"Properties and Images of Tangent Functions" Trigonometric Functions PPT

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"Properties and Images of Tangent Functions" Trigonometric Functions PPT

Part One: Explanation of Curriculum Standards

1. Be able to draw the graph of the function y=tan x with the help of the tangent line in the unit circle.

2. Master the domain, range, periodicity, odd-evenness, and monotonicity of the tangent function.

3. Be able to use the image and properties of the tangent function to solve related problems.

Properties of tangent functions and image PPT, part 2 content: independent preview

1. Graph of tangent function

1. Based on the quotient relationship in the basic relationship between congruent trigonometric functions, can you infer that y=tan x is a periodic function?

Tip: Because tan x=sinx/cosx,

So tan(x+π)=(sin"(" x+π")" )/(cos"(" x+π")" )=("-" sinx)/("-" cosx)=tan x ,

So y=tan x is a periodic function.

2. Fill in the blanks

(1) The graph of the tangent function (as shown in the figure):

(2) The graph of the tangent function is called a tangent curve.

(3) Graphical characteristics of the tangent function: The tangent curve is composed of straight lines that are parallel to each other.

It consists of infinite curves separated by x=π/2+kπ,k∈Z.

3. Determine right or wrong

(1) The periods of the functions y=|tan x| and y=tan x are equal, both are π. ()

(2) The minimum positive period of the function y=tan|x| is π/2. ()

Answer: (1)√ (2)×

2. Properties of tangent function

1. Observe the tangent curve and think: What is the value range of the tangent function? Is the tangent function an increasing function in the entire domain? Will the tangent function be a decreasing function in a certain interval? The graph of the tangent function is about something Is the line symmetrical? Is it symmetrical about some point?

Tip: The value range of the tangent function is R; the tangent function is not an increasing function in the entire domain; the tangent function will not be a decreasing function in a certain interval; the graph of the tangent function cannot be symmetrical about a certain straight line; about some points is symmetrical.

2. Fill in the blanks

3. Do it

(1) The domain of the function y=tan(2x+π/3) is ______;

(2) The monotonically increasing interval of the function y=tan(x"-" π/4) is ____________.

Analysis: (1) From 2x+π/3≠kπ+π/2, k∈Z,

The solution is x≠kπ/2+π/12(k∈Z),

So the function domain is {x├|x≠kπ/2+π/12 "," k"∈" Z┤}.

(2) From kπ-π/2

The solution is kπ-π/4

So the monotonically increasing interval of the function is

(kπ"-" π/4 "," kπ+3π/4)(k∈Z).

Answer: (1){x├|x≠kπ/2+π/12 "," k"∈" Z┤}(2)(kπ"-" π/4 "," kπ+3π/4 )(k∈Z)

Properties of tangent functions and image PPT, part three content: inquiry learning

The definition domain and range problem of tangent function

Example 1 Find the domain and range of the following functions:

(1)f(x)=tan(1/2 x"-" π/3);

(2)f(x)=√(√3 "-" tanx).

Analysis: Solve according to the definition domain and value range of the tangent function and combined with the graph of the tangent function.

Solution: (1) According to the question, 1/2x-π/3≠kπ+π/2,k∈Z,

So x≠2kπ+5π/3,k∈Z.

So the domain of the function is {x├|x≠2kπ+5π/3 "," k"∈" Z┤}.

From the value range of the tangent function, we can know that the value range of the function is (-∞, +∞).

(2) According to the meaning of the question √3-tan x≥0, so tan x≤√3.

Combined with the image of y=tan x, it can be seen that,

On ("-" π/2 "," π/2), the angle x that satisfies tan x≤√3 should satisfy -π/2

Therefore, the domain of the function y=√(√3 "-" tanx) is {x├|kπ"-" π/2

Reflection on the methods and points to note when finding the domain of the tangent function:

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≠ +kπ, k∈Z. For the constructed triangle inequality , often solved using the graph of the tangent function. Steps for solving inequalities of the form tan x>a:

The properties and image PPT of the tangent function, the fourth part: thinking analysis

Mistaking the center of symmetry of the tangent function graph leads to errors

Typical example: A symmetry center of the image y=tan(2x+θ) is (π/3 "," 0). If -π/2<θ<π/2, then θ=__________.

The symmetry center of the wrong solution function y=tan x is (kπ,0), where k∈Z, then let 2x+θ=kπ,k∈Z. When x=π/3, the solution is θ=kπ-2π/ 3. k∈Z, from -π/2<θ<π/2, we get θ=π/3.

Where is the misinterpretation? Can you spot it? How to avoid such mistakes?

Tip: In the wrong solution, the symmetry center (kπ/2 "," 0) (k∈Z) of the tangent function y=tan x image is mistaken for (kπ,0) (k∈Z), resulting in the value of θ Seek error.

Properties and image PPT of tangent function, part 5: practice in class

1. The minimum positive period of f(x)=tan("-" 2x+π/3) is ()

A.π/4 B.π/2 C.π D.2π

Analysis: T=π/("|-" 2"|" )=π/2.

Answer:B

2. Function f(x)=sin xtan x()

A. It is an odd function B. It is an even function

C. Is a non-odd or even function D. Is both an odd function and an even function

Analysis: The domain is {x├|x≠kπ+π/2 "," k"∈" Z┤}, which is symmetrical about the origin.

From f(-x)=sin (-x)·tan(-x)=(-sin x)·(-tan x)=sin xtan x=f(x), then f(x) is an even function. Therefore Choose B.

Answer:B

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Update Time: 2024-09-08

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