INSTRUCTIONS AND PROPHECIES OF THE Blessed MOTHER ALIPIA GOLOSEEVSKY, Kyiv...
Sinus acute angleα of a right triangle is the ratio opposite catheter to the hypotenuse.
It is denoted as follows: sin α.
Cosine acute angle α of a right triangle is the ratio of the adjacent leg to the hypotenuse.
It is denoted as follows: cos α.
Tangent acute angle α is the ratio of the opposite leg to the adjacent leg.
It is denoted as follows: tg α.
Cotangent acute angle α is the ratio of the adjacent leg to the opposite one.
It is designated as follows: ctg α.
The sine, cosine, tangent and cotangent of an angle depend only on the magnitude of the angle.
Rules:
Basic trigonometric identities in a right triangle:
(α - acute angle opposite the leg b and adjacent to the leg a . Side With - hypotenuse. β - the second acute angle).
b | sin 2 α + cos 2 α = 1 | |
a | 1 | |
b | 1 | |
a | 1 1 | |
sinα |
As the acute angle increasessinα andtg α increase, andcos α decreases.
For any acute angle α:
sin (90° - α) = cos α
cos (90° - α) = sin α
Explanatory example:
Let in a right triangle ABC
AB = 6,
BC = 3,
angle A = 30º.
Find the sine of angle A and the cosine of angle B.
Solution .
1) First, we find the value of angle B. Everything is simple here: since in a right triangle the sum of acute angles is 90º, then angle B \u003d 60º:
B \u003d 90º - 30º \u003d 60º.
2) Calculate sin A. We know that the sine is equal to the ratio of the opposite leg to the hypotenuse. For angle A, the opposite leg is side BC. So:
BC 3 1
sin A = -- = - = -
AB 6 2
3) Now we calculate cos B. We know that the cosine is equal to the ratio of the adjacent leg to the hypotenuse. For angle B, the adjacent leg is the same side BC. This means that we again need to divide BC into AB - that is, perform the same actions as when calculating the sine of angle A:
BC 3 1
cos B = -- = - = -
AB 6 2
The result is:
sin A = cos B = 1/2.
sin 30º = cos 60º = 1/2.
From this it follows that in a right triangle the sine of one acute angle is equal to the cosine of another acute angle - and vice versa. This is exactly what our two formulas mean:
sin (90° - α) = cos α
cos (90° - α) = sin α
Let's check it out again:
1) Let α = 60º. Substituting the value of α into the sine formula, we get:
sin (90º - 60º) = cos 60º.
sin 30º = cos 60º.
2) Let α = 30º. Substituting the value of α into the cosine formula, we get:
cos (90° - 30º) = sin 30º.
cos 60° = sin 30º.
(For more on trigonometry, see the Algebra section)
Lesson on the topic "Sine, cosine and tangent of an acute angle of a right triangle"
Lesson Objectives:
educational - to introduce the concept of sine, cosine, tangent of an acute angle in a right triangle, to explore the dependencies and relationships between these quantities;
developing - the formation of the concept of sine, cosine, tangent as functions of an angle, the domain of definition of trigonometric functions, development logical thinking, the development of correct mathematical speech;
educational - development of the skill of independent work, culture of behavior, accuracy in keeping records.
Course of the lesson:
“Education is not the number of lessons listened to, but the number of understood ones. So, if you want to go forward, then hurry slowly and be careful.
2. Lesson motivation.
One wise man said: “The highest manifestation of the spirit is the mind. The highest manifestation of the mind is geometry. The geometry cell is a triangle. It is as inexhaustible as the universe. The circle is the soul of geometry. Know the circumference, and you will not only know the soul of geometry, but you will elevate your soul.”
Together we will try to do a little research. Let's share any ideas that come to your mind, and don't be afraid to make a mistake, any thought can give us a new direction to search. Let our achievements not seem big to someone, but they will be our own achievements!
3. Actualization of basic knowledge.
What are the angles?
What are triangles?
What are the main elements that define a triangle?
What are triangles based on sides?
What are triangles based on angles?
What is a cathet?
What is a hypotenuse?
What are the sides of a right triangle called?
What are the relationships between the sides and angles of this triangle?
Why do you need to know the relationship between sides and angles?
What tasks from life can lead to the need to calculate unknown sides in a triangle?
The term "hypotenuse" comes from the Greek word "iponeinous", meaning "stretching over something", "pulling". The word originates from the image of the ancient Greek harps, on which the strings are stretched at the ends of two mutually perpendicular stands. The term "katetos" comes from the Greek word "katetos", which means the beginning of "plumb line", "perpendicular".
Euclid said: "The legs are the sides that make a right angle."
AT Ancient Greece a method for constructing a right-angled triangle on the ground was already known. For this, a rope was used, on which 13 knots were tied, at the same distance from each other. During the construction of the pyramids in Egypt, this is how right triangles were made. This is probably why a right-angled triangle with sides 3,4,5 was called the Egyptian triangle.
4. Learning new material.
In ancient times, people followed the luminaries and, based on these observations, kept a calendar, calculated the sowing dates, the time of the flood of rivers; ships on the sea, caravans on land were guided along the way by the stars. All this led to the need to learn how to calculate the sides in a triangle, two of whose vertices are on the ground, and the third is represented by a point in the starry sky. Based on this need, a science arose - trigonometry - a science that studies the relationships between the sides in a triangle.
What do you think, are the relations already known to us sufficient for solving such problems?
The purpose of today's lesson is to explore new connections and dependencies, to derive relationships, using which in the following geometry lessons, you can solve such problems.
Let's feel ourselves in the role of scientists and, following the geniuses of antiquity Thales, Euclid, Pythagoras, we will follow the path of searching for truth.
For this we need a theoretical basis.
Highlight corner A and leg BC in red.
Highlight in green cathet AS.
Let us calculate what part is the opposite leg for an acute angle A to its hypotenuse, for this we compose the ratio of the opposite leg to the hypotenuse:
This ratio has a special name - such that every person in every point of the planet understands that we are talking about a number representing the ratio of the opposite leg of an acute angle to the hypotenuse. The word is sine. Write it down. Since the word sine without the name of the angle loses all meaning, the mathematical notation is as follows:
Now make the ratio of the adjacent leg to the hypotenuse for acute angle A:
This ratio is called cosine. His mathematical notation:
Consider one more relation for an acute angle A: the ratio of the opposite leg to the adjacent leg:
This ratio is called tangent. His mathematical notation:
5. Consolidation of new material.
Let's consolidate our intermediate discoveries.
Sinus is...
Cosine is...
Tangent is...
| sin A = | sin O = | sin A 1 = |
| cos A = | cos O = | cos A 1 = |
| tan A = | tg O = | tg A 1 = |
Solve verbally No. 88, 889, 892 (work in pairs).
Using the acquired knowledge to solve a practical problem:
“From the tower of the lighthouse, 70 m high, a ship is visible at an angle of 3 to the horizon. What is
distance from the lighthouse to the ship?
The task is solved frontally. During the discussion, we make a drawing and the necessary notes on the board and in notebooks.
When solving the problem, Bradis tables are used.
Consider the solution of the problem p.175.
Solve #902(1).
6. Fizminutka for the eyes.
Without turning your head, look around the classroom wall clockwise along the perimeter, the chalkboard around the perimeter counterclockwise, the triangle depicted on the stand clockwise and its equal triangle counterclockwise. Turn your head to the left and look at the horizon line, and now at the tip of your nose. Close your eyes, count to 5, open your eyes and...
We put our hands to our eyes,
Let's set our legs strong.
Turning to the right
Let's look majestic.
And to the left too
Look from under the palms.
And - to the right! And further
Over the left shoulder!
and now we will continue to work.
7. Independent work students.
Solve no.
8. The results of the lesson. Reflection. D / s.
What did you learn new? On the lesson:
have you considered...
did you analyze...
You received …
you concluded...
you have replenished your vocabulary with the following terms ...
World science began with geometry. A person cannot truly develop culturally and spiritually if he has not studied geometry at school. Geometry arose not only from the practical, but also the spiritual needs of man.
This is how she poetically explained her love for geometry
I love geometry...
I study geometry because I love
Geometry is needed, without it we are nowhere.
Sine, cosine, circle - everything is important here,
Everything is needed here
You just have to be very clear and understand everything.
Complete assignments and checklists on time.
Where the tasks for solving a right-angled triangle were considered, I promised to present a technique for memorizing the definitions of sine and cosine. Using it, you will always quickly remember which leg belongs to the hypotenuse (adjacent or opposite). I decided not to put it off indefinitely, necessary material below, please see
The fact is that I have repeatedly observed how students in grades 10-11 have difficulty remembering these definitions. They remember very well that the leg refers to the hypotenuse, but which one- forget and confused. The price of a mistake, as you know in the exam, is a lost score.
The information that I will present directly to mathematics has nothing to do. It is associated with figurative thinking, and with the methods of verbal-logical connection. That's right, I myself, once and for all remembereddefinition data. If you still forget them, then with the help of the presented techniques it is always easy to remember.
Let me remind you the definitions of sine and cosine in a right triangle:
Cosine acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse:
Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:
So, what associations does the word cosine evoke in you?
Probably everyone has their ownRemember the link:
Thus, you will immediately have an expression in your memory -
«… ratio of ADJACENT leg to hypotenuse».
The problem with the definition of cosine is solved.
If you need to remember the definition of the sine in a right triangle, then remembering the definition of the cosine, you can easily establish that the sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse. After all, there are only two legs, if the adjacent leg is “occupied” by the cosine, then only the opposite side remains for the sine.
What about tangent and cotangent? Same confusion. Students know that this is the ratio of legs, but the problem is to remember which one refers to which - either opposite to adjacent, or vice versa.
Definitions:
Tangent an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one:
Cotangent acute angle in a right triangle is the ratio of the adjacent leg to the opposite:
How to remember? There are two ways. One also uses a verbal-logical connection, the other - a mathematical one.
MATHEMATICAL METHOD
There is such a definition - the tangent of an acute angle is the ratio of the sine of an angle to its cosine:
* Remembering the formula, you can always determine that the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one.
Likewise.The cotangent of an acute angle is the ratio of the cosine of an angle to its sine:
So! Remembering these formulas, you can always determine that:
- the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent
- the cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite one.
VERBAL-LOGICAL METHOD
About tangent. Remember the link:
That is, if you need to remember the definition of the tangent, using this logical connection, you can easily remember what it is
"... the ratio of the opposite leg to the adjacent"
If it comes to cotangent, then remembering the definition of tangent, you can easily voice the definition of cotangent -
"... the ratio of the adjacent leg to the opposite"
There is an interesting technique for memorizing tangent and cotangent on the site " Mathematical tandem " , look.
METHOD UNIVERSAL
You can just grind.But as practice shows, thanks to verbal-logical connections, a person remembers information for a long time, and not only mathematical.
I hope the material was useful to you.
Sincerely, Alexander Krutitskikh
P.S: I would be grateful if you tell about the site in social networks.
We begin our study of trigonometry with a right triangle. Let's define what the sine and cosine are, as well as the tangent and cotangent of an acute angle. These are the basics of trigonometry.
Recall that right angle is an angle equal to 90 degrees. In other words, half of the unfolded corner.
Sharp corner- less than 90 degrees.
Obtuse angle- greater than 90 degrees. In relation to such an angle, "blunt" is not an insult, but a mathematical term :-)
Let's draw a right triangle. A right angle is usually denoted . Note that the side opposite the corner is denoted by the same letter, only small. So, the side lying opposite the angle A is denoted.
The angle is indicated by the corresponding Greek letter.
Hypotenuse A right triangle is the side opposite the right angle.
Legs- sides opposite sharp corners.
The leg opposite the corner is called opposite(relative to angle). The other leg, which lies on one side of the corner, is called adjacent.
Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:
Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:
Tangent acute angle in a right triangle - the ratio of the opposite leg to the adjacent:
Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of an angle to its cosine:
Cotangent acute angle in a right triangle - the ratio of the adjacent leg to the opposite (or, equivalently, the ratio of cosine to sine):
Pay attention to the basic ratios for sine, cosine, tangent and cotangent, which are given below. They will be useful to us in solving problems.
Let's prove some of them.
Okay, we have given definitions and written formulas. But why do we need sine, cosine, tangent and cotangent?
We know that the sum of the angles of any triangle is.
We know the relationship between parties right triangle. This is the Pythagorean theorem: .
It turns out that knowing two angles in a triangle, you can find the third one. Knowing two sides in a right triangle, you can find the third. So, for angles - their ratio, for sides - their own. But what to do if in a right triangle one angle (except for a right one) and one side are known, but you need to find other sides?
This is what people faced in the past, making maps of the area and the starry sky. After all, it is not always possible to directly measure all the sides of a triangle.
Sine, cosine and tangent - they are also called trigonometric functions of the angle- give the ratio between parties and corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.
We will also draw a table of sine, cosine, tangent and cotangent values for "good" angles from to.
Notice the two red dashes in the table. For the corresponding values of the angles, the tangent and cotangent do not exist.
Let's analyze several problems in trigonometry from the Bank of FIPI tasks.
1. In a triangle, the angle is , . Find .
The problem is solved in four seconds.
Because the , .
2. In a triangle, the angle is , , . Find .
Let's find by the Pythagorean theorem.
Problem solved.
Often in problems there are triangles with angles and or with angles and . Memorize the basic ratios for them by heart!
For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.
A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.
We considered problems for solving right triangles - that is, for finding unknown sides or angles. But that's not all! In the variants of the exam in mathematics, there are many tasks where the sine, cosine, tangent or cotangent of the outer angle of the triangle appears. More on this in the next article.
The sine is one of the basic trigonometric functions, the application of which is not limited to geometry alone. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and the calculation of the sine is sometimes necessary to solve various tasks. In general, the calculation of the sine will help to consolidate drawing skills and knowledge of trigonometric identities.
Ruler and pencil games
A simple task: how to find the sine of an angle drawn on paper? To solve, you need a regular ruler, a triangle (or a compass) and a pencil. The simplest way to calculate the sine of an angle is by dividing the far leg of a triangle with a right angle by the long side - the hypotenuse. Thus, first you need to complete the acute angle to the figure of a right triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the angle. It will be necessary to observe an angle of exactly 90 °, for which we need a clerical triangle.
Using a compass is a bit more precise, but will take longer. On one of the rays, you need to mark 2 points at a certain distance, set a radius on the compass approximately equal to the distance between the points, and draw semicircles with centers at these points until these lines intersect. By connecting the points of intersection of our circles with each other, we will get a strict perpendicular to the ray of our angle, it remains only to extend the line until it intersects with another ray.
In the resulting triangle, you need to measure the side opposite the corner and the long side on one of the rays with a ruler. The ratio of the first measurement to the second will be the desired value of the sine of the acute angle.
Find the sine for an angle greater than 90°
For obtuse angle the task is not much more difficult. Draw a ray from a vertex to opposite side using a ruler to form a straight line with one of the rays of the angle of interest to us. With the resulting acute angle, you should proceed as described above, the sines of adjacent angles, forming together a developed angle of 180 °, are equal.
Calculating the sine from other trigonometric functions
Also, the calculation of the sine is possible if the values of other trigonometric functions of the angle or at least the length of the sides of the triangle are known. Trigonometric identities will help us with this. Let's look at common examples.
How to find the sine with a known cosine of an angle? The first trigonometric identity, coming from the Pythagorean theorem, says that the sum of the squares of the sine and cosine of the same angle is equal to one.
How to find the sine with a known tangent of an angle? The tangent is obtained by dividing the far leg by the near one or by dividing the sine by the cosine. Thus, the sine will be the product of the cosine and the tangent, and the square of the sine will be the square of this product. We replace the squared cosine with the difference between unity and the square sine according to the first trigonometric identity and, through simple manipulations, we bring the equation to calculate the square sine through the tangent, respectively, to calculate the sine, you will have to extract the root from the result obtained.
How to find the sine with a known cotangent of an angle? The value of the cotangent can be calculated by dividing the length of the near one from the leg angle by the length of the far one, and also dividing the cosine by the sine, that is, the cotangent is the inverse function of the tangent with respect to the number 1. To calculate the sine, you can calculate the tangent using the formula tg α \u003d 1 / ctg α and use the formula in the second option. You can also derive a direct formula by analogy with the tangent, which will look like this.
How to find the sine of the three sides of a triangle
There is a formula for finding the length of the unknown side of any triangle, not just a right triangle, given two known sides using trigonometric function cosine of the opposite angle. She looks like this.
