Presentation "Function y=cosx, its properties and graph". Graphs and properties of trigonometric functions of sine and cosine. Sine function their properties and graphs presentation

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Function y \u003d sin x, its properties and graph. Lesson objectives: Repeat and systematize the properties of the function y \u003d sin x. Learn how to plot a function y \u003d sin x.

y = sin x The domain of definition is the set R of all real numbers: D(f) = (- ∞; + ∞) Property 1.

y = sin x Since sin (-x) = - sin x, then y = sin x is an odd function, which means that its graph is symmetrical about the origin. Property 2.

y = sin x The function y = increases on the interval and decreases on the interval [ π /2; π]. Property 3. 0 π /2 π

y = sin x The function y = sin x is bounded both from below and from above: - 1 ≤ sin x ≤ 1 Property 4.

y = sin x y max = -1 y max = 1 Property 5 . 0 π /2 π

Let's build a graph of the function y = sin x in a rectangular coordinate system Oxy.

y 0 π /2 π x

First, let's build a part of the graph on the segment . -2 π -3 π /2 - π - π /2 0 π /2 π 3 π /2 2 π X 1 -1 Y x 0 π /6 π /3 π /2 2 π /3 5 π /6 π y 0 1/2 √ 3/2 1 √ 3/2 1/2 0 Now let's build a part of the graph on the segment [ - π ; 0 ], given the oddness of the function y= sin x . On the segment [ π ; 2 π ] the graph of the function looks like this again: And on the segment [ -2 π ; - π ] the graph of the function looks like this: Thus, the entire graph is a continuous line, which is called a sinusoid. Sine wave arch Half-wave sine wave

No. 168 - orally. -3 π -5 π /2 -2 π -3 π /2 - π - π /2 0 π /2 π 3 π /2 2 π 5 π /2 3 π X Y 1 -1

Solve exercises 170, 172, 173 (a, b). Homework: No. 171, 173 (c, d)

On the topic: methodological developments, presentations and notes

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One of the important terms in trigonometry is cosine. In this presentation, the cosine function will be considered, its graph will be built. All properties that it possesses will be given in detail.

On the first slide, before starting to consider the function itself, one of the cast formulas is recalled. It was previously demonstrated in detail along with the proof.

This formula says that the cosine function can be replaced by a sine with certain changes in the argument. Thus, having already studied sinusoids, schoolchildren will be able to build this function. As a result, they will get a graph of the cosine function.

The graph of the function can be seen on the second slide. It can be noted that the sinusoid has only shifted to Pi/2. Thus, unlike a sine wave, the graph of the cosine function does not pass through the point (0; 0).

The first step would be to consider the scope of the function. This is an important point and the analysis of any function in mathematics begins with this. The scope of this function is the entire numerical axis. This is clearly seen in the graph of the function.

Unlike the sine, the cosine function is even. That is, if you change the sign of the argument, the sign of the function will not change. Evenness is determined by the property of the sine.

At certain intervals, the function increases, at certain intervals, it decreases. This suggests that the cosine function is monotonic. These intervals are shown on the next slide. On the graph, you can clearly see the increase and decrease of the function.

The fifth property is limitation. The cosine function is bounded both above and below. The minimum value is -1 and the maximum is +1.

Since there are no breakpoints and sharp peaks, the cosine function, like the sine function, is continuous.

The last slide shows a summary of all the properties that were discussed in the presentation. These are a number of basic characteristics that the cosine function has. By memorizing them, you can easily cope with a number of equations that contain cosine. It will be easiest to master these properties in the case of a complete understanding of the essence.

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Lesson Objectives:

To develop students' ability to draw a graph of a function y=sinx, according to the schedule to read its properties. Create conditions for monitoring the assimilation of knowledge and skills.
Developing - to promote the formation of skills to apply techniques: comparisons, generalizations, identifying the main thing, transferring knowledge to a new situation, developing mathematical horizons, thinking and speech, attention and memory.
Educational - to promote the education of interest in mathematics and its applications, activity, mobility, communication skills, general culture.

Teaching methods: partially search. Checking the level of knowledge, working on a generalizing scheme, solving cognitive generalizing tasks, systemic generalizations, self-testing, perception of new material, mutual testing.

Lesson organization forms: individual, frontal, work in pairs.

Equipment and sources of information: Screen; multimedia projector; notebook. Cards with mathematical dictation, answers to questions of mathematical dictation, cards with prescribed properties of a function y=sinx.

Lesson plan:

Organizational moment.
Repetition of the studied material.
Test work on knowledge control topic: "Reduction formulas".
Systematization of theoretical material on plotting the function y=sinx and its properties.
Explanation of new material.
Consolidation of new material.
Summing up the lesson.
Homework.

During the classes

I. Organizational moment.

(slide 2)

The French writer Anatole France (1844-1924) once remarked: "Learning can only be fun ... To digest knowledge, you must absorb it with appetite." So, let's follow this advice of the writer today in the lesson, we will be active, attentive, we will absorb knowledge with great desire, because they will be useful to you in your later life. * (MOU secondary school No. 256, Fokino).

Today we have the first lesson on the topic of trigonometric functions. We will look at their graphs and properties. Let's start with the topic: "The function y=sinx, its properties and graph". We are faced with the task of applying our knowledge and skills in constructing graphs of functions.

II. Repetition of the studied material.

(slide 3)

Topic: " Cast formulas»

Target: Repeat the rule for applying reduction formulas. Focus on the rule model: quarter, sign, function.

1. Consider examples: , , , , .

III. Verification work.

(slide 4)

Topic: " Cast formulas»

Target: Knowledge control and bringing into the knowledge system by reduction formulas.

The work is carried out in two versions, tasks are projected onto the screen. Two students perform the same task at the boards on the cards.

Option 1	Option 2

The work is over, the students exchange notebooks for mutual verification, two students mark their answers on the screen, the class comments on the correctness of the assignments. Students control the correctness of the test work and give the neighbor an assessment. "5" - 5 completed tasks, "4" - 4 tasks, "3" - 3 tasks. Assemble notebooks with verification work and homework done. The assessment will be announced at the next lesson, taking into account the completeness of the homework done.

IV. Systematization of theoretical material.

(slide 5)

Topic: " Properties of function graphs»

Target: Repetition of the description of the properties of the function according to the finished graph.

domain;
function zeros;
intervals of sign constancy;
increase, decrease of function;
limitation;
even, odd;
range of values;
find the largest and smallest value of the function on the interval .

V. Explanation of new material.

(Slide 6-8)

Purpose: to consider the graph of a function; formulate the properties of the function.

Students in notebooks depict a coordinate unit circle and a coordinate system for parallel consideration of the sine values \u200b\u200bon the unit circle and plotting points in the prepared coordinate system. After the students realize the principle of constructing a curve, the teacher comments on this work through the "cells". The points are built according to the scheme through:

“on the axis”, “cell corner”, “almost one”, “one”, then the movement occurs in the reverse order: “almost one”, “cell corner”, “on the axis”.

The teacher says that this curve is called a sinusoid.

(Slide 9.)

After plotting the graph, the students, similarly to the work done with the previous function, write down the properties of the function . In all properties, we assume that .

Function Properties

function zeros: x=πk,

>0 on (2πk, π+ 2πk),

<0 на (-π+ 2πk, 2πk),

- increases by

- decreases to

, ,

odd function

VI. Consolidation of the material covered.

(Slide 10)

Purpose: Applying the acquired knowledge: finding the values of the function.

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"Functions of tangent and cotangent" - Properties of the function y \u003d tgx. Solutions. Equation roots. Schedule. Building a graph. Function properties. Meaning. Fraction. Basic properties of the function. Function y = tgx. Basic properties. y=ctgx. Graph of the function y=ctgx. Numbers.

"Conversion of trigonometric graphs" - Sine function. Converting graphs of trigonometric functions. Characteristic of the graph of harmonic oscillation. Graph of the function y=f(x)+m. cosine function. Graph of the function y=f(|x|). Graph of the function y=|f(x)|. Characterization of transformations of graphs of functions. Y=f(x). Tangent function. Sections of the resulting graph.

"Arcfunctions" - Functional-graphical method for solving equations. Arctgx. Function. trigonometric functions. Properties of arc functions. Y \u003d arcctgx. Arcctg t = a. Arccosx. Graphical method for solving equations. Value area. Equality. Definitions. Expression. Definition. Arctg t. Arccos t. The set of real numbers.

"Algebra "Trigonometric functions"" - Trigonometric functions of the angular argument. Table of values of trigonometric functions of some angles. Handbook of Algebra and the Beginnings of Analysis. Solution of trigonometric inequalities. Solution of trigonometric equations. Converting sums of trigonometric functions to products. Trigonometry.

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