful, that is, x≠π2+kπ, k∈Z.

②When finding the domain of the tangent function y=Atan(ωx+φ)(A≠0,ω>0), “ωx+φ” should be regarded as a “whole”. Let ωx+φ≠kπ+π2, k∈Z, and solve for x.

(2) Method to find the value range of tangent function

①For the value domain of y=Atan(ωx+φ), ωx+φ can be regarded as a whole, combined with the image, and use monotonicity to evaluate the domain.

②For the quadratic function related to y=tan x, tan x can be regarded as a whole, and the matching method can be used to calculate the value domain.

Monotonicity of tangent function and its applications

(1) Find the monotonic interval of y=tan12x+π4.

(2) Compare the sizes of tan 65π and tan-137π.

regular method

(1) Method of comparing magnitude using monotonicity of tangent function

①Use the periodicity or induction formula of the function to reduce the angle to the same monotonic interval.

② Use monotonicity to compare large and small relationships.

(2) Method to find the monotonic interval of the function y = Atan (ωx + φ) (A, ω, and φ are all constants)

① If ω＞0, since y=tan That’s it.

②If ω<0, you can use the induction formula to first transform y=Atan(ωx+φ) into y=Atan[-(-ωx-φ)]=-Atan(-ωx-φ), that is, x Change the coefficient of x to a positive value, and then use the idea of ​​"whole substitution" to find the range of x.

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

1. Function f(x)＝|tan 2x|is ()

A． Odd functions with period π

B. Even function with period π

C． An odd function with period π2

D. Even function with period π2

2. Compare sizes: tan 13π4________tan17π5.

3. Find the domain and period of the function y=tan(3x-π3), and point out its monotonic interval.

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Update Time: 2024-10-02

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