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Lesson objectives:
- Form students' ability to draw a graph of a function y = sinx, according to the schedule to read its properties. Create conditions for controlling the assimilation of knowledge and skills.
- Developing - to contribute to the formation of skills to apply techniques: comparison, generalization, identifying the main thing, transferring knowledge to a new situation, developing a mathematical outlook, thinking and speech, attention and memory.
- Educational - to contribute to fostering interest in mathematics and its applications, activity, mobility, communication skills, general culture.
Teaching methods: partial search. Testing 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 = sinx.
Lesson plan:
- Organizational moment.
- Repetition of the material learned.
- Testing work on the control of knowledge topic: "Formulas of reduction".
- Systematization of theoretical material on the construction of the graph of the function y = sinx and its properties.
- Explanation of the new material.
- Securing new material.
- Summing up the lesson.
- 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 tutorial on trigonometric functions. We will look at their graphs and properties. And let's start the study with the topic: "Function y = sinx, its properties and graph." Our task is to apply our knowledge and skills when constructing graphs of functions.
II. Repetition of the material learned.
(Slide 3)
Topic: " Casting Formulas "
Target: Repeat the rule of applying the casting formulas. Focus on the rule model: quarter, sign, function.
1. Consider examples:,,,,.
III. Verification work.
(Slide 4)
Topic: " Casting Formulas "
Target: Knowledge control and bringing it into the knowledge system according to the reduction formulas.
The work is carried out in two versions, the tasks are projected on the screen. Two students carry out the same task over 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. Students monitor the correctness of the test and give the neighbor a grade. "5" - 5 completed tasks, "4" - 4 tasks, "3" - 3 tasks. Collect notebooks with test work and homework completed. The assessment will be announced in the next lesson, taking into account the completeness of the completed homework.
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 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 of the sine values on the unit circle and plotting points in 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 = πk, |
| > 0 on (2πk, π + 2πk), |
| <0 на (-π+ 2πk, 2πk), |
- increases by  ,
|
- decreases by  ,
|
| , , |
| , , |
| function is odd |
Vi. Consolidation of the passed material.
(Slide 10)
Purpose: Application of the knowledge gained: finding the values of the function.
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Slide captions:
Function y = sin x, its properties and graph. Lesson objectives: To review and systematize the properties of the function y = sin x. Learn to plot the function y = sin x.
y = sin x Domain of definition - 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 its graph is symmetric about the origin. Property 2.
y = sin x The function y = increases on the segment and decreases on the segment [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 naim = -1 y naib = 1 Property 5. 0 π / 2 π
Let us construct a graph of the function y = 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 = 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 will be able to 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 is 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 can be clearly seen in the graph of the function.
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. The graph clearly shows 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 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 in the case of a complete understanding of the essence.
The section in the mathematics of trigonometry includes the study of concepts such as sine, cosine, tangent, and cotangent. Separately, schoolchildren will need to consider each function, study the nature of the behavior on the graph, consider the frequency, scope, range of values and other parameters.
So the sine function. The first slide shows the general view of the function. The variable t is used as an argument.
The first step, as with every function, is the scope, which indicates what values the argument can take. In the case of sine, this is the entire number axis. You can see this later on the function graph.
The second property, which is considered using sine as an example, is parity. The sinusoid is odd. This is because the function of -x will be equal to the function with a minus sign. In order to recall this material, you can return to previous presentations and view.
This property is demonstrated on the unit circle that appears on the left side of the slide. Thus, the property is proved geometrically as well.
The third property that must also be considered is the property of monotony. On some segments the function increases, on some it decreases. This enables us to call the sinusoid a monotonic function. Since the intervals of increase and decrease are infinite, this is noted by periodicity.
The fourth property is limitation. The sinusoid is bounded at both the top and bottom. The minimum value, in this case, is 1, the maximum is +1. Thus, the sine function is bounded both above and below.
The definition of a sinusoid is given, which must be filled. Further, various deformations of a sinusoid at different values are considered.
After the definition is given, consideration of the properties of the sine function continues. It is continuous. This can be clearly seen on the graph of the function. No breakpoints exist.
The last slide shows how you can graphically solve an equation that contains a sine function. This method will simplify the solution and make it clearer.
