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

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

Part One: Learning Objectives

1. Master the maximum and minimum values ​​of y=sin x, y=cos x, and be able to find the range and maximum value of simple trigonometric functions. (main difficulty)

2. Master the monotonicity of y=sin x, y=cos x, and be able to use monotonicity to compare sizes. (emphasis)

3. Be able to find the monotonic intervals of the functions y=Asin(ωx+φ) and y=Acos(ωx+φ). (Key point, easy to confuse)

core competencies

1. Improve mathematical operation literacy through monotonicity and optimal value calculations.

2. Combine with function images to cultivate intuitive imagination literacy.

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

A preliminary exploration of new knowledge

Analytical formula y＝sin x y＝cos x

Value range _________ _________

Monotonicity: Monotonically increasing on -π2+2kπ, π2+2kπ, k∈Z, monotonically decreasing on π2+2kπ, 3π2+2kπ, k∈Z

It increases monotonically on [-π+2kπ, 2kπ], k∈Z, and decreases monotonically on [2kπ, π+2kπ], k∈Z

Maximum value x=π2+2kπ, when k∈Z, ymax=1; x=-π2+2kπ, when k∈Z, ymin=-1

When x=2kπ, k∈Z, ymax=1; x=π+2kπ, when k∈Z, ymin=-1

Thinking: y=sin x and y=cos x are both decreasing functions in the interval (m, n) (where 0＜m＜n＜2π). Can you determine the minimum value of m and the maximum value of n?

Tip: From the monotonicity of the sine function and the cosine function, we know that m=π2, n=π.

First try

1. The function y=-cos x is () on the interval -π2, π2

A． increasing function

B. subtract function

C． Decrease first and then increase function

D. First increase and then decrease function

2. The value range of function y=sin xπ4≤x≤5π6 is ________.

3. When the function y=2-sin x obtains the maximum value, the value set of x is _________.

4. If cos x=m-1 is meaningful, then the value range of m is ________.

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

Monotonicity of sine and cosine functions

[Example 1] (1) The function y = cos x is an increasing function on the interval -π, a', then the value range of a is ________.

(2) Given the function f(x)=2sinπ4+2x+1, find the monotonically increasing interval of the function f(x).

[Ideas Enlightenment] (1) Determine the range of a→y=cos x is in the interval [-π, a] is an increasing function→y=cos x is in the interval[ -π, 0] is an increasing function, and on the interval 0, π] is a decreasing function → the range of a.

(2) Determine the increasing interval → let u=π4+2x→y=2sin the monotonically increasing interval of u.

regular method

1. To find the monotonic interval of a function of the form y=Asin(ωx+φ)+b or y=Acos(ωx+φ)+b (where A≠0, ω>0, and b is a constant), you can use the monotonicity of the sine function and the cosine function. The interval is found by solving the inequality.

2. Pay attention to two points when solving the specific problem: ① Treat ωx + φ as a whole. If ω<0, first use the induction formula to deform the formula and make the coefficient of x positive; ② When A>0, ω>0, "ωx+φ" is substituted into the monotonic interval of the sine (or cosine) function, and the monotonic interval consistent with the monotonicity can be obtained; when A<0, ω>0, the same method can be used to obtain the monotonic interval that is opposite to the monotonicity of the sine (cosine) function. Monotone interval.

Reminder: The monotonicity of composite functions follows the law of "same increase and different decrease".

Using the monotonicity of trigonometric functions to compare sizes

[Example 2] Use the monotonicity of trigonometric functions to compare the sizes of the following groups of numbers.

(1) sin-π18 and sin-π10;

(2) sin 196° and cos 156°;

(3) cos-235π and cos-174π.

[Ideas Enlightenment] Use the induced formula to simplify → Use the monotonicity of the function to deduce the size of the corresponding function value from the size of the independent variable

regular method

Strategies for Comparing Trigonometric Function Values

1 Using the induction formula, for the sine function, the two angles are generally converted into -π2, π2 or π2, 3π2; for the cosine function, the two angles are generally converted into -π, 0&# 093; or [0, within π].

2 Functions with different names are converted into functions with the same name.

3. If the independent variables are not in the same monotonic interval, use the monotonicity of the sine and cosine functions to compare the magnitude.

Class summary

1. There are many methods to determine the monotonic interval of a trigonometric function, such as the substitution method, the list method, the image method, etc. When solving the problem, you need to choose appropriately. At the same time, you must pay attention to finding the monotonic interval of the function must be carried out within the definition domain of the function.

2. The most basic application of function monotonicity is to compare the size and evaluation domain. There are many methods to find the value domain of trigonometric functions. If the functional formula contains multiple trigonometric function formulas, the functional formula must be deformed first.

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

1. Thinking and analysis

(1)y=sin x is an increasing function on (0, π). ()

(2)cos 1>cos 2>cos 3.()

(3) Function y=-12sin x, x∈0, the maximum value of π2 is 0.()

2. The value range of y=2cos x2 is ()

A． [-2,2]

B. [0,2]

C． [-2,0]

D. R

3. sin2π7________sin-15π8 (fill in “＞” or “＜”).

4. The monotonically increasing interval of the function y=1-sin 2x.

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For more information about the PPT courseware "Graphics and Properties of Trigonometric Functions Trigonometric Functions Monotonicity and Maximum Values ​​of Sine and Cosine Functions" PPT courseware, please click on the ppt tag of Trigonometric Functions ppt Graphics and Properties of Trigonometric Functions ppt Monotonicity and Maximum Values ​​of Sine and Cosine Functions .

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Update Time: 2024-06-29

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