Home Grape Presentation of the sine function, its properties and graph. Graphs and properties of trigonometric functions of sine and cosine. VI. Consolidation of the material covered

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Presentation of the sine function, its properties and graph. Graphs and properties of trigonometric functions of sine and cosine. VI. Consolidation of the material covered

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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

An interactive test that contains 5 tasks with the choice of one correct answer out of four proposed, taking into account the time spent on passing the test; The test was created in PowerPoint-2007 with...

Graphs and properties of the trigonometric functions of sine and cosine Graph of the function y = sinx Graph of the function y = sinx Properties of the function y = sinx Properties of the function y = sinx Graph of the function y = cosx Graph of the function y = cosx Properties of the function y = cosx Properties of the function y = cosx Comparison of properties functions y = sinx and y = cosx Comparison of properties of functions y = sinx and y = cosx

Properties of the function y = sinx 6. Constancy intervals of the function y = sinx: sinx > 0 for x (2k; +2k), sinx 0 for x (2k; +2k), sinx 0 for x (2k; +2k), sinx 0 for x (2k; +2k), sinx 0 for x (2k; +2k), sinx title="(!LANG:Properties of the function y = sinx 6. Constancy intervals of the function y = sinx: sinx > 0 for x (2k; +2k), sinx

Properties of the function y = cosx k cosx 0 for x (-/2+k;/2+k), k cosx 0 for x (-/2+k;/2+k), k cosx 0 for x (-/2+k;/2 +k), k cosx title="(!LANG:Properties of the function y = cosx

Comparison of properties of functions y = sinx and y = cosx Function y = sinxy = cosx Domain D(sinx) = D(cosx) = Set of values E(sinx) = [-1,1]E(cosx) = [-1,1] Even and odd odd even Zeros of the function x = k, k x = /2+k, k Intervals of constant sign y(x)>0 x (2k; +2k)x (- /2+k; /2+k) k y(x ) 0 x (2k; +2k)x (- /2+k; /2+k) k y(x)

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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 development 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 problems, systemic generalizations, self-examination, perception of new material, mutual verification.

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 checking, 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. Notebooks with test work and completed homework are collected. 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