Trigonometric functions, their properties and presentation graphs. Trigonometric function graphs - presentation. Inverse trigonometric functions

Slide 6

Double angle trigonometric functions:

sin 2x \u003d 2sinx cosx cos 2x \u003d cos2x -sin2x tg 2x \u003d 2tg x / (1-tg2x) ctg 2x \u003d ctg2x-1 / (2 ctg x)

Slide 7

Half Angle Trigonometric Functions

Often useful are formulas expressing the powers of sin and cos of a simple argument through sin and cos of a multiple, for example: Formulas for cos2x and sin2x can be used to find the values \u200b\u200bof T. f. half argument

Slide 8

Trigonometric functions of the sum of angles

sin (x + y) \u003d sin x cos y + cos x sin y sin (xy) \u003d sin x cos y - cos x sin y cos (x + y) \u003d cos x cos y - sin x sin y cos (xy) \u003d cos x cos y + sin x sin y

Slide 9

For large values \u200b\u200bof the argument, you can use the so-called reduction formulas that allow you to express T. f. any argument through T. f. argument x, which simplifies the compilation of tables T. f. and their use, as well as plotting. These formulas have the form: in the first three formulas n can be any integer, with the upper sign corresponding to the value n \u003d 2k, and the lower one - to the value n \u003d 2k + 1; in the latter, n can only be an odd number, with the upper sign taken at n \u003d 4k + 1, and the lower one at n \u003d 4k - 1.

Slide 10

The most important trigonometric formulas are addition formulas expressing T. f. the sum or difference of the values \u200b\u200bof the argument through T. f. these values: the signs on the left and right sides of all formulas are consistent, that is, the upper (lower) sign on the left corresponds to the upper (lower) sign on the right. From them, in particular, formulas for T. f. multiple arguments, for example:

Slide 11

Derivatives of all Trigonometric functions are expressed in terms of Trigonometric functions

Slide 12

The graph of the function y \u003d sinx is:

Slide 13

The graph of the function y \u003d cosx is:

Slide 14

The graph of the function y \u003d tgx looks like:

Slide 15

The graph of the function y \u003d ctgx looks like:

Slide 16

The history of the emergence of trigonometric functions

T. f. arose for the first time in connection with research in astronomy and geometry. The ratios of segments in a triangle and a circle, which are essentially T. f., Are found already in the 3rd century. BC e. in the works of the mathematicians of Ancient Greece - Euclid, Archimedes, Apollonius of Perga, and others. However, these relations are not an independent object of research for them, so that T. f. as such they were not studied. T. f. were considered initially as segments and in this form were used by Aristarchus (late 4th - 2nd half of the 3rd centuries BC)

Slide 17

Hipparchus (2nd century BC), Menelaus (1st century AD) and Ptolemy (2nd century AD) when solving spherical triangles. Ptolemy compiled the first table of chords for acute angles in 30 "with an accuracy of 10-6. The expansion of the thermal function in power series was obtained by I. Newton (1669). He owns the definition of the TF for real and complex arguments, the currently accepted symbolism, the establishment of a connection with the exponential function, the orthogonality of the system of sines and cosines

View all slides

Trigonometric functions

x \u003d cost. Presentation on the topic: "Trigonometric functions". Number circle. All numbers with denominator 4 correspond to Cartesian coordinates. Accurate to the sign, depending on the quarter in which the point is located. Sine, cosine, tangent and cotangent. Quarter Signs: Properties of sine, cosine, tangent and cotangent. Basic trigonometric formulas. Relationship between trigonometric functions of angular and numeric argument. arc length AM - numeric argument, Angle. - Angular argument. Values \u200b\u200bof trigonometric functions. Training exercises. Point P divides the third quarter in a ratio of 1: 5. Find the length of the arc CP, PD, AP. - Trigonometric functions.ppt

Examples of trigonometric functions

Trigonometric functions. Trigonometric functions of an acute angle. Right-angled triangle ABC. For some angles, exact values \u200b\u200bcan be recorded. Connection of acute angle trigonometric functions. Double angle trigonometric functions. Half-angle trigonometric functions. Trigonometric functions of the sum of angles. You can use the so-called reduction formulas. The most important trigonometric formulas are addition formulas. Derivatives of all trigonometric functions. Graph of the function y \u003d sinx. Function graph y \u003d cosx. Function graph y \u003d tgx. - Examples of trigonometric functions.ppt

Basic trigonometric functions

Trigonometric functions. Mathematical model. Determination of evenness and oddness of a function. Domain. The set of values \u200b\u200bof trigonometric functions. Find the scope of the function. Function definition area. The set of function values. Periodicity. Which function is even. Function g (x). Value. Positive period. Function properties. Function graph. Properties of the function y \u003d sin x. Points. X values. Gaps. Range of values. Plot the function. Properties of the function y \u003d tg (x). Function y \u003d tg (x). Find the scope. Use a formula to define a function. - Basic trigonometric functions.ppt

Algebra "Trigonometric functions"

A guide to algebra and the beginnings of analysis. Content. Trigonometry. Sine and cosine. Tangent and Cotangent. Trigonometric functions of a numeric argument. Trigonometric functions of an angular argument. Casting formulas. Table of values \u200b\u200bof trigonometric functions of some angles. Transformation formulas for trigonometric functions. Formulas for converting a product of trigonometric functions into a sum. Converting sums of trigonometric functions into products. Complementary angle formula. Arcsine. Solving the simplest trigonometric equations. Homogeneous trigonometric equations. - Algebra "Trigonometric functions" .ppt

Properties of trigonometric functions

Properties of trigonometric functions. Mathematical cafe. Crossword. Defining each property of a function. Exercise for the eyes. Read the graph of the function. Reading the graph of a function. Physical education. List the properties. The task. - Properties of trigonometric functions.ppt

Trigonometric functions and their properties

What are the similarities and differences between trigonometric functions? Problem question: Educational project on the topic: You, me and trigonometry. Trigonometric functions. Definition. Trigonometric functions Number circle. The equation of the number circle: x2 + y2 \u003d 1. The movement along the number circle is counterclockwise. Trigonometric functions Sine and cosine. Trigonometric functions tangent and cotangent. Trigonometric functions of a numeric argument. Trigonometric functions Function y \u003d sin x. The line serving as the graph of the function y \u003d sin x is called a sinusoid. - Trigonometric functions and their properties.ppt

Angular Trigonometric Functions

The values \u200b\u200bof the trigonometric functions of the angular argument. Summarize and systematize educational material on the topic. Trigonometric functions of a numeric argument. The cosine of the angle A (cos A) is the abscissa (x) of the point. The values \u200b\u200bof the trigonometric functions of the angles of the unit circle. Values \u200b\u200bof trigonometric functions of basic angles. Values \u200b\u200bof trigonometric functions of other corners of the table. Signs of trigonometric functions in the quarters of the unit circle. Casting formulas. The task. Independent work. - Angular Argument Trigonometric Functions.ppt

Trigonometric function graphs

Trigonometric function graphs. Trigonometric functions. The graph of the function y \u003d sin x is a sinusoid. y \u003d sin x. Properties of the function y \u003d sin x. y \u003d sin x. Properties of the function y \u003d sin x. 6. Intervals of monotony: the function increases on intervals of the form: [-p / 2 + 2pn; p / 2 + 2pn], n? Z. Monotonic intervals: the function decreases on intervals of the form:, n? Z. Properties of the function y \u003d sin x. 7. Extremum points: Xmax \u003d p / 2 + 2pn, n? Z Xmin \u003d -p / 2 + 2pn, n? Z. 8. Range of values: E (y) \u003d [-1; 1]. Conversion of graphs of trigonometric functions. Plot the Function y \u003d sin (x + p / 4). - Graphs of trigonometric functions.ppt

Converting trigonometric plots

Conversion of graphs of trigonometric functions. Characteristic of transformations of graphs of functions. Stretching. Function graph. Compression. Graph of the function y \u003d f (x). Parallel transfer. Graph of the function y \u003d f (x) + m. Transfer. Y \u003d f (x). Graph of the function y \u003d f (| x |). Part of the schedule. Graph of the function y \u003d | f (x) |. Plots of the resulting schedule. Graph of the function y \u003d | f (| x |) |. Characteristic of the harmonic oscillation graph. Sine function. Cosine function. Tangent function. Cotangent function. - Convert trigonometric plots.ppt

Plotting trigonometric functions

Converting charts. Knowledge formation. Application of MS Excel program. Function graphs. Plotting a function. Parallel chart transfer. Building a graph. Moving the chart along the Ox axis. Y2 \u003d sinx + 2.Y1 \u003d sinx. Y \u003d sin (x + 1.5) +2. Construction. Y \u003d af (x). Y2 \u003d 2sinx. Y \u003d 2sin (x + 1.5) + 2. Build your own graphs. Y \u003d sin (x - 0.75) + 2.Y \u003d 2.5cos (x + 1.5) -1. Graph of the function y \u003d f (x + t) + m. - Plotting trigonometric functions.ppt

Converting Trigonometric Functions Plots

Lesson-presentation “Graphs of trigonometric functions. Converting Graphs ”. Lesson equipment: computer, projector, screen. Objectives: To generalize knowledge and skills. Develop the ability to observe, compare, generalize. Cultivate cognitive activity, perseverance in achieving goals. Introductory word of the teacher. Let's take a closer look at the graphs of trigonometric functions. "Graphs of trigonometric functions". Overview of trigonometric functions. Y \u003d sinx Y \u003d cosx. The first student. 1.Sine function. 2.Cosine function. Second student. Overview of trigonometric functions. y \u003d tgx y \u003d ctgx. - Convert graphs of trigonometric functions .ppt

Y sinx function

Properties and graph of the SINUS function. Oral warm-up. cos90 °. sin90 °. sin (? / 4). cos180 °. sin270 °. sin (? / 3). cos (? / 6). cos360 °. ctg (? / 6). tg (? / 4). sin (3? / 2). cos (2?). cos (-? / 2). cos (? / 3). cos (??). Name the functions whose graphs are shown in the figure. y \u003d cosx. Plotting y \u003d sin x. y \u003d \u003d sinx. P - six cells. Plot the function y \u003d sinx using the trigonometric circle. P - three cells. Create a template for the graph of the function y \u003d sinx. Sinus axis. sin0 \u003d 0. sinp \u003d 0. sin (-p) \u003d 0. Basic properties of the function y \u003d sinx. Domain. - The set R of all real numbers. - Function y sinx.pptx

Function y \u003d cos x

Function y \u003d cos x. Plotting the function y \u003d cos x. Building a graph. How to use periodicity and parity when plotting. Let's find some points for plotting. Let us extend the resulting graph along the entire number line. Function graph. How to find the scope. Domain. Lots of meanings. Periodicity. Even, odd. Increase, decrease. Function zeros, positive and negative values. Properties of the function y \u003d cos x. Transforming the graph of the function y \u003d cos x. Y \u003d cos x + A. Y \u003d cos x + A (properties). Y \u003d k cos x. Y \u003d kcos x (properties). - Function y \u003d cos x.ppt

Tangent function

Properties of the function y \u003d tg x and its graph. Lesson objectives. Region definitions. The function y \u003d tg x is increasing. Plotting the function y \u003d tg x. Properties of the function y \u003d tg x. The function y \u003d tgx is undefined. The set of function values. Find all roots of the equation. Find all solutions to the inequality. - tangent function.ppt

Tangent and Cotangent Functions

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

ArcFunctions

Inverse trigonometric functions. Function. Equality. Trigonometric functions. Domain. Function definition area. Definition. Arccos t. Arctg t. Arcctg t \u003d a. Definitions. Range of values. Lots of real numbers. Y \u003d arcctgx. Arccosx. Expression. Find the values \u200b\u200bof the expressions. Arctgx. Properties of arc functions. Graphical method for solving equations. Functional-graphic method for solving equations. - ArcFunctions.ppt

Inverse trigonometric functions

Inverse trigonometric functions. From the history of trigonometric functions. Ancient Greece. III century BC e. Euclid, Apolonius of Perga. Side ratios in a right-angled triangle. OK. 190 BC e Hipparchus of Nicea. Abu al-Waf introduced the trigonometric functions tangent and cotangent. Karl Scherfer introduced modern notation for inverse trigonometric functions. Y \u003d arcsinx is strictly ascending. Properties of the function y \u003d arcsin x. The arccosine of a number m is such an angle x for which: The function y \u003d arccosx is strictly decreasing. Properties of the function y \u003d arccos x. - Inverse trigonometric functions.ppt

Properties of inverse trigonometric functions

Elective course in mathematics. Inverse trigonometric functions. Solving equations. Research work. Calculate. Oral exercises. Specify the scope of the function. Specify the range of values \u200b\u200bof the function. Find the meaning of the expression. Decision. Let's solve the system of equations. Term. Initial equation. The triple satisfies the original equation. Reiteration. Arc functions. Working in groups. Solve equations. -

Contents 1. Introduction slide 2. Start learning slide 3. Stages of learning slide 4. Function groups slide 5. Definition and graph of sine slide 6. Definition and graph of cosine slide 7. Definition and graph of tangent slide 8. Definition and graph of cotangent slide 9. Inverse three functions slide 10. Basic formulas slide 11. The meaning of trigonometry slide 12. Literature used slide 13. Author and compiler of the slide

In ancient times, trigonometry arose in connection with the needs of astronomy, surveying and construction, that is, it was purely geometric in nature and represented mainly the "calculus of chords". Over time, some analytical moments began to be interspersed into it. In the first half of the 18th century, there was a sharp change, after which trigonometry took a new direction and shifted towards mathematical analysis. It was at this time that trigonometric dependencies began to be considered as functions. This is of not only mathematical and historical, but also methodological and pedagogical interest. In ancient times, trigonometry arose in connection with the needs of astronomy, surveying and construction, that is, it was purely geometric in nature and represented mainly the "calculus of chords". Over time, some analytical moments began to be interspersed into it. In the first half of the 18th century, there was a sharp change, after which trigonometry took a new direction and shifted towards mathematical analysis. It was at this time that trigonometric dependencies began to be considered as functions. This is of not only mathematical and historical, but also methodological and pedagogical interest.

At present, much attention is paid to the study of trigonometric functions precisely as functions of a numerical argument in the school course of algebra and the beginnings of analysis. There are several different approaches to teaching this topic in a school course, and a teacher, especially a beginner, can easily get confused about which approach is most appropriate. But trigonometric functions are the most convenient and visual means for studying all the properties of functions (before using the derivative), and especially such a property of many natural processes as periodicity. Therefore, their study should be given close attention.

In addition, great difficulties in studying the topic "Trigonometric functions" in the school course arise due to the discrepancy between a sufficiently large amount of content and the relatively small number of hours allocated for the study of this topic. Thus, the problem of this research work is the need to eliminate this discrepancy through careful selection of the content and the development of effective methods of presenting this material. The object of the research is the process of studying the functional line in the high school course. The subject of the research is the method of studying trigonometric functions in the algebra course and the beginning of analysis in the classroom.

Trigonometric functions Trigonometric functions are mathematical functions of an angle. They are important in the study of geometry, as well as in the study of periodic processes. Usually trigonometric functions are defined as the ratios of the sides of a right-angled triangle or the lengths of certain segments in the unit circle. More modern definitions express trigonometric functions in terms of the sums of series or as solutions of some differential equations, which makes it possible to expand the domain of definition of these functions to arbitrary real numbers and even to complex numbers.

In the study of trigonometric functions, the following stages can be distinguished: I. First acquaintance with trigonometric functions of an angular argument in geometry. The argument value is considered in the range (0о; 90о). At this stage, students will learn that sin, cos, tg and ctg of the angle depend on its degree measure, get acquainted with tabular values, the basic trigonometric identity and some reduction formulas. II. Generalization of the concepts of sine, cosine, tangent and cotangent for angles (0о; 180о). At this stage, the relationship between trigonometric functions and the coordinates of a point on the plane is considered, theorems of sines and cosines are proved, and the question of solving triangles using trigonometric relations is considered. III. Introduction of the concepts of trigonometric functions of a numeric argument. IV. Systematization and expansion of knowledge about trigonometric functions of a number, consideration of graphs of functions, research, including with the help of a derivative.

There are several ways to define trigonometric functions. They can be divided into two groups: analytical and geometric. 1. The analytical methods include the definition of the function y \u003d sin x as a solution to the differential equation f (x) \u003d - c * f (x) or as the sum of the power series sin x \u003d x - x3 / 3! + X5 / 5! -… 2. Geometric methods include the definition of trigonometric functions based on projections and coordinates of the radius vector, determination through the aspect ratio of the sides of a right-angled triangle and determination using a number circle. In the school course, preference is given to geometric methods due to their simplicity and clarity.

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

X y 1 y \u003d cosx Individual survey (review of materials from the previous day)

On the site I found an interesting material "Model of biorhythms" To build a model of biorhythms, you must enter the date of birth of a person, the date of reference (day, month, year) and the duration of the forecast (number of days). As you can see, the graph is a sinusoid.

On the site I found material that the trajectory of a bullet coincides with a sinusoid. It is seen from the figure that the projections of the vectors on the X and Y axes are, respectively, υ x \u003d υ o cos α υ y \u003d υ o sin α

On the site math.ru/load/shkolnaja_matematika/alge bra_10_klass / grafiki_trigon / there is material about the 360 \u200b\u200b° rotation of the Earth in 365 days. Interestingly, this can be represented as a sinusoid. math.ru/load/shkolnaja_matematika/alge bra_10_klass / grafiki_trigon /

In physics lessons, we studied the oscillatory motion of a pendulum. On the site I found material that the pendulum oscillates along a curve called the cosine

Anatole France Learning can only be fun ... To digest knowledge, you need to absorb it with appetite. Lunch.

Properties of the function 1. D (tan х) \u003d R, except for х \u003d П / 2 + Пn, 2. E (tan х) \u003d R. 3. Periodic function with the main period T \u003d П. 4. Odd function. 5. Increases over the entire domain of definition 6. Zeros of the function: y (x) \u003d 0 for x \u003d Pn, 7. Not bounded either above or below. 8. There is no highest or lowest value. Function graph y \u003d tg x.

Properties of the function y \u003d ctg x 1. D (ctg x) \u003d R, except for x \u003d Pn, 2. E (ctg x) \u003d R. 3. Periodic function with the main period T \u003d P. 4. Odd function. 5. Decreases on the entire domain of definition 6. Zeros of the function: y (x) \u003d 0 for x \u003d 2/2 + n, 7. Not bounded either above or below. 8. There is no highest or lowest value.

It might be helpful to read:

• Presentation "Strong and weak electrolytes" presentation of a lesson for an interactive whiteboard in chemistry (grade 9) on the topic Presentation on the use of electrolytes in medicine;
• Presentation on the topic: "Acids - as electrolytes" Presentation on the use of electrolytes in medicine;
• Presentation vladimir dal presentation for a reading lesson on the topic Literary presentation for grade 4;
• More about telephoto lenses;
• Creativity of Korney Ivanovich Chukovsky presentation for a reading lesson (grade 2) on the topic;
• Presentation on the topic "Biography of Paustovsky. Presentation on the topic of Paustovsky biography;
• Migratory and wintering birds presentation Migratory and wintering birds presentation for children;
• Traumatic brain injury Presentation on the topic of closed traumatic brain injury;