Instructions

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note

When calculating the sides of a right triangle, knowledge of its characteristics can play a role:
1) If the leg right angle lies opposite an angle of 30 degrees, then it is equal to half the hypotenuse;
2) The hypotenuse is always longer than any of the legs;
3) If a circle is circumscribed around a right triangle, then its center must lie in the middle of the hypotenuse.

The hypotenuse is the side in a right triangle that is opposite the 90 degree angle. In order to calculate its length, it is enough to know the length of one of the legs and the size of one of the acute angles of the triangle.

Instructions

Let us know one of the legs and the angle adjacent to it. To be specific, let these be the side |AB| and angle α. Then we can use the formula for the trigonometric cosine - cosine ratio of the adjacent leg to. Those. in our notation cos α = |AB| / |AC|. From this we obtain the length of the hypotenuse |AC| = |AB| / cos α.
If we know the side |BC| and angle α, then we will use the formula to calculate the sine of the angle - the sine of the angle is equal to the ratio of the opposite leg to the hypotenuse: sin α = |BC| / |AC|. We find that the length of the hypotenuse is |AC| = |BC| / cos α.

For clarity, let's look at an example. Let the length of the leg |AB| be given. = 15. And angle α = 60°. We get |AC| = 15 / cos 60° = 15 / 0.5 = 30.
Let's look at how you can check your result using the Pythagorean theorem. To do this, we need to calculate the length of the second leg |BC|. Using the formula for the tangent of the angle tan α = |BC| / |AC|, we get |BC| = |AB| * tan α = 15 * tan 60° = 15 * √3. Next, we apply the Pythagorean theorem, we get 15^2 + (15 * √3)^2 = 30^2 => 225 + 675 = 900. Check completed.

Helpful advice

After calculating the hypotenuse, check whether the resulting value satisfies the Pythagorean theorem.

Sources:

• Table of prime numbers from 1 to 10000

Legs are the two short sides of a right triangle that make up the vertex whose size is 90°. The third side in such a triangle is called the hypotenuse. All these sides and angles of the triangle are interconnected by certain relationships that make it possible to calculate the length of the leg if several other parameters are known.

Instructions

Use the Pythagorean theorem for leg (A) if you know the length of the other two sides (B and C) of the right triangle. This theorem states that the sum of the squared lengths of the legs is equal to the square of the hypotenuse. From this it follows that the length of each of the legs is equal square root from the lengths of the hypotenuse and the second leg: A=√(C²-B²).

Use the definition of the direct trigonometric function “sine” for an acute angle if you know the magnitude of the angle (α) lying opposite the leg being calculated and the length of the hypotenuse (C). This states that the sine of this known ratio of the length of the desired leg to the length of the hypotenuse. This means that the length of the desired leg is equal to the product of the length of the hypotenuse and the sine of the known angle: A=C∗sin(α). For the same known quantities, you can also use the cosecant and calculate the required length by dividing the length of the hypotenuse by the cosecant of the known angle A=C/cosec(α).

Use the definition of the direct trigonometric cosine function if, in addition to the length of the hypotenuse (C), the magnitude of the acute angle (β) adjacent to the desired one is also known. The cosine of this angle is the ratio of the lengths of the desired leg and the hypotenuse, and from this we can conclude that the length of the leg is equal to the product of the length of the hypotenuse and the cosine of the known angle: A=C∗cos(β). You can use the definition of the secant function and calculate the desired value by dividing the length of the hypotenuse by the secant of the known angle A=C/sec(β).

Derive the required formula from a similar definition for the derivative of the trigonometric function tangent, if in addition to the value of the acute angle (α) lying opposite the desired leg (A), the length of the second leg (B) is known. The tangent of the angle opposite to the desired leg is the ratio of the length of this leg to the length of the second leg. This means that the desired value will be equal to the product of the length of the known leg and the tangent of the known angle: A=B∗tg(α). From these same known quantities, another formula can be derived if we use the definition of the cotangent function. In this case, to calculate the length of the leg, it will be necessary to find the ratio of the length of the known leg to the cotangent of the known angle: A=B/ctg(α).

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The word “kathet” came into Russian from Greek. In exact translation, it means a plumb line, that is, perpendicular to the surface of the earth. In mathematics, legs are the sides that form a right angle of a right triangle. The side opposite this angle is called the hypotenuse. The term “cathet” is also used in architecture and welding technology.

Both legs are connected to each other and by a cotangent. In this case, the tangent will be the ratio of side a to side b, that is, the opposite side to the adjacent side. This relationship can be expressed by the formula tgCAB=a/b. Accordingly, the inverse ratio will be the cotangent: ctgCAB=b/a.

The relationship between the sizes of the hypotenuse and both legs was determined by the ancient Greek Pythagoras. People still use the theorem and his name. It says that the square of the hypotenuse is equal to the sum of the squares of the legs, that is, c2 = a2 + b2. Accordingly, each leg will be equal to the square root of the difference between the squares of the hypotenuse and the other leg. This formula can be written as b=√(c2-a2).

The length of the leg can also be expressed through the relationships known to you. According to the theorems of sines and cosines, a leg is equal to the product of the hypotenuse and one of these functions. It can be expressed as and or cotangent. Leg a can be found, for example, using the formula a = b*tan CAB. In exactly the same way, depending on the given tangent or , the second leg is determined.

The term "cathet" is also used in architecture. It is applied to the Ionic capital and plumb through the middle of its back. That is, in this case, this term is perpendicular to a given line.

In welding technology there is a “fillet weld leg”. As in other cases, this is the shortest distance. Here we're talking about about the gap between one of the parts being welded to the boundary of the seam located on the surface of the other part.

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Sources:

• what are leg and hypotenuse in 2019

The ratio of the opposite side to the hypotenuse is called sinus of an acute angle right triangle.

\sin \alpha = \frac(a)(c)

Cosine of an acute angle of a right triangle

The ratio of the adjacent leg to the hypotenuse is called cosine of an acute angle right triangle.

\cos \alpha = \frac(b)(c)

Tangent of an acute angle of a right triangle

The ratio of the opposite side to the adjacent side is called tangent of an acute angle right triangle.

tg \alpha = \frac(a)(b)

Cotangent of an acute angle of a right triangle

The ratio of the adjacent side to the opposite side is called cotangent of an acute angle right triangle.

ctg \alpha = \frac(b)(a)

Sine of an arbitrary angle

The ordinate of a point on the unit circle to which the angle \alpha corresponds is called sine of an arbitrary angle rotation \alpha .

\sin \alpha=y

Cosine of an arbitrary angle

The abscissa of a point on the unit circle to which the angle \alpha corresponds is called cosine of an arbitrary angle rotation \alpha .

\cos \alpha=x

Tangent of an arbitrary angle

The ratio of the sine of an arbitrary rotation angle \alpha to its cosine is called tangent of an arbitrary angle rotation \alpha .

tan \alpha = y_(A)

tg \alpha = \frac(\sin \alpha)(\cos \alpha)

Cotangent of an arbitrary angle

The ratio of the cosine of an arbitrary rotation angle \alpha to its sine is called cotangent of an arbitrary angle rotation \alpha .

ctg\alpha =x_(A)

ctg \alpha = \frac(\cos \alpha)(\sin \alpha)

An example of finding an arbitrary angle

If \alpha is some angle AOM, where M is a point of the unit circle, then

\sin \alpha=y_(M) , \cos \alpha=x_(M) , tg \alpha=\frac(y_(M))(x_(M)), ctg \alpha=\frac(x_(M))(y_(M)).

For example, if \angle AOM = -\frac(\pi)(4), then: the ordinate of point M is equal to -\frac(\sqrt(2))(2), abscissa is equal to \frac(\sqrt(2))(2) and that's why

\sin \left (-\frac(\pi)(4) \right)=-\frac(\sqrt(2))(2);

\cos \left (\frac(\pi)(4) \right)=\frac(\sqrt(2))(2);

tg;

ctg \left (-\frac(\pi)(4) \right)=-1.

Table of values ​​of sines of cosines of tangents of cotangents

The values ​​of the main frequently occurring angles are given in the table:

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. It was proved by Pythagoras in completely immemorial times, and since then it has brought a lot of benefit to those who know it. 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 ones Pythagorean pants and let's 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:

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 of ​​the 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:

The sine of an acute angle is equal to the ratio of the 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

1. - 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 have all equal angles!

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:

One of the areas of mathematics that students struggle with the most is trigonometry. It is not surprising: in order to freely master this area of ​​​​knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to use trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to derive complex logical chains.

Origins of trigonometry

Getting acquainted with this science should begin with the definition of sine, cosine and tangent of an angle, but first you need to understand what trigonometry does in general.

Historically, the main object of study in this section mathematical science were right triangles. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values ​​of all parameters of the figure in question using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy and even in art.

First stage

Initially, people talked about the relationship between angles and sides exclusively using the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in Everyday life this branch of mathematics.

The study of trigonometry in school today begins with right triangles, after which students use the acquired knowledge in physics and solving abstract trigonometric equations, which begin in high school.

Spherical trigonometry

Later, when science reached the next level of development, formulas with sine, cosine, tangent, and cotangent began to be used in spherical geometry, where different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence at least because earth's surface, and the surface of any other planet is convex, which means that any surface marking will be “arc-shaped” in three-dimensional space.

Take the globe and the thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken on the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy and other theoretical and applied fields.

Right triangle

Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.

The first step is to understand the concepts related to right triangle. First, the hypotenuse is the side opposite the 90 degree angle. It is the longest. We remember that according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.

For example, if the two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.

The two remaining sides, which form a right angle, are called legs. In addition, we must remember that the sum of the angles in a triangle in a rectangular coordinate system is equal to 180 degrees.

Definition

Finally, with a firm understanding of the geometric basis, one can turn to the definition of sine, cosine and tangent of an angle.

The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent side to the hypotenuse.

Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means their ratio will always be less than one. Thus, if in your answer to a problem you get a sine or cosine with a value greater than 1, look for an error in the calculations or reasoning. This answer is clearly incorrect.

Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. Dividing the sine by the cosine will give the same result. Look: according to the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relationship as in the definition of tangent.

Cotangent, accordingly, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing one by the tangent.

So, we have looked at the definitions of what sine, cosine, tangent and cotangent are, and we can move on to formulas.

The simplest formulas

In trigonometry you cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? But this is exactly what is required when solving problems.

The first formula you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you need to know the size of the angle rather than the side.

Many students cannot remember the second formula, which is also very popular when solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation makes the trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, transformation rules and several basic formulas, you can at any time derive the required more complex formulas on a sheet of paper.

Formulas for double angles and addition of arguments

Two more formulas that you need to learn are related to the values ​​of sine and cosine for the sum and difference of angles. They are presented in the figure below. Please note that in the first case, sine and cosine are multiplied both times, and in the second, the pairwise product of sine and cosine is added.

There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a training try to get them yourself by taking the alpha angle equal to the angle beta.

Finally, note that double angle formulas can be rearranged to reduce the power of sine, cosine, tangent alpha.

Theorems

The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of ​​the figure, and the size of each side, etc.

The sine theorem states that dividing the length of each side of a triangle by the opposite angle results in the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all the points of a given triangle.

The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product multiplied by the double cosine of the adjacent angle - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.

Careless mistakes

Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's take a look at the most popular ones.

Firstly, you shouldn't convert fractions to decimals until you get the final result - you can leave the answer as common fraction, unless otherwise stated in the conditions. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author’s idea, should be reduced. In this case, you will waste your time on unnecessary mathematical operations. This is especially true for values ​​such as the root of three or the root of two, because they are found in problems at every step. The same goes for rounding “ugly” numbers.

Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but you will also demonstrate a complete lack of understanding of the subject. This is worse than a careless mistake.

Thirdly, do not confuse the values ​​for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to confuse them, as a result of which you will inevitably get an erroneous result.

Application

Many students are in no hurry to start studying trigonometry because they do not understand its practical meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts with which you can calculate the distance to distant stars, predict the fall of a meteorite, or send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on a surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.

Finally

So you're sine, cosine, tangent. You can use them in calculations and successfully solve school problems.

The whole point of trigonometry comes down to the fact that using the known parameters of a triangle you need to calculate the unknowns. There are six parameters in total: the length of three sides and the size of three angles. The only difference in the tasks lies in the fact that different input data are given.

You now know how to find sine, cosine, tangent based on the known lengths of the legs or hypotenuse. Since these terms mean nothing more than a ratio, and a ratio is a fraction, main goal The trigonometric problem becomes finding the roots of an ordinary equation or a system of equations. And here regular school mathematics will help you.