Sine is one of the basic trigonometric functions, the use of which is not limited to geometry alone. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and calculating the sine is sometimes necessary to solve various tasks. In general, calculating the sine will help consolidate drawing skills and knowledge of trigonometric identities.
Games with ruler and pencil
A simple task: how to find the sine of an angle drawn on paper? To solve, you will need a regular ruler, a triangle (or 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, you first need to complete the acute angle to the shape of a right triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the angle. We will need to maintain an angle of exactly 90°, for which we need a clerical triangle.
Using a compass is a little more accurate, but will take more time. On one of the rays you need to mark 2 points at a certain distance, adjust the radius on the compass, approximately equal to distance between points, and draw semicircles with centers at these points until the intersections of these lines are obtained. By connecting the intersection points of our circles with each other, we get a strict perpendicular to the ray of our angle; all that remains is to extend the line until it intersects with another ray.
In the resulting triangle, you need to use a ruler to measure the side opposite the corner and the long side on one of the rays. The ratio of the first dimension to the second will be the desired value of the sine acute angle.
Find the sine for an angle greater than 90°
For obtuse angle the task is not much more difficult. You need to draw a ray from the vertex to the opposite side using a ruler to form a straight line with one of the rays of the angle we are interested in. The resulting acute angle should be treated as described above, sines adjacent corners, forming together a reverse angle of 180°, are equal.
Calculating sine using other trigonometric functions
Also, calculating the sine is possible if the values of other trigonometric functions of the angle or at least the lengths 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, based on the Pythagorean theorem, states 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 side by the near side or 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 one and the square sine according to the first trigonometric identity and through simple manipulations we reduce the equation to the calculation of the square sine through the tangent; accordingly, to calculate the sine, you will have to extract the root of 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 leg closest to the angle by the length of the far one, and also by dividing the cosine by the sine, that is, the cotangent is a function, reciprocal of tangent relative to the number 1. To calculate the sine, you can calculate the tangent using the formula tg α = 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 in the following way.
How to find the sine of three sides of a triangle
There is a formula for finding the length of the unknown side of any triangle, not just a rectangular one, from two known parties using the trigonometric function of the cosine of the opposite angle. She looks like this.
Lesson on the topic “Sine, cosine and tangent of an acute angle of a right triangle”
Lesson objectives:
educational - introduce the concept of sine, cosine, tangent of an acute angle in a right triangle, 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, development of correct mathematical speech;
educational – development of skills of independent work, culture of behavior, accuracy in record keeping.
Lesson progress:
“Education is not the number of lessons taken, but the number of understood. So, if you want to go forward, then hurry up slowly and be careful."
2. Lesson motivation.
One wise man said: “ Supreme manifestation spirit is the mind. The highest manifestation of reason is geometry. The geometry cell is a triangle. It is as inexhaustible as the Universe. The circle is the soul of geometry. Know the circle, and you will not only know the soul of geometry, but you will elevate your soul.”
We will try to do a little research together with you. Let's share your ideas that come to your mind, and don't be afraid to make mistakes, any thought can give us a new direction to search. Our achievements may not seem great to someone, but they will be our own achievements!
3. Updating of basic knowledge.
What angles can there be?
What are triangles?
What are the main elements that define a triangle?
What types of triangles are there depending on the sides?
What types of triangles are there depending on the angles?
What is a leg?
What is a hypotenuse?
What are the sides of a right triangle called?
What relationships between the sides and angles of this triangle do you know?
Why do you need to know the relationships between sides and angles?
What tasks in life may lead to the need to calculate unknown parties in a triangle?
The term "hypotenuse" comes from Greek word“hypoinouse”, meaning “stretching over something”, “contracting”. The word originates from the image of ancient Greek harps, on which the strings are stretched at the ends of two mutually perpendicular stands. The term "cathetus" comes from the Greek word "kathetos", which means the beginning of a "plumb line", "perpendicular".
Euclid said: “The legs are the sides that enclose a right angle.”
IN Ancient Greece a method for constructing a right triangle on the ground was already known. To do this, they used a rope on which 13 knots were tied, at the same distance from each other. During the construction of the pyramids in Egypt, right triangles were made in this way. This is probably why a right triangle with sides 3,4,5 was called Egyptian triangle.
4. Studying new material.
In ancient times, people watched the stars and, based on these observations, kept a calendar, calculated sowing dates, and the time of river floods; ships at sea and caravans on land navigated their journey 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, the science of trigonometry arose - a science that studies the connections between the sides of a triangle.
Do you think the relationships we already know are enough to solve such problems?
The purpose of today's lesson is to explore new connections and dependencies, to derive relationships, using which in the next geometry lessons you will be able to solve such problems.
Let's feel like we're in the role scientific workers and following the geniuses of antiquity Thales, Euclid, Pythagoras let's walk the path search for truth.
For this we need theoretical basis.
Highlight angle A and leg BC in red.
Highlight green leg AC.
Let's calculate what part is the opposite leg for an acute angle A to its hypotenuse, for this we create the ratio opposite side to the hypotenuse:
This relationship has a special name - such that every person in every point of the planet understands that we're talking about about a number representing the ratio of the opposite side of an acute angle to the hypotenuse. This 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 a relation adjacent leg to the hypotenuse for acute angle A:
This ratio is called cosine. Its mathematical notation:
Let's consider another ratio for an acute angle A: the ratio of the opposite side to the adjacent side:
This ratio is called tangent. Its mathematical notation:
5. Consolidation of new material.
Let's consolidate our intermediate discoveries.
Sine is...
Cosine is...
Tangent is...
| sin A = | sin ABOUT = | sin A 1 = |
| cos A = | cos ABOUT = | cos A 1 = |
| tan A = | tg ABOUT = | tan A 1 = |
Solve orally No. 88, 889, 892 (work in pairs).
Using the acquired knowledge to solve practical problem:
“From the lighthouse tower, 70 m high, a ship is visible at an angle of 3° to the horizon. What's it like
distance from the lighthouse to the ship?
The problem 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 to problem p. 175.
Solve No. 902(1).
6. Exercise for the eyes.
Without turning your head, look around the classroom wall around the perimeter clockwise, the chalkboard around the perimeter counterclockwise, the triangle depicted on the stand clockwise and the 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'll put our palms to our eyes,
Let's spread our strong legs.
Turning to the right
Let's look around majestically.
And you need to go left too
Look from under your palms.
And - to the right! And further
Over your left shoulder!
Now let's continue working.
7. Independent work students.
Solve no.
8. Lesson summary. Reflection. D/z.
What new things have you learned? At the lesson:
have you considered...
you analyzed...
You received …
you have concluded...
you have replenished lexicon 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 from the spiritual needs of man.
This is how she poetically explained her love for geometry
I love geometry...
I teach geometry because I love it
We need geometry, without it we can’t get anywhere.
Sine, cosine, circumference - everything is important here,
Everything is needed here
You just need to learn and understand everything very clearly,
Complete assignments and tests on time.
Sine and cosine originally arose from the need to calculate quantities in right triangles. It was noticed that if the degree measure of the angles in a right triangle is not changed, then the aspect ratio, no matter how much these sides change in length, always remains the same.
This is how the concepts of sine and cosine were introduced. The sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the side adjacent to the hypotenuse.
Theorems of cosines and sines
But cosines and sines can be used for more than just right triangles. To find the value of an obtuse or acute angle or side of any triangle, it is enough to apply the theorem of cosines and sines.
The cosine theorem is quite simple: “The square of the side of a triangle equal to the sum the squares of the other two sides minus twice the product of these sides by the cosine of the angle between them.”
There are two interpretations of the sine theorem: small and extended. According to the small one: “In a triangle, the angles are proportional opposing parties». This theorem often expanded due to the property of the circumscribed circle of a triangle: “In a triangle, the angles are proportional to the opposite sides, and their ratio is equal to the diameter of the circumscribed circle.”
Derivatives
The derivative is a mathematical tool that shows how quickly a function changes relative to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.
When solving problems, you need to know the tabular values of the derivatives of trigonometric functions: sine and cosine. The derivative of a sine is a cosine, and a cosine is a sine, but with a minus sign.
Application in mathematics
Sines and cosines are especially often used in solving right triangles and problems related to them.
The convenience of sines and cosines is also reflected in technology. It was easy to evaluate angles and sides using the theorems of cosines and sines, breaking down complex figures and objects into “simple” triangles. Engineers often deal with aspect ratio calculations and degree measures, spent a lot of time and effort to calculate the cosines and sines of non-tabular angles.
Then Bradis tables came to the rescue, containing thousands of values of sines, cosines, tangents and cotangents different angles. IN Soviet time some teachers forced their students to memorize pages of Bradis tables.
Radian - angular magnitude arcs, length equal to the radius or 57.295779513° degrees.
Degree (in geometry) - 1/360th part of a circle or 1/90th part right angle.
π = 3.141592653589793238462… ( approximate value Pi numbers).
Cosine table for angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°.
| Angle x (in degrees) | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° | 210° | 225° | 240° | 270° | 300° | 315° | 330° | 360° |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Angle x (in radians) | 0 | π/6 | π/4 | π/3 | π/2 | 2 x π/3 | 3 x π/4 | 5 x π/6 | π | 7 x π/6 | 5 x π/4 | 4 x π/3 | 3 x π/2 | 5 x π/3 | 7 x π/4 | 11 x π/6 | 2 x π |
| cos x | 1 | √3/2 (0,8660) | √2/2 (0,7071) | 1/2 (0,5) | 0 | -1/2 (-0,5) | -√2/2 (-0,7071) | -√3/2 (-0,8660) | -1 | -√3/2 (-0,8660) | -√2/2 (-0,7071) | -1/2 (-0,5) | 0 | 1/2 (0,5) | √2/2 (0,7071) | √3/2 (0,8660) | 1 |
Average level
Right triangle. The Complete Illustrated Guide (2019)
RIGHT TRIANGLE. FIRST LEVEL.
In problems, the right angle is not at all necessary - the lower left, so you need to learn to recognize a right triangle in this form,
and in this
and in this 
What's good about a right triangle? Well... first of all, there are special beautiful names for his sides.
Attention to the drawing!
Remember and don't confuse: there are two legs, and there is only one hypotenuse(one and only, unique and longest)!
Well, we’ve discussed the names, now the most important thing: the Pythagorean Theorem.
Pythagorean theorem.
This theorem is the key to solving many problems involving a right triangle. Pythagoras proved it completely time immemorial, and since then she has brought a lot of benefit to those who know her. And the best thing about it is that it is simple.
So, Pythagorean theorem:
Do you remember the joke: “Pythagorean pants are equal on all sides!”?
Let's draw these same Pythagorean pants and look at them. 
Doesn't it look like some kind of shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, or more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:
"Sum areas of squares, built on the legs, is equal to square area, built on the hypotenuse."
Does it really sound a little different? And so, when Pythagoras drew the statement of his theorem, this is exactly the picture that came out.
In this picture, the sum of the areas of the small squares is equal to the area of the large square. And so that children can better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty came up with this joke about Pythagorean pants.
Why are we now formulating the Pythagorean theorem?
Did Pythagoras suffer and talk about squares?
You see, in ancient times there was no... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to remember everything in words??! And we can rejoice that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to remember it better:
It should be easy now:
| The square of the hypotenuse is equal to the sum of the squares of the legs. |
Well, the most important theorem about right triangles has been discussed. If you are interested in how it is proven, read the following levels of theory, and now let's move on... to dark forest... trigonometry! To the terrible words sine, cosine, tangent and cotangent.
Sine, cosine, tangent, cotangent in a right triangle.
In fact, everything is not so scary at all. Of course, the “real” definition of sine, cosine, tangent and cotangent should be looked at in the article. But I really don’t want to, do I? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:
Why is everything just about the corner? Where is the corner? In order to understand this, you need to know how statements 1 - 4 are written in words. Look, understand and remember!
1.
Actually it sounds like this:
What about the angle? Is there a leg that is opposite the corner, that is, an opposite (for an angle) leg? Of course have! This is a leg!
What about the angle? Look carefully. Which leg is adjacent to the corner? Of course, the leg. This means that for the angle the leg is adjacent, and
Now, pay attention! Look what we got:
See how cool it is:
Now let's move on to tangent and cotangent.
How can I write this down in words now? What is the leg in relation to the angle? Opposite, of course - it “lies” opposite the corner. What about the leg? Adjacent to the corner. So what have we got?
See how the numerator and denominator have swapped places?
And now the corners again and made an exchange:
Summary
Let's briefly write down everything we've learned.
 |
Pythagorean theorem: |
The main theorem about right triangles is the Pythagorean theorem.
Pythagorean theorem
By the way, do you remember well what legs and hypotenuse are? If not very good, then look at the picture - refresh your knowledge
It is quite possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true? How can I prove it? Let's do like the ancient Greeks. Let's draw a square with a side.
See how cleverly we divided its sides into lengths and!
Now let's connect the marked dots
Here we, however, noted something else, but you yourself look at the drawing and think why this is so.
What is the area equal to? larger square? Right, . What about a smaller area? Certainly, . The total area of the four corners remains. Imagine that we took them two at a time and leaned them against each other with their hypotenuses. What happened? Two rectangles. This means that the area of the “cuts” is equal.
Let's put it all together now.
Let's transform:
So we visited Pythagoras - we proved his theorem in an ancient way.
Right triangle and trigonometry
For a right triangle, the following relations hold:
Sine of an acute angle equal to the ratio opposite side to the hypotenuse
The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.
The tangent of an acute angle is equal to the ratio of the opposite side to the adjacent side.
The cotangent of an acute angle is equal to the ratio of the adjacent side to the opposite side.
And once again all this in the form of a tablet:
It is very comfortable!
Signs of equality of right triangles
I. On two sides
II. By leg and hypotenuse
III. By hypotenuse and acute angle
IV. Along the leg and acute angle
a) 
b) 
Attention! It is very important here that the legs are “appropriate”. For example, if it goes like this:
THEN TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.
Need to in both triangles the leg was adjacent, or in both it was opposite.
Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles? Take a look at the topic “and pay attention to the fact that for equality of “ordinary” triangles, three of their elements must be equal: two sides and the angle between them, two angles and the side between them, or three sides. But for the equality of right triangles, only two corresponding elements are enough. Great, right?
The situation is approximately the same with the signs of similarity of right triangles.
Signs of similarity of right triangles
I. Along an acute angle
II. On two sides
III. By leg and hypotenuse
Median in a right triangle
Why is this so?
Instead of a right triangle, consider a whole rectangle.
Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What do you know about the diagonals of a rectangle?
And what follows from this?
So it turned out that
- - median:
Remember this fact! Helps a lot!
What’s even more surprising is that the opposite is also true.
What good can be obtained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture
Look carefully. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But there is only one point in the triangle, the distances from which from all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCLE. So what happened?
So let's start with this “besides...”.
Let's look at and.
But similar triangles all angles are equal!
The same can be said about and
Now let's draw it together:
What benefit can be derived from this “triple” similarity?
Well, for example - two formulas for the height of a right triangle.
Let us write down the relations of the corresponding parties:
To find the height, we solve the proportion and get the first formula "Height in a right triangle":
So, let's apply the similarity: .
What will happen now?
Again we solve the proportion and get the second formula:
You need to remember both of these formulas very well and use the one that is more convenient. Let's write them down again
Pythagorean theorem:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: .
Signs of equality of right triangles:
- on two sides:
- by leg and hypotenuse: or
- along the leg and adjacent acute angle: or
- along the leg and the opposite acute angle: or
- by hypotenuse and acute angle: or.
Signs of similarity of right triangles:
- one acute corner: or
- from the proportionality of two legs:
- from the proportionality of the leg and hypotenuse: or.
Sine, cosine, tangent, cotangent in a right triangle
- The sine of an acute angle of a right triangle is the ratio of the opposite side to the hypotenuse:
- The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
- The tangent of an acute angle of a right triangle is the ratio of the opposite side to the adjacent side:
- The cotangent of an acute angle of a right triangle is the ratio of the adjacent side to the opposite side: .
Height of a right triangle: or.
In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .
Area of a right triangle:
- via legs:
Teachers believe that every student should be able to carry out calculations, know trigonometric formulas, but not every teacher explains what sine and cosine are. What is their meaning, where are they used? Why are we talking about triangles, but the textbook shows a circle? Let's try to connect all the facts together.
School subject
The study of trigonometry usually begins in grades 7-8 high school. At this time, students are explained what sine and cosine are and are asked to solve geometric problems using these functions. More appear later complex formulas and expressions that need to be transformed algebraically (formulas of double and half angle, power functions), work is carried out with a trigonometric circle.
However, teachers are not always able to clearly explain the meaning of the concepts used and the applicability of the formulas. Therefore, the student often does not see the point in this subject, and memorized information is quickly forgotten. However, it is worth explaining once to a high school student, for example, the connection between function and oscillatory motion, And logical connection will be remembered for many years, and jokes about the uselessness of the item will become a thing of the past.
Usage
For the sake of curiosity, let's look into various branches of physics. Do you want to determine the range of a projectile? Or are you calculating the friction force between an object and a certain surface? Swinging the pendulum, watching the rays passing through the glass, calculating the induction? Trigonometric concepts appear in almost any formula. So what are sine and cosine?
Definitions
The sine of an angle is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the same hypotenuse. There is absolutely nothing complicated here. Perhaps students are usually confused by the meanings they see in trigonometric table, because square roots appear there. Yes, getting decimals from them is not very convenient, but who said that all numbers in mathematics must be equal?
In fact, you can find a funny hint in trigonometry problem books: most of the answers here are even and in worst case contain the root of two or three. The conclusion is simple: if your answer turns out to be a “multi-story” fraction, double-check the solution for errors in calculations or reasoning. And you will most likely find them.
What to remember
Like any science, trigonometry has data that needs to be learned.
First, you should remember numeric values for sines, cosines of a right triangle 0 and 90, as well as 30, 45 and 60 degrees. These indicators occur in nine out of ten school tasks. By looking at these values in a textbook, you will lose a lot of time, and there will be nowhere to look at them at all during a test or exam.
It must be remembered that the value of both functions cannot exceed one. If anywhere in your calculations you get a value outside the 0-1 range, stop and try the problem again.
The sum of the squares of sine and cosine is equal to one. If you have already found one of the values, use this formula to find the remaining one.
Theorems
There are two basic theorems in basic trigonometry: sines and cosines.
The first states that the ratio of each side of a triangle to the sine of the opposite angle is the same. The second is that the square of any side can be obtained by adding the squares of the two remaining sides and subtracting their double product multiplied by the cosine of the angle lying between them.
Thus, if we substitute the value of an angle of 90 degrees into the cosine theorem, we get... the Pythagorean theorem. Now, if you need to calculate the area of a figure that is not a right triangle, you don’t have to worry anymore - the two theorems discussed will significantly simplify the solution of the problem.
Goals and objectives
Learning trigonometry will become much easier when you realize one simple fact: all the actions you perform are aimed at achieving just one goal. Any parameters of a triangle can be found if you know the bare minimum of information about it - this could be the value of one angle and the length of two sides or, for example, three sides.
To determine the sine, cosine, tangent of any angle, these data are sufficient, and with their help you can easily calculate the area of the figure. Almost always, the answer requires one of the mentioned values, and they can be found using the same formulas.
Inconsistencies in learning trigonometry
One of the puzzling questions that schoolchildren prefer to avoid is discovering the connection between different concepts in trigonometry. It would seem that triangles are used to study the sines and cosines of angles, but for some reason the symbols are often found in the figure with a circle. In addition, there is a completely incomprehensible wave-like graph called a sine wave, which has no external resemblance to either a circle or triangles.
Moreover, angles are measured either in degrees or in radians, and the number Pi, written simply as 3.14 (without units), for some reason appears in the formulas, corresponding to 180 degrees. How is all this connected?
Units
Why is Pi exactly 3.14? Do you remember what this meaning is? This is the number of radii that fit in an arc on half a circle. If the diameter of the circle is 2 centimeters, the circumference will be 3.14 * 2, or 6.28.
Second point: you may have noticed the similarity between the words “radian” and “radius”. The fact is that one radian is numerically equal to the value the angle subtended from the center of the circle onto an arc one radius long.
Now we will combine the acquired knowledge and understand why “Pi in half” is written on top of the coordinate axis in trigonometry, and “Pi” is written on the left. This is an angular value measured in radians, because a semicircle is 180 degrees, or 3.14 radians. And where there are degrees, there are sines and cosines. It is easy to draw a triangle from the desired point, setting aside segments to the center and to the coordinate axis.
Let's look into the future
Trigonometry, studied in school, deals with rectilinear system coordinates, where, no matter how strange it may sound, a straight line is a straight line.
But there is more complex ways working with space: the sum of the angles of the triangle here will be more than 180 degrees, and the straight line in our view will look like a real arc.
Let's move from words to action! Take an apple. Make three cuts with a knife so that when viewed from above you get a triangle. Take out the resulting piece of apple and look at the “ribs” where the peel ends. They are not straight at all. The fruit in your hands can be conventionally called round, but now imagine how complex the formulas must be with which you can find the area of the cut piece. But some specialists solve such problems every day.
Trigonometric functions in life
Have you noticed that the shortest route for an airplane from point A to point B on the surface of our planet has a pronounced arc shape? The reason is simple: the Earth is spherical, which means you can’t calculate much using triangles - you have to use more complex formulas.
You cannot do without the sine/cosine of an acute angle in any questions related to space. Interestingly, there are a lot of factors coming together here: trigonometric functions are required when calculating the motion of planets in circles, ellipses and various trajectories of more than complex shapes; the process of launching rockets, satellites, shuttles, undocking research vehicles; monitoring distant stars and the study of galaxies that humans will not be able to reach in the foreseeable future.
In general, the field of activity for a person who knows trigonometry is very wide and, apparently, will only expand over time.
Conclusion
Today we learned, or at least repeated, what sine and cosine are. These are concepts that you don’t need to be afraid of - just want them and you will understand their meaning. Remember that trigonometry is not a goal, but only a tool that can be used to satisfy real human needs: build houses, ensure traffic safety, even explore the vastness of the universe.
Indeed, science itself may seem boring, but as soon as you find in it a way to achieve your own goals and self-realization, the learning process will become interesting, and your personal motivation will increase.
As homework Try to find ways to apply trigonometric functions in an area of activity that interests you personally. Imagine, use your imagination, and then you will probably find that new knowledge will be useful to you in the future. And besides, mathematics is useful for general development thinking.