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Lesson objectives:

1. Developing skills and abilities to use trigonometric formulas to simplify trigonometric expressions.
2. Implementation of the principle of an activity approach in teaching students, developing students’ communication skills and tolerance, the ability to listen and hear others and express their opinions.
3. Increasing students' interest in mathematics.

Lesson type: training.

Lesson type: lesson on skills and abilities.

Form of study: group

Type of groups: group sitting together. Students of different levels of training, awareness of a given subject, compatible students, which allows them to complement and enrich each other.

Equipment: board; chalk; table "Trigonometer"; route sheets; cards with letters (A, B, C.) for completing the test; plates with crew names; score sheets; tables with names of stages of the journey; magnets, multimedia complex.

During the classes

Students sit in groups: 4 groups of 5-6 people. Each group is a crew of a car with names corresponding to the names of trigonometric functions, led by a steering wheel. Each crew is given a route sheet and a goal is determined: to complete the given route successfully, without errors. The lesson is accompanied by a presentation.

I. Organizational moment.

The teacher informs the topic of the lesson, the purpose of the lesson, the course of the lesson, the work plan of the groups, the role of the helmsmen.

Teacher's opening remarks:

– Guys! Write down the number and topic of the lesson: “Trigonometric functions of a numerical argument.”

Today in class we will learn:

1. Calculate values ​​of trigonometric functions;
2. Simplify trigonometric expressions.

To do this you need to know:

1. Definitions of trigonometric functions
2. Trigonometric relations (formulas).

It has long been known that one head is good, but two are better, so today you work in groups. It is also known that the one who walks will master the road. But we live in an age of speed and time is precious, which means we can say this: “The road will be mastered by those who drive,” so today our lesson will be held in the form of a game “Mathematical Rally.” Each group is a vehicle crew, led by a steering wheel.

Purpose of the game:

• successfully complete the route for each crew;
• identify rally champions.

The name of the crews corresponds to the make of the car you are driving.

The crews and their helmsmen are introduced:

• Crew – “sine”
• Crew – “cosine”
• Crew - "tangent"
• Crew – “cotangent”

The motto of the race: “Hurry up slowly!”

You have to run through a “mathematical terrain” with many obstacles.

Route sheets were issued to each crew. Crews who know definitions and trigonometric formulas will be able to overcome obstacles.

During the run, each helmsman guides the crew, assisting, and assessing the contribution of each crew member to overcome the route in the form of “pros” and “cons” on the score sheet. For each correct answer the group receives a “+” and an incorrect answer “-”.

You have to overcome the following stages of the journey:

Stage I. SDA (traffic rules).
Stage II. Technical inspection.
Stage III. Cross-country race.
Stage IV. A sudden stop is an accident.
V stage. Halt.
Stage VI. Finish.
VII stage. Results.

And so off we go!

Stage I. SDA (traffic rules).

1) In each crew, the helmsmen distribute tickets with theoretical questions to each crew member:

1. Explain the definition of the sine of t and its signs by quarters.
2. Explain the definition of the cosine of the number t and its signs by quarters.
3. State the smallest and largest values ​​of sin t and cos t.
4. Explain the definition of the tangent of the number t and its signs by quarters.
5. Explain the definition of the cotangent of the number t and its signs by quarters.
6. Tell us how to find the value of the sin t function from a known number t.

2) Collect the “scattered” formulas. There is a table on the secret board (see below). The crews must align the formulas. Each team writes the answer on the board in the form of a line of corresponding letters (in pairs).

Answer: ab, vg, de, hedgehog, zi, yk.

Stage II. Technical inspection.

Oral work: test.

On the secret board it is written: task: simplify the expression.

The answer options are written next to them. Crews determine the correct answers in 1 minute. and pick up the corresponding set of letters.

Answer: C V A.

Stage III. Cross-country race.

The crews have 3 minutes for a meeting to decide the task, and then the crew representatives write the decision on the board. When the crew representatives finish writing down the solution to the first task, all students (together with the teacher) check the correctness and rationality of the solutions and write them down in a notebook. The helmsmen evaluate the contribution of each crew member using the “+” and “–” signs on the evaluation sheets.

Tasks from the textbook:

• Crew – “sine”: No. 118 g;
• Crew – “cosine”: No. 122 a;
• Crew – “tangent”: No. 123 g;
• Crew – “cotangent”: No. 125

Stage IV. A sudden stop is an accident.

– Your car has broken down. Your car needs to be repaired.

Statements are given for each crew, but there are mistakes in them. Find these mistakes and explain why they were made. The statements are used trigonometric functions, corresponding to the brands of your cars.

V stage. Halt.

You are tired and need to rest. While the crew is resting, the helmsmen sum up preliminary results: they count the “pros” and “cons” of the crew members and the crew as a whole.

For students:

3 or more “+” – score “5”;
2 “+” – rating “4”;
1 “+” – rating “3”.

For crews:“+” and “-” cancel each other out. Only the remaining characters are counted.

Guess the charade.

From the numbers you take my first syllable,
The second is from the word “proud”.
And you will drive the third horses,
The fourth will be the bleating of a sheep.
My fifth syllable is the same as the first
The last letter in the alphabet is the sixth,
And if you guess everything correctly,
Then in mathematics you will get a section like this.
(Trigonometry)

The word "trigonometry" (from the Greek words "trigonon" - triangle and "metreo" - measure) means "measurement of triangles." The emergence of trigonometry is associated with the development of geography and astronomy - the science of the movement of celestial bodies, the structure and development of the Universe.

As a result of the astronomical observations made, the need arose to determine the position of the luminaries, calculate distances and angles. Since some distances, for example, from the Earth to other planets, could not be measured directly, scientists began to develop techniques for finding relationships between the sides and angles of a triangle, in which two vertices are located on the earth, and the third is a planet or star. Such relationships can be derived by studying various triangles and their properties. This is why astronomical calculations led to the solution (i.e., finding the elements) of the triangle. This is what trigonometry does.

The beginnings of trigonometry were discovered in ancient Babylon. Babylonian scientists were able to predict solar and lunar eclipses. Some information of a trigonometric nature is found in ancient monuments of other ancient peoples.

Stage VI. Finish.

To successfully cross the finish line, all you have to do is strain yourself and make a “sprint.” It is very important in trigonometry to be able to quickly determine the values ​​of sin t, cost, tgt, ctg t, where 0 ≤ t ≤ . Close textbooks.

Crews alternately name the values ​​of the functions sin t, cost, tgt, ctg t if:

VII stage. Results.

Results of the game.

The helmsmen hand over evaluation sheets. The crew that became the champion of the “Mathematical Rally” is determined and the work of the remaining groups is characterized. Next are the names of those who received grades “5” and “4”.

Lesson summary.

- Guys! What did you learn in class today? (simplify trigonometric expressions; find values ​​of trigonometric functions). What do you need to know for this?

• definitions and properties sin t, cos t, tg t, ctg t;
• relations connecting the values ​​of various trigonometric functions;
• signs of trigonometric functions on the quarters of the number circle.
• values ​​of trigonometric functions of the first quarter of the number circle.

– I think you understand that you need to know the formulas well in order to apply them correctly. You also realized that trigonometry is a very important part of mathematics, as it is used in other sciences: astronomy, geography, physics, etc.

Homework:

• for students who received “5” and “4”: §6, No. 128a, 130a, 134a.
• for other students: §6, No. 119g, No. 120g, No. 121g.

Trigonometric functions of a numeric argument.

Trigonometric functions of numeric argumentt are functions of the form y= cos t,
y= sin t, y= tg t, y= ctg t.

Using these formulas, through the known value of one trigonometric function, you can find the unknown values ​​of other trigonometric functions.

Explanations.

1) Take the formula cos 2 t + sin 2 t = 1 and use it to derive a new formula.

To do this, divide both sides of the formula by cos 2 t (for t ≠ 0, that is, t ≠ π/2 + π k). So:

cos 2 t sin 2 t 1
--- + --- = ---
cos 2 t cos 2 t cos 2 t

The first term is equal to 1. We know that the ratio of sine to conis is tangent, which means the second term is equal to tg 2 t. As a result, we get a new (and already known to you) formula:

2) Now divide cos 2 t + sin 2 t = 1 by sin 2 t (for t ≠ π k):

cos 2 t sin 2 t 1
--- + --- = ---, where t ≠ π k + π k, k– integer
sin 2 t sin 2 t sin 2 t

The ratio of cosine to sine is the cotangent. Means:

Knowing the basic principles of mathematics and having learned the basic formulas of trigonometry, you can easily derive most of the other trigonometric identities on your own. And this is even better than just memorizing them: what you learn by heart is quickly forgotten, but what you understand is remembered for a long time, if not forever. For example, it is not necessary to memorize what the sum of one and the square of the tangent is equal to. If you forgot, you can easily remember if you know the simplest thing: tangent is the ratio of sine to cosine. In addition, apply the simple rule of adding fractions with different denominators and get the result:

sin 2 t 1 sin 2 t cos 2 t + sin 2 t 1
1 + tg 2 t = 1 + --- = - + --- = ------ = ---
cos 2 t 1 cos 2 t cos 2 t cos 2 t

In the same way, you can easily find the sum of one and the square of the cotangent, as well as many other identities.

Trigonometric functions of angular argument.

In functionsat = cost, at = sint, at = tgt, at = ctgt variablet can be more than just a numeric argument. It can also be considered a measure of the angle - that is, the angular argument.

Using the number circle and coordinate system, you can easily find the sine, cosine, tangent, and cotangent of any angle. To do this, two important conditions must be met:
1) the vertex of the angle must be the center of the circle, which is also the center of the coordinate axis;

2) one of the sides of the angle must be a positive axis beam x.

In this case, the ordinate of the point at which the circle and the second side of the angle intersect is the sine of this angle, and the abscissa of this point is the cosine of this angle.

Explanation. Let's draw an angle, one side of which is the positive ray of the axis x, and the second side comes out from the origin of the coordinate axis (and from the center of the circle) at an angle of 30º (see figure). Then the point of intersection of the second side with the circle corresponds to π/6. We know the ordinate and abscissa of this point. They are also the cosine and sine of our angle:

√3 1
--; --
2 2

And knowing the sine and cosine of an angle, you can easily find its tangent and cotangent.

Thus, the number circle, located in a coordinate system, is a convenient way to find the sine, cosine, tangent, or cotangent of an angle.

But there is an easier way. You don’t have to draw a circle and a coordinate system. You can use simple and convenient formulas:

Example: find the sine and cosine of an angle equal to 60º.

Solution :

π 60 π √3
sin 60º = sin --- = sin -- = --
180 3 2

π 1
cos 60º = cos -- = -
3 2

Explanation: we found out that the sine and cosine of an angle of 60º correspond to the values ​​of a point on a circle π/3. Next, we simply find the values ​​of this point in the table - and thus solve our example. The table of sines and cosines of the main points of the number circle is in the previous section and on the “Tables” page.

Definition. Trigonometric functions of a numerical argument are the same-named trigonometric functions of an angle equal to radians.

Let us explain this definition with specific examples.

Example 1. Let's calculate the value. Here we mean an abstract irrational number. According to the definition. So, .

Example 2. Let's calculate the value. Here, by 1.5 we mean an abstract number. As defined (see Appendix II).

Example 3. Calculate the value We obtain the same as above (see Appendix II).

So, in the future, by the argument of trigonometric functions we will understand an angle (arc) or just a number, depending on the problem we are solving. And in some cases, the argument can be a quantity that has another dimension, for example time, etc. By calling an argument an angle (arc), we can mean by it the number with which it is measured in radians.

The main trigonometric identity in Russian mathematics textbooks is the relation sin 2 ⁡ α + cos 2 ⁡ α = 1

We have looked at the most basic trigonometric functions (do not be fooled, in addition to sine, cosine, tangent and cotangent, there are many other functions, but more on them later), but for now let’s look at some basic properties of the functions already studied.

Trigonometric functions of numeric argument

Whatever real number t is taken, it can be associated with a uniquely defined number sin(t) . True, the matching rule is quite complex and consists of the following.

To find the value of sin(t) from the number t, you need:

1. position the number circle on the coordinate plane so that the center of the circle coincides with the origin of coordinates, and the starting point A of the circle falls at point (1; 0);
2. find a point on the circle corresponding to the number t;
3. find the ordinate of this point.
4. this ordinate is the desired sin(t) .

In fact, we are talking about the function s = sin(t) , where t is any real number. We can calculate some values ​​of this function (for example, sin(0) = 0, \(sin \frac (\pi)(6) = \frac(1)(2) \) etc.), we know some of its properties.

In the same way, we can consider that we have already received some ideas about three more functions: s = cos(t) s = tan(t) s = ctg(t) All these functions are called trigonometric functions of the numerical argument t.

Relationship between trigonometric functions

As you, I hope, can guess, all trigonometric functions are interconnected and even without knowing the meaning of one, it can be found through another.

For example, the most important formula in all trigonometry is basic trigonometric identity:

\[ sin^(2) t + cos^(2) t = 1 \]

As you can see, knowing the value of the sine, you can find the value of the cosine, and also vice versa. Also very common formulas connecting sine and cosine with tangent and cotangent:

\[ \boxed (\tan\; t=\frac(\sin\; t)(\cos\; t), \qquad t \neq \frac(\pi)(2)+ \pi k) \]

\[ \boxed (\cot\; t=\frac(\cos\; )(\sin\; ), \qquad t \neq \pi k) \]

From the last two formulas one can derive another trigometric identity, this time connecting tangent and cotangent:

\[ \boxed (\tan \; t \cdot \cot \; t = 1, \qquad t \neq \frac(\pi k)(2)) \]

Now let's see how these formulas work in practice.

EXAMPLE 1. Simplify the expression: a) \(1+ \tan^2 \; t \), b) \(1+ \cot^2 \; t \)

a) First of all, let’s write the tangent, keeping the square:

\[ 1+ \tan^2 \; t = 1 + \frac(\sin^2 \; t)(\cos^2 \; t) \]

\[ 1 + \frac(\sin^2 \; t)(\cos^2 \; t)= \sin^2\; t + \cos^2\; t + \frac(\sin^2 \; t)(\cos^2 \; t) \]

Now let’s put everything under a common denominator, and we get:

\[ \sin^2\; t + \cos^2\; t + \frac(\sin^2 \; t)(\cos^2 \; t) = \frac(\cos^2 \; t + \sin^2 \; t)(\cos^2 \; t ) \]

And finally, as we see, the numerator can be reduced to one by the main trigonometric identity, as a result we get: \[ 1+ \tan^2 \; = \frac(1)(\cos^2 \; t) \]

b) With the cotangent we perform all the same actions, only the denominator will no longer be a cosine, but a sine, and the answer will be like this:

\[ 1+ \cot^2 \; = \frac(1)(\sin^2 \; t) \]

Having completed this task, we derived two more very important formulas that connect our functions, which we also need to know like the back of our hands:

\[ \boxed (1+ \tan^2 \; = \frac(1)(\cos^2 \; t), \qquad t \neq \frac(\pi)(2)+ \pi k) \]

\[ \boxed (1+ \cot^2 \; = \frac(1)(\sin^2 \; t), \qquad t \neq \pi k) \]

You must know all the formulas presented by heart, otherwise further study of trigonometry without them is simply impossible. In the future there will be more formulas and there will be a lot of them and I assure you that you will definitely remember all of them for a long time, or maybe you won’t remember them, but EVERYONE should know these six things!