"Images and Properties of Trigonometric Functions" Trigonometric Functions PPT (Third Lesson: Monotonicity and Maximum Values ​​of Sine and Cosine Functions)

Description

"Images and Properties of Trigonometric Functions" Trigonometric Functions PPT (Third Lesson: Monotonicity and Maximum Values ​​of Sine and Cosine Functions)

Part One: Learning Objectives

Understand the monotonicity of sine functions and cosine functions, and be able to find the monotonic intervals of functions

Can use the monotonicity of trigonometric functions to compare the values ​​of trigonometric functions

Can use the monotonicity of trigonometric functions to find the maximum value and range of the function

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

Problem guide

Preview textbooks P204-P207 and think about the following questions:

1. Are the monotonic intervals of sine and cosine functions the same? What are they?

2. What are the maximum values ​​of the sine and cosine functions?

A preliminary exploration of new knowledge

Graphs and properties of sine and cosine functions

■Instructions from famous teachers

The sine and cosine functions are not monotonic functions in the domain of definition. It is also wrong to say "the sine function is an increasing function in the first quadrant" because there are infinitely many monotonic increasing intervals in the first quadrant, and in each monotonic increasing interval , y=sin x all increases from 0 to 1, but it cannot be regarded as a monotonic interval.

self-test

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

(1) The maximum value of function y=12sin x is 1.()

(2)∃x0∈[0, 2π], satisfying cos x0=2.()

(3) The sine function and the cosine function are both monotonic functions within the domain of definition. ()

In the following interval, what makes the function y=sin x an increasing function is ()

A． [0,π]B. π2, 3π2

C． -π2,π2 D. [π,2π]

The minimum and maximum values ​​of the function y=1-2cosπ2x are ()

A． -1,3 B. -1,1

C． 0,3 D． 0,1

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

Monotonicity of sine and cosine functions

Find the monotonically decreasing interval of the following function:

(1)y=12cos2x+π3;

(2)y=2sinπ4-x.

Solving strategy

Strategies for finding monotonic intervals of positive and cosine functions

(1) Combine the graphs of sine and cosine functions and memorize their monotonic intervals.

(2) When finding the monotonic interval of a function of the form y=Asin(ωx+φ)(A>0,ω>0), the "element substitution method" should be used for the overall substitution, and "ωx+φ" should be regarded as a whole "z ”, that is, finding the monotonic interval of the original function by finding the monotonic interval of y=Asin z. The same as above to find the monotonic interval of a function of the form y=Acos(ωx+φ)(A>0,ω>0).

Track training

1. Function y=sinx+π2, x∈R in ()

A． -π2, π2 is an increasing function

B. [0, π] is a decreasing function

C． [-π, 0] is a decreasing function

D. [-π, π] is a decreasing function

2. Find the monotonically increasing interval of the function y=sinx+π4.

Compare trigonometric function values

Compare the sizes of the following sets of numbers.

(1) sin 1017π and sin 1117π;

(2) cos-7π8 and cos 6π7;

(3)sin 194° and cos 160°.

regular method

Steps to Compare Trigonometric Function Values

(1) Functions with different names are transformed into functions with the same name;

(2) Use the induction formula to transform the angle into the same monotonic interval;

(3) Use the monotonicity of functions to compare sizes.

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

1. Among the following functions, the one that is always positive and an increasing function on the interval π2, π is ()

A． y＝sin x　B． y＝cosx

C． y=-sin x D. y=-cos x

2. The function y=3cos12x-π4 takes the maximum value when x=________.

3. sin21π5________sin425π (fill in ">" or "<").

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Update Time: 2024-11-14

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

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