Presentation "Function y \u003d cosx, its properties and graph". Graphs and properties of sine and cosine trigonometric functions. Sine function their properties and graphics presentation

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Function y \u003d sin x, its properties and graph. Lesson objectives: To review and organize the properties of the function y \u003d sin x. Learn to plot the function y \u003d sin x.

y \u003d sin x Domain of definition - the set R of all real numbers: D (f) \u003d (- ∞; + ∞) Property 1.

y \u003d sin x Since sin (-x) \u003d - sin x, then y \u003d sin x is an odd function, so its graph is symmetric about the origin. Property 2.

y \u003d sin x The function y \u003d increases on the segment and decreases on the segment [π / 2; π]. Property 3.0 π / 2 π

y \u003d sin x The function y \u003d sin x is bounded both from below and from above: - 1 ≤ sin x ≤ 1 Property 4.

y \u003d sin x y naim \u003d -1 y naib \u003d 1 Property 5. 0 π / 2 π

Let us plot the function y \u003d sin x in the rectangular coordinate system Oxy.

y 0 π / 2 π x

First, let's build a part of the graph on a 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 draw a part of the graph on the segment [- π; 0], taking into account the oddness of the function y \u003d sin x. On the segment [π; 2 π] the graph of the function looks like this again: And on the interval [-2 π; - π] the graph of the function looks like this: Thus, the whole graph is a continuous line, which is called a sinusoid. Sine arc Half 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 Х У 1 -1

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


On the subject: methodological developments, presentations and notes

Interactive test, which 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 and ...

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

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

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


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

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


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


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 value is + 1.


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

The last slide summarizes all the properties that were discussed in the presentation. These are some of the main characteristics that the cosine function has. Having memorized them, you can easily cope with a number of equations that contain a cosine. It will be easiest to master these properties if you fully understand the essence.











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Lesson objectives:

  1. Form students' ability to draw a graph of a function y \u003d sinx, according to the schedule to read its properties. Create conditions for controlling the assimilation of knowledge and skills.
  2. Developing - to promote the formation of skills to apply techniques: comparison, generalization, identification of the main thing, transfer of knowledge to a new situation, development of mathematical horizons, thinking and speech, attention and memory.
  3. Educational - to promote the development of interest in mathematics and its applications, activity, mobility, communication skills, general culture.

Teaching methods:partial search. Checking the level of knowledge, working according to 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 \u003d sinx.

Lesson plan:

  1. Organizational moment.
  2. Repetition of the material studied.
  3. Testing work on the control of knowledge topic: "Formulas of reduction".
  4. Systematization of theoretical material on the construction of the graph of the function y \u003d sinx and its properties.
  5. Explanation of the new material.
  6. Securing new material.
  7. Summing up the lesson.
  8. Homework.

During the classes

I. Organizational moment.

(Slide 2)

The French writer Anatole France (1844-1924) once remarked: "You can only learn fun ... To digest knowledge, you need to 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 a great desire, because they will be useful to you in your future life. * (School № 256, Fokino).

Today we have our first lesson on the topic trigonometric functions... We will look at their charts and properties. And let's start the study with the topic: "Function y \u003d sinx, its properties and graph."Our task is to apply our knowledge and skills when building graphs of functions.

II. Repetition of the material studied.

(Slide 3)

Subject: "Casting formulas "

Goal:Repeat the rule of applying the casting formulas. Focus on the rule model: quarter, sign, function.

1. Consider examples:,,,,.

III. Verification work.

(Slide 4)

Subject: "Casting formulas "

Goal: Knowledge control and bringing it into the knowledge system according to reduction formulas.

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

Option 1 Option 2

The work is over, the students change notebooks for mutual checking, on the screen two students mark their answers, the class comments on the correctness of the assignments. Pupils monitor the correctness of the test work and give a grade to a neighbor. "5" - 5 completed tasks, "4" - 4 tasks, "3" - 3 tasks. Collecting notebooks with verification work and homework done. The assessment will be announced in the next lesson, taking into account the completeness of the homework completed.

IV. Systematization of theoretical material.

(Slide 5)

Subject: "Properties of function graphs "

goal: Repetition of the description of the properties of the function according to the finished schedule.

  • domain;
  • function zeros;
  • intervals of constancy;
  • increasing, decreasing function;
  • limitation;
  • even, odd;
  • range of values;
  • find the largest and smallest value of the function on the segment.

V. Explanation of the new material.

(Slide 6-8)

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

Pupils in notebooks depict the coordinate unit circle and coordinate system for parallel consideration on the unit circle of the sine values \u200b\u200band plotting points into the prepared coordinate system. After the students understand the principle of constructing the curve, the teacher comments on this work through the "cells". Points are drawn according to the scheme through:

"On the axis", "corner of the cell", "almost one", "one", then the movement occurs in the reverse order: "almost one", "corner of the cell", "on the axis".

The teacher says that this curve is called a sinusoid.

(Slide 9.)

After building the graph, 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
zeros of the function: x \u003d πk,
\u003e 0 on (2πk, π + 2πk),
<0 на (-π+ 2πk, 2πk),
- increases by ,
- decreases by ,
, ,
, ,
function odd

Vi. Consolidation of the passed material.

(Slide 10)

Purpose: Application of the knowledge gained: finding the values \u200b\u200bof the function.

"Properties of inverse trigonometric functions" - Inverse trigonometric functions. Oral exercises. Let's solve the system of equations. Elective course in mathematics. Initial equation. Arc functions. Solve equations. Working in groups. Research. Reiteration. Solving equations. Term. Calculate. Specify the scope of the function. Decision.

"Function y \u003d cos x" - Y \u003d k · cos x (properties). Y \u003d - cos x. Increase, decrease. Y \u003d cos (-x) (properties). Plotting the function y \u003d cos x. Y \u003d | cos x | (properties). Properties of the function y \u003d cos x. Y \u003d k cos x. Y \u003d | cos x |. How to find the scope. Y \u003d - cos x (properties). Function zeros, positive and negative values.

"Arcfunctions" - Arccos t. Y \u003d arcctgx. Find the values \u200b\u200bof the expressions. Function. Graphical method for solving equations. Expression. Equality. Inverse trigonometric functions. Domain. Trigonometric functions. Arccosx. Function definition area. Definitions. Range of values. Definition. Functional-graphic method for solving equations.

"Algebra" Trigonometric functions "" - Solving homogeneous trigonometric equations. Casting formulas. Converting sums of trigonometric functions into products. Transformation formulas for trigonometric functions. Formulas for converting a product of trigonometric functions into a sum. Homogeneous trigonometric equations. Sine and cosine.

"Converting trigonometric graphs" - Parallel transfer. Stretching. Compression. Graph of the function y \u003d f (| x |). Y \u003d f (x). Part of the schedule. Cotangent function. Graph of the function y \u003d | f (| x |) |. Characteristic of the harmonic oscillation graph. Plots of the resulting schedule. Graph of the function y \u003d f (x). Convert graphs of trigonometric functions. Graph of the function y \u003d | f (x) |.

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

There are 18 presentations in total

"Function y \u003d cos x" - Zeros of the function, positive and negative values. Let's find some points for plotting. Y \u003d cos (x - a). Transforming the graph of the function y \u003d cos x. Function y \u003d cos x. Y \u003d cos x + A (properties). Properties. Symmetrical reflection about the abscissa axis. Function graph. Even, odd.

"Properties of inverse trigonometric functions" - Specify the range of values \u200b\u200bof the function. Solve equations. Find the meaning of the expression. Solving equations. Working in groups. Elective course in mathematics. Arc functions. Let's solve the system of equations. Research. Specify the scope of the function. Reiteration. The triple satisfies the original equation.

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

"Convert trigonometric graphs" - Sine function. Conversion of graphs of trigonometric functions. Characteristic of the harmonic oscillation graph. Graph of the function y \u003d f (x) + m. Cosine function. Graph of the function y \u003d f (| x |). Graph of the function y \u003d | f (x) |. Characteristic of transformations of graphs of functions. Y \u003d f (x). Tangent function. Plots of the resulting schedule.

"Arcfunctions" - Functional-graphic method for solving equations. Arctgx. Function. Trigonometric functions. Properties of arc functions. Y \u003d arcctgx. Arcctg t \u003d a. Arccosx. Graphical method for solving equations. Range of values. Equality. Definitions. Expression. Definition. Arctg t. Arccos t. Lots of real numbers.

"Algebra" Trigonometric functions "" - Trigonometric functions of an angular argument. Table of values \u200b\u200bof trigonometric functions of some angles. A guide to algebra and the beginnings of analysis. Solution of trigonometric inequalities. Solving trigonometric equations. Converting sums of trigonometric functions into products. Trigonometry.

 

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