Presentation on the topic Schedule Kosinus. Presentation "Function Y \u003d SINX, its properties and a schedule". V. Explanation of the new material

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Function y \u003d sin x, its properties and graph. Objectives of the lesson: repeat and systematize the properties of the function y \u003d sin x. Learn to build a function graph y \u003d sin x.

y \u003d SIN X The definition area is the set R of all valid 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, it means that its graph is symmetrical relative to the start of coordinates. 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 limited and below, and from above: - 1 ≤ sin x ≤ 1 Property 4.

y \u003d sin x y nym \u003d -1 y Naib \u003d 1 property 5. 0 π / 2 π

We construct a graph of the function y \u003d sin x in the rectangular coordinate system okhu.

in 0 π / 2 π x

First, we 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 in 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 we will now construct a part of the graph on the segment [- π; 0], given the oddness of the function y \u003d sin x. On the segment [π; 2 π] the graph of the function looks again like this: and on the segment [-2 π; - π] The graph of the function looks like this: thus, the entire schedule is a continuous line called a sinusoid. Arch sinusoids half-wave sinusoids

№ 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

Decide exercises 170, 172, 173 (A, B). Homework: № 171, 173 (B, D)


On the topic: Methodical development, presentations and abstracts

An interactive test that contains 5 tasks with a choice of one right response from four offered, taking into account the time spent on the passage of the test; The test is created in the PowerPoint-2007 C and ...

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

On the first slide, before you begin consideration of the function directly, one of the formulas of the lead is reminded. Earlier, it was demonstrated in detail with the proof.

This formula suggests that the cosine function can be replaced with sinus with certain changes in the change in the argument. Thus, after studying the sinusoids, schoolchildren can build this feature. As a result, they will receive a graph of the cosine function.


The graph of the function can be seen on the second slide. You can pay attention to that sinusoid only shifted on Pi / 2. Thus, in contrast to the sinusoid, the graph of the cosine function does not pass through the point (0; 0).

First of all, it would be worth considering the field definition area. This is an important point and the analysis of any function in mathematics begins. The area of \u200b\u200bdefinition of this function is the entire number axis. This is clearly visible on the function graph.


Unlike sinus, the cosine function is even. That is, if you change the argument sign, the function sign will not change. The parity is determined by the sinus property.


At certain intervals, the function increases, on certain - decreases. This suggests that the cosine function is monotonous. These intervals are shown on the following slide. On the graph, you can clearly see the increasing and decrease of the function.


Fifth property is limited. The cosine function has limitations from above, and below. The minimum value is -1, and the maximum - + 1.


Since there are no points of rupture and sharp peaks - the cosine function, as well as the sinus function, is continuous.

On the last slide demonstrates generally all the properties that were considered in the presentation. This is a number of basic characteristics that have a cosine function. By remembering them, you can easily cope with a number of equations that contain cosine. The easiest way to master these properties in the case of a complete understanding of the essence.

The trigonometry section in mathematics includes the study of such concepts as sinus, cosine, tangent and catangent. In detachable, schoolchildren will need to consider each function, learn the nature of behavior on the chart, consider the frequency, the definition area, the range of values \u200b\u200band other parameters.

So, the function of sinus. The first slide displays a general view of the function. A variable T is used as an argument.

First of all, as at each function, the definition area is considered, which indicates which values \u200b\u200ban argument may receive. In the case of sinus, this is the whole number axis. You can see this later on the function graph.


The second property, which is considered on the example of sinus is parity. The sinusoid is odd. This is explained by the fact that the function from the function will be equal to the minus sign. In order to recall this material, you can return to previous presentations and view.


This property is demonstrated on a single circle that appears in the left side of the slide. Thus, the property is proved and geometrically.


The third property that needs to be considered is the property of monotony. On some segments, the function increases, on some - decreases. This gives us the opportunity to name the sinusoid monotonous function. Since increasing and descending intervals are infinite number, it is observed by periodicity.


Fourth property - limitations. The sinusoid is limited and from above, and below. The minimum value, with this, - 1, the maximum +1. Thus, the function of sine is limited and from above, and below.


The definition of the sinusoids that must be filled out. Further considers various deformations of sinusoids at different values.

After the definition is given, the properties of the sinus function continues. It is continuous. This is clearly visible on the function graph. There are no discontinuity points.

The last slide shows how the equation can be solved in graphically, which contains the sinus function. This method will simplify the decision and make it more visual.

 

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