Lesson on the topic trigonometric functions of the angular argument. Trigonometric functions of angular argument

Whatever real number t is taken, it can be assigned a uniquely defined number sin t. True, the correspondence rule is rather complicated; as we saw above, it consists in the following.

To find the value of sin t by the number t, you need:

1) position the number circle in the coordinate plane so that the center of the circle coincides with the origin, and the starting point A of the circle hits the point (1; 0);

2) find a point on the circle corresponding to the number t;

3) find the ordinate of this point.

This ordinate is sin t.

In fact, we are talking about the function u = sin t, where t is any real number.

All these functions are called trigonometric functions of the numerical argument t.

There are a number of relationships connecting the values ​​of various trigonometric functions, we have already received some of these relationships:

sin 2 t + cos 2 t = 1

From the last two formulas, it is easy to obtain a relation connecting tg t and ctg t:

All of these formulas are used in cases where, knowing the value of a trigonometric function, it is required to calculate the values ​​of the remaining trigonometric functions.

The terms "sine", "cosine", "tangent" and "cotangent" were actually familiar, however, they were still used in a slightly different interpretation: in geometry and physics, they considered sine, cosine, tangent and cotangent g l a(but not

numbers, as it was in the previous paragraphs).

It is known from geometry that the sine (cosine) of an acute angle is the ratio of the leg of a right triangle to its hypotenuse, and the tangent (cotangent) of an angle is the ratio of the legs of a right triangle. A different approach to the concepts of sine, cosine, tangent and cotangent was developed in the previous paragraphs. In fact, these approaches are interrelated.

Let's take an angle with a degree measure b o and arrange it in the model "numerical circle in a rectangular coordinate system" as shown in Fig. fourteen

corner top compatible with center

circles (with the origin of the coordinate system),

and one side of the corner is compatible with

the positive ray of the x-axis. Point

intersection of the other side of the angle with

the circle will be denoted by the letter M. Ordina-

Figure 14 b o , and the abscissa of this point is the cosine of the angle b o .

To find the sine or cosine of the angle b o it is not at all necessary to make these very complex constructions each time.

It suffices to note that the arc AM is the same part of the length of the numerical circle as the angle b o is from the angle of 360°. If the length of the arc AM is denoted by the letter t, then we get:

Thus,

For example,

It is believed that 30 ° is a degree measure of an angle, and is a radian measure of the same angle: 30 ° = rad. Generally:

In particular, I'm glad from where, in turn, we get.

So what is 1 radian? There are various measures of segment lengths: centimeters, meters, yards, etc. There are also various measures to indicate the magnitude of the angles. We consider the central angles of the unit circle. An angle of 1° is a central angle based on an arc that is part of a circle. An angle of 1 radian is a central angle based on an arc of length 1, i.e. on an arc whose length is equal to the radius of the circle. From the formula, we get that 1 rad \u003d 57.3 °.

Considering the function u = sin t (or any other trigonometric function), we can consider the independent variable t as a numerical argument, as was the case in the previous paragraphs, but we can also consider this variable as a measure of the angle, i.e. angular argument. Therefore, speaking of a trigonometric function, in a certain sense it is indifferent to consider it a function of a numerical or angular argument.

Lesson and presentation on the topic: "The trigonometric function of the angular argument, the degree measure of the angle and radians"

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What will we study:
1. Let's remember the geometry.
2. Definition of the angular argument.
3. Degree measure of an angle.
4. Radian measure of an angle.
5. What is a radian?
6. Examples and tasks for independent solution.

Geometry repetition

Guys, in our functions:

y= sin(t), y= cos(t), y= tg(t), y= ctg(t)

The variable t can take not only numeric values, that is, be a numeric argument, but it can also be considered as a measure of an angle - an angular argument.

Let's remember geometry!
How did we define sine, cosine, tangent, cotangent there?

The sine of an angle is the ratio of the opposite leg to the hypotenuse

Cosine of an angle - the ratio of the adjacent leg to the hypotenuse

The tangent of an angle is the ratio of the opposite leg to the adjacent one.

The cotangent of an angle is the ratio of the adjacent leg to the opposite one.

Definition of the trigonometric function of the angular argument

Let's define trigonometric functions as functions of an angle argument on a number circle:
With the help of a numerical circle and a coordinate system, we can always easily find the sine, cosine, tangent and cotangent of an angle:

We place the vertex of our angle α at the center of the circle, i.e. to the center of the coordinate axis, and position one of the sides so that it coincides with the positive direction of the x-axis (OA)
Then the second side intersects the number circle at the point M.

Ordinate points M: sine of angle α
Abscissa points M: cosine of angle α

Note that the length of the arc AM is the same part of the unit circle as our angle α from 360 degrees: where t is the length of the arc AM.

Degree measure of an angle

1) Guys, we got a formula for determining the degree measure of an angle through the length of an arc of a numerical circle, let's take a closer look at it:

Then we write the trigonometric functions in the form:

For example:

Radian measure of angles

When calculating the degree or radian measure of an angle, remember! :
For example:

By the way! Designation rad. you can drop!

What is a radian?

Dear friends, we have come across a new concept - Radian. So what is it?

There are various measures of length, time, weight, for example: meter, kilometer, second, hour, gram, kilogram and others. So the Radian is one of the measures of the angle. It is worth considering the central angles, that is, located in the center of the numerical circle.
An angle of 1 degree is a central angle based on an arc equal to 1/360 of the circumference.

An angle of 1 radian is a central angle based on an arc equal to 1 in a unit circle, and in an arbitrary circle on an arc equal to the radius of the circle.

Examples:

Examples of converting from a degree measure of an angle to a radian, and vice versa

Tasks for independent solution

1. Find the radian measure of angles:
a) 55° b) 450° c) 15° d) 302°

2. Find:
a) sin(150°) b) cos(45°) c) tg(120°)

3. Find the degree measure of angles:

The video lesson "Trigonometric functions of the angular argument" is a visual material for conducting a mathematics lesson on the relevant topic. The video is composed in such a way that the studied material is presented as convenient as possible for students to understand, is easy to remember, well reveals the connection between the available information about trigonometric functions from the section on studying triangles and their definition using a unit circle. It can become an independent part of the lesson, as it fully covers this topic, supplemented by important comments during the scoring.

Animation effects are used to visually demonstrate the relationship between different definitions of trigonometric functions. Highlighting the text in color, clear understandable constructions, supplementing with comments helps to quickly master, remember the material, and achieve the goals of the lesson faster. The connection between the definitions of trigonometric functions is clearly demonstrated using animation effects and color highlighting, contributing to understanding and memorization of the material. The manual is aimed at improving the effectiveness of training.

The lesson starts with a topic introduction. Then the definitions of sine, cosine, tangent and cotangent of an acute angle of a right triangle are recalled. The definition highlighted in the box recalls that the sine and cosine are formed as the ratio of the leg to the hypotenuse, the tangent and cotangent are formed by the ratio of the legs. Students are also reminded of the recently studied material that when considering a point belonging to a unit circle, the abscissa of the point is the cosine, and the ordinate is the sine of the number corresponding to this point. The connection of these concepts is demonstrated using construction. A unit circle is displayed on the screen, placed so that its center coincides with the origin. A ray is constructed from the origin of coordinates, making an angle α with the positive semi-axis of the abscissa. This ray intersects the unit circle at point O. Perpendiculars descend from the point to the abscissa and y-axis, demonstrating that the coordinates of this point determine the cosine and sine of the angle α. It is noted that the length of the arc AO from the point of intersection of the unit circle with the positive direction of the abscissa axis to the point O is the same part of the entire arc as the angle α from 360°. This allows you to make the proportion α/360=t/2π, which is displayed right there and highlighted in red for memorization. The value t=πα/180° is derived from this proportion. Taking this into account, the relationship between the definitions of sine and cosine sinα°= sint= sinπα/180, cosα°=cost=cosπα/180 is determined. For example, finding sin60 ° is given. Substituting the degree measure of the angle into the formula, we obtain sin π 60°/180°. Reducing the fraction by 60, we get sin π/3, which is equal to √3/2. It is noted that if 60° is the degree measure of an angle, then π/3 is called the radian measure of the angle. There are two possible records of the ratio of the degree measure of the angle to the radian: 60°=π/3 and 60°=π/3 rad.

The concept of an angle of one degree is defined as a central angle based on an arc whose length 1/360 represents part of the circumference. The following definition reveals the concept of an angle of one radian - a central angle based on an arc of length one, or equal to the radius of a circle. Definitions are marked as important and highlighted for memorization.

To convert one degree measure of an angle to a radian and vice versa, the formula α ° \u003d πα / 180 rad is used. This formula is highlighted in a frame on the screen. From this formula it follows that 1°=π/180 rad. In this case, one radian corresponds to an angle of 180°/π≈57.3°. It is noted that when finding the values ​​of trigonometric functions of the independent variable t, it can be considered both a numerical argument and an angular one.

Further, examples of using the acquired knowledge in the course of solving mathematical problems are demonstrated. In example 1, it is required to convert values ​​from degrees to radians 135° and 905°. On the right side of the screen, there is a formula that displays the relationship between a degree and a radian. After substituting the value into the formula, we get (π/180) 135. After reducing this fraction by 45, we get the value 135°=3π/4. To convert an angle of 905° to radians, the same formula is used. After substituting the value into it, it turns out (π / 180) 905 \u003d 181π / 36 rad.

In the second example, the inverse problem is solved - the degree measure of angles expressed in radians π/12, -21π/20, 2.4π is found. On the right side of the screen, the studied formula for the relationship between the degree and radian measure of the angle 1 rad \u003d 180 ° / π is recalled. Each example is solved by substituting the radian measure into the formula. Substituting π/12, we get (180°/π)·(π/12)=15°. Similarly, the values ​​of the remaining angles -21π/20=-189° and 2.4π=432° are found.

The video lesson "Trigonometric functions of the angular argument" is recommended to be used in traditional mathematics lessons to increase the effectiveness of learning. The material will help to provide visualization of learning during distance learning on this topic. A detailed, understandable explanation of the topic, solving problems on it can help the student master the material on his own.

TEXT INTERPRETATION:

"Trigonometric functions of angular argument".

We already know from geometry that the sine (cosine) of an acute angle of a right triangle is the ratio of the leg to the hypotenuse, and the tangent (cotangent) is the ratio of the legs. And in algebra, we call the abscissa of a point on the unit circle a cosine, and the ordinate of this point a sine. We will make sure that all this is closely interconnected.

Let's place an angle with a degree measure α° (alpha degrees), as shown in Figure 1: the vertex of the angle is compatible with the center of the unit circle (with the origin of the coordinate system), and one side of the angle is compatible with the positive ray of the x-axis. The second side of the angle intersects the circle at point O. The ordinate of point O is the sine of the angle alpha, and the abscissa of this point is the cosine of alpha.

Note that the arc AO is the same part of the length of the unit circle as the angle alpha is from the angle of three hundred and sixty degrees. Let us denote the length of the AO arc through t(te), then we will make up the proportion =

(alpha refers to trusts of sixty as te to two pi). From here we find te: t = = (te equals pi alpha divided by one hundred and eighty).

Thus, to find the sine or cosine of the angle alpha degrees, you can use the formula:

sin α ° \u003d sint \u003d sin (the sine of alpha degrees is equal to the sine of te and is equal to the sine of private pi alpha to one hundred and eighty),

cosα° \u003d cost \u003d cos (the cosine of alpha degrees is equal to the cosine of te and is equal to the cosine of private pi alpha to one hundred and eighty).

For example, sin 60 ° \u003d sin \u003d sin \u003d (the sine of sixty degrees is equal to the sine of pi by three, according to the table of basic values ​​​​of sines, it is equal to the root of three by two).

It is believed that 60 ° is a degree measure of an angle, and (pi by three) is a radian measure of the same angle, that is, 60 ° = glad(sixty degrees equals pi times three radians). For brevity, we have agreed notation glad omit, that is, the following notation is allowed: 60°= (show abbreviations radian measure = rad.)

An angle of one degree is a central angle that is supported by an arc that is (one three hundred and sixtieth) part of the arc. An angle of one radian is a central angle that rests on an arc of length one, that is, on an arc whose length is equal to the radius of a circle (we consider the central angles of a unit circle to show an angle in pi radians on a circle).

Let's remember the important formula for converting a degree measure into a radian:

α° = glad. (alpha equals pi alpha divided by one hundred and eighty radians) In particular, 1° = glad(one degree equals pi divided by one hundred and eighty radians).

From this we can find that one radian is equal to the ratio of one hundred and eighty degrees to pi and is approximately equal to fifty-seven point three tenths of a degree: 1 glad= ≈ 57.3°.

From the above: when we talk about any trigonometric function, for example, about the function s \u003d sint (es is equal to sinus te), the independent variable t (te) can be considered both a numerical argument and an angular argument.

Consider examples.

EXAMPLE 1. Convert from degrees to radians: a) 135°; b) 905°.

Decision. Let's use the formula for converting degrees to radians:

a) 135° = 1° ∙ 135 = glad ∙ 135 = glad

(one hundred and thirty-five degrees is equal to pi times one hundred and eighty radians times one hundred thirty-five, and after reduction is three pi times four radians)

b) Similarly, using the formula for converting a degree measure into a radian, we obtain

905° = glad ∙ 905 = glad.

(nine hundred and five degrees equals one hundred and eighty-one pi times thirty-six radians).

EXAMPLE 2. Express in degrees: a) ; b) -; c) 2.4π

(pi times twelve; minus twenty-one pi times twenty; two point four tenths of a pi).

Decision. a) Express in degrees pi by twelve, use the formula for translating the radian measure of the angle into the degree measure in 1 glad=, we get

glad = 1 glad∙ = ∙ = 15°

Similarly b) - = 1 glad∙ (-) \u003d ∙ (-) \u003d - 189 ° (minus twenty-one pi by twenty equals minus one hundred eighty-nine degrees),

c) 2.4π = 1 glad∙ 2.4π = ∙ 2.4π = 432° (two point four of pi equals four hundred and thirty-two degrees).

Trigonometric functions of a numeric argument we parsed. We took point A on the circle and looked for sines and cosines from the resulting angle β.

We denoted the point as A, but in algebra it is often denoted as t and all formulas/functions are given with it. We also will not deviate from the canons. Those. t - it will be a certain number, and therefore numeric function(e.g. sint)

It is logical that since we have a circle with a radius of one, then

Trigonometric functions of angular argument we also successfully parsed it - according to the canons, we will write for such functions: sin α °, meaning by α ° any angle with the number of degrees we need.

The ray of this angle will give us the second point on the circle (OA - point A) and the corresponding points C and B for the numerical argument function, if we need it: sin t = sin α°

Lines of sines, cosines, tangents and cotangents

Never forget that the y-axis is the sine line, the x-axis is the line of cosines! The points obtained from the circle are marked on these axes.

BUT the lines of tangents and cotangents are parallel to them and pass through the points (1; 0) and (0; 1) respectively.