Instruction

Let it be known one of the cathets of the rectangular triangle. Suppose | BC | \u003d b. Then we can use the Pytagora theorem, according to the hypotenuse equal to the sum of the squares of the cathets: a ^ 2 + b ^ 2 \u003d C ^ 2. From this equation we find unknown catat | AB | \u003d a \u003d √ (C ^ 2 - b ^ 2).

Let it be known one of the corners of the rectangular triangle, suppose ∟α. Then AB and BC of the ABC rectangular triangle can be found using trigonometric functions. So we obtain: sinus ∟α is equal to the ratio of an opposite Cate SIN α \u003d b / c, cosine ∟α is equal to the ratio of the adjacent category to the COS α \u003d A / C hypotenneus. From here we find the required lengths of the parties: | AB | \u003d A \u003d C * COS α, | BC | \u003d B \u003d C * SIN α.

Let the ratio of cathets k \u003d a / b be known. We also solve the task using trigonometric functions. The ratio A / B has nothing like Cotangent ∟α: the adjacent CTG category α \u003d A / B. In this case, from this equality, express A \u003d B * CTG α. And we substitute in the Pytagora theorem a ^ 2 + b ^ 2 \u003d C ^ 2:

b ^ 2 * Ctg ^ 2 α + b ^ 2 \u003d C ^ 2. We carry out b ^ 2 for brackets, we obtain b ^ 2 * (Ctg ^ 2 α + 1) \u003d C ^ 2. And hence it easily we get the length of the category B \u003d C / √ (CTG ^ 2 α + 1) \u003d C / √ (k ^ 2 + 1), where k is the specified ratio of cathets.

By analogy, if the ratio of B / A cathets is known, we solve the task using tangent Tg α \u003d b / a. We substitute the value B \u003d A * TG α in the theorem of the Pythagore A ^ 2 * TG ^ 2 α + A ^ 2 \u003d C ^ 2. From here a \u003d C / √ (Tg ^ 2 α + 1) \u003d C / √ (k ^ 2 + 1), where k is a given ratio of cathets.

Consider private cases.

∟α \u003d 30 °. Then | AB | \u003d a \u003d c * cos α \u003d c * √3 / 2; | BC | \u003d B \u003d C * sin α \u003d C / 2.

∟α \u003d 45 °. Then | AB | \u003d | BC | \u003d a \u003d b \u003d C * √2 / 2.

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note

Square roots are extracted with a positive sign, because Length cannot be a negative value. It seems obvious, but this error is very common if you solve the task on the machine.

Helpful advice

To find the cathets of a rectangular triangle, it is convenient to use the formulas of bringing: sin β \u003d sin (90 ° - α) \u003d COS α; COS β \u003d COS (90 ° - α) \u003d sin α.

Sources:

• Bradys tables for finding the values \u200b\u200bof trigonometric functions

The relationship between the sides and corners of the rectangular triangle is considered in the section of mathematics, which is called trigonometry. To find the sides of the rectangular triangle, it is enough to know the pepagora theorem, the definition of trigonometric functions, and have any means to find the values \u200b\u200bof trigonometric functions, for example, a Calculator or Bradys table. Consider below the main cases of the tasks of finding the sides of the rectangular triangle.

You will need

• Calculator, Bradys tables.

Instruction

If one of the sharp corners are given, for example, A, and hypotenuse, then the katenets can be found from the definitions of the main trigonometrics:

a \u003d C * SIN (A), B \u003d C * COS (A).

If one of the sharp angles is specified, for example, A, and one of the cathettes, for example, A, then hypotenuse and other catat are calculated from the relations: B \u003d A * TG (A), C \u003d A * SIN (A).

Helpful advice

In the event that you do not know the value of the sine or cosine of some of the necessary to calculate the angles, you can use the brady's tables, the values \u200b\u200bof trigonometric functions are given to them for a large number of corners. In addition, most modern calculators are able to calculate the sines and cosines of the corners.

Sources:

• how to calculate the side of the rectangular triangle in 2019

Tip 3: how to find an angle if the sides of the rectangular triangle are known

Tre. galnik, one of the corners of which is direct (equal to 90 °), called rectangular. His longer side always lies in front of the direct angle and is called hypotenuse, and the other two parties Catestriate. If the lengths of these three sides are known, then find the values \u200b\u200bof all the corners galnikand will not be difficult, as it will actually need to be calculated only one of the corners. You can do this in several ways.

Instruction

Use to calculate the values \u200b\u200b(α, β, γ) of the definition of trigonometric functions through the rectangular tre. Such, for example, for the sinus of an acute angle as the ratio of the length of the opposite catech to the length of the hypotenuse. It means if the length of the cathets (a and b) and hypotenuses (C), then find, for example, the sine of the angle α, lying opposite the category A can, dividing the length parties And on the length parties C (hypotenuses): Sin (α) \u003d A / C. Having learned the value of the sinus of this angle, it is possible to find its value in degrees, using the reverse sinus function - arxinus. That is, α \u003d arcsin (sin (α)) \u003d arcsin (A / C). In the same way, you can find the magnitude of the acute angle in the galnike, but this is not necessary. Since the sum of all the corners galnika is 180 °, and in three galnike One of the angles is 90 °, the value of the third angle can be calculated as a difference between 90 ° and the value of the angle of the angle: β \u003d 180 ° -90 ° -α \u003d 90 ° -α.

Instead of determining sinus, it is possible to determine the cosine of an acute angle, which is formulated as the ratio of the length of the adjacent category of the catech to the length of the hypotenuse: COS (α) \u003d b / c. And here, use the inverse trigonometric function (arquosine) to find the corner value in degrees: α \u003d arccos (COS (α)) \u003d ArcCOS (B / C). After that, as in the previous step, it will remain to find the magnitude of the missing angle: β \u003d 90 ° -α.

You can use a similar tangent - it is expressed by the ratio of the length of the opposite class of the category to the length of the adjacent category: TG (α) \u003d A / B. The magnitude of the angle in degrees again determine through a reverse trigonometric function -: α \u003d arctg (Tg (α)) \u003d arctg (A / B). The formula of the magnitudes of the missing angle will remain unchanged: β \u003d 90 ° -α.

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Tip 4: how to find the length of the side of the rectangular triangle

This triangle is considered rectangular, which has one of the corners direct. Side trianglelocated opposite the direct angle, called hypotenuse, and the other two parties - Cateties. To find the lengths of the sides of the rectangular triangleYou can use in several ways.

Instruction

You can learn the third parties, knowing the length of two other sides triangle. This can be performed using the Pythagorean theorem, which states that the square is rectangular triangle The sum of the squares of his cathets. (A² \u003d B² + C²). From here you can express the lengths of all sides of the rectangular triangle:
b² \u003d A² - C²;
c² \u003d A² - B²
For example, at rectangular triangle The length of hypotenuse A (18 cm) is known and one of the cathets, for example C (14 cm). To length Another category is required to perform 2 algebraic actions:
c² \u003d 18² - 14² \u003d 324 - 196 \u003d 128 cm
c \u003d √128 cm
Answer: Cate length is √128 cm or, approximately 11.3 cm

You can resort to if the length of the hypotenuse and the magnitude of one of the sharp rectangular value is known. triangle. Let C bes, one of the sharp corners is equal to α. In this case, find 2 others parties rectangular triangle It will be possible with the following formulas:
a \u003d c * sinα;
b \u003d C * COSα.
It can be given: the length of the hypotenuse is 15 cm, one of the sharp corners is 30 degrees. To find the lengths of the two other sides, you need to perform 2 steps:
a \u003d 15 * sin30 \u003d 15 * 0.5 \u003d 7.5 cm
b \u003d 15 * COS30 \u003d (15 * √3) / 2 \u003d 13 cm (approximately)

The most nontrivial way to find length parties rectangular triangle - It is to express it from the perimeter of this figure:
P \u003d a + b + c, where p is the perimeter of rectangular triangle. From this expression it is easy to express length any of the sides of the rectangular triangle.

Tip 5: how to find an angle of a rectangular triangle, knowing all sides

Knowledge of all three sides in right coal The triangle is more than enough to calculate any of its corners. This information is so much that you even have the ability to choose from the parties to use in the calculations to use the most likely trigonometric function.

Instruction

If you prefer to deal with arxinus, use in the calculation of the length of the hypotenuse (C) - the longest parties - and that category (a), which lies opposite the desired angle (α). The division of the length of this category on the length of the hypotenuse will give the size of the sinus of the desired angle, and the inverse sinus function - the arxinus - from the resulting value will restore the angle value of the corner. Therefore, use the following: α \u003d arcsin (A / C).

To replace the ArcCosinus Arksinus, use the calculations of the length of the sides, which form the desired angle (α). One of them will be hypotenuse (C), and the other - cathelet (B). By definition of the cosine - the length of the adjacent category of the category to the length of the hypotenuse, and the angle from the cosine value is the function of the Arkkosinus. Use such a formula of calculations: α \u003d ArcCOS (B / C).

You can use in the calculations. To do this, you need the length of two short sides - cathets. Tangent of acute angle (α) in straight coal The triangle is determined by the ratio of the length of the category (A), which is lying opposite it, to the length of the adjacent category (B). By analogy with the options described above, use such a formula: α \u003d arctg (A / B).

Formula

What triangle is called rectangular?

There are several types of triangles. Ond all the corners are sharp, others have one stupid and two sharp, thirds are two sharp and straight. On this basis, each type of these geometric shapes and was called: acute-angled, stupid and rectangular. That is, the rectangular is called such a triangle, in which one of the corners is 90 °. There is another, similar to the first. Rectangular is called a triangle, in which two sides are perpendicular.

Hypotenuse and kartets

In the acute and stupid triangles, segments connecting the peaks of the corners are called simply by the parties. The sides have other names. Those who adjoin direct corner are called customers. The side, opposing the direct corner, is called hypotenuse. Translated from the Greek word "hypotenuse" means "stretched", and "catat" - "perpendicular".

Relations between hypotenuse and custom

The sides of the rectangular triangle are interconnected by certain ratios that greatly facilitate calculations. For example, knowing the size of the cathets, one can calculate the length of the hypotenuse. This ratio named by the name of the Pythagoreian theorem discovered and looks like this:

c2 \u003d A2 + B2, where C is hypotenuse, a and b - kartets. That is, hypotenuse will be equal to square root from the sum of squares of cathets. To find any of the cathets, sufficiently from the square of the hypotenuses, to subtract the square of another category and remove the square root from the difference.

Prudent and opposing catat

Instruct the rectangular triangle of the DC. The letter C is usually denoted by the vertex of a direct angle, and in the tops of sharp corners. The parties oppose each corner are conveniently called a, b and s, by names of the corners lying opposite them. Consider the angle A. Kartat and for him will be opposite, roll B - adjacent. The ratio of the opposite catech to the hypotenuse is called. It is possible to calculate this trigonometric function by the formula: sina \u003d A / C. The ratio of the adjacent catech to the hypotenuse is called cosine. It is calculated by the formula: cosa \u003d b / c.

Thus, knowing the angle and one of the parties, it is possible to calculate the other side according to these formulas. Both catetes are connected by trigonometric ratios. The attitude of the opposite to the adjacent is called Tangent, and the adjacent to the opposing - Kotangent. You can express these relations with TGA \u003d A / B or CTGA \u003d B / a formulas.

Cates are called two sides of a rectangular triangle that form a straight angle. The opposite direct corner is the longest side of the triangle is called hypotenuse. In order to discover the hypotenuse, you need to know the length of the cathets.

Instruction

1. The lengths of cathets and hypotenuses are associated with the relation, which is described by the Pythagora theorem. Algebraic wording: "In a rectangular triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length of the cathets." The formula of the Pythagora looks like this: C2 \u003d A2 + B2, where C is the length of the hypotenuse, a and b - the length of the cathettes.

2. Knowing the length of cathets, according to the Pythagore theorem, is allowed to detect a rectangular hypothenus: C \u003d? (A2 + B2).

3. Example. The length of one of the cathets is 3 cm, the length of another is 4 cm. The sum of their squares is 25 cm?: 9 cm? + 16 cm? \u003d 25 cm?. The hypotenuse. In our case, is equal to square root from 25 cm? - 5 cm. It became, the length of the hypotenuse is 5 cm.

The hypotenuse is called the side in a rectangular triangle, which is the opposite of an angle of 90 degrees. In order to calculate its length, it is enough to know the length of one of the cathets and the magnitude of one of the sharp corners of the triangle.

Instruction

1. With a famous cathet and acute corner of a rectangular triangle, then the size of the hypotenuse can be equal to the ratio of the cote to the cosine / sinus of this angle, if this angle is the opposite / adjacent: H \u003d C1 (either C2) / SIN ?; H \u003d C1 (or C2 ) /cos?. For example: Let the ABC rectangular triangle with a hypothenuisa AB and a direct angle of C. Let the angle B are 60 degrees, and the angle A 30 degrees of the BC Cate length 8 cm. You need to detect the length of the AB hypotenuse. To do this, it is allowed to use any of the methods proposed above: AB \u003d BC / COS60 \u003d 8 cm.ab \u003d Bc / sin30 \u003d 8 cm.

Hypotenuse - the longest side of the rectangular triangle . It is located opposite to straight corner. The method of finding a rectangular hypotenuse triangle It depends on what initial data you own.

Instruction

1. If you win a rectangular cattiet triangle , then the length of the hypotenuse of rectangular triangle It can be detected with a subband Pythagoree theorem - the square of the hypotenuse length is equal to the sum of the squares of the spectal lengths: C2 \u003d A2 + B2, where a and b - the length of the rolls of rectangular triangle .

2. If we serve one of the cathettes and a sharp angle, then the formula for finding the hypotenuse will depend on what a given angle with respect to the watchet is adjacent (located near the category) or the opposite (located on the contrary. In the case of the adjacent angle, the hypotenuse is equal to the ratio of the category On the cosine of this angle: C \u003d A / COS ?; E the angle of the opposite, hypotenuse is equal to the ratio of the category of the corner: C \u003d A / SIN?.

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The hypotenuse is called the side of the rectangular triangle lying on the contrary direct angle. It is the greatest side of the rectangular triangle. It is permitted by the Pythagora theorem or with support for the formulas of trigonometric functions.

Instruction

1. Cates are called the sides of the rectangular triangle, adjacent to the straight corner. In the picture, the cathets are indicated as AB and BC. Let the length of both cathets are specified. Denote them as | AB | and | BC |. In order to detect the length of hypotenuses | AC |, we use the Pythagora theorem. According to this theorem, the sum of the squares of the cathets is equal to the square of the hypotenuse, i.e. In the notation of our drawing | AB | ^ 2 + | BC | ^ 2 \u003d | AC | ^ 2. From the formula we get that the length of the AC hypotenuse is like | AC | \u003d? (| AB | ^ 2 + | BC | ^ 2).

2. Let us see an example. Let the length of the cathets are set | AB | \u003d 13, | BC | \u003d 21. According to Pythagora theorem, we obtain that | AC | ^ 2 \u003d 13 ^ 2 + 21 ^ 2 \u003d 169 + 441 \u003d 610. In order to obtain the length of the hypotenuse, it is necessary to remove the square root from the sum of the squares of the cathets, i.e. from among 610: | AC | \u003d? 610. Using the table of squares of integers, we find out that the number 610 is not a complete square of some integer. In order to obtain the final value of the hypotenuse length, try to transfer a full square from the root sign. To do this, decompose the number 610 for multipliers. 610 \u003d 2 * 5 * 61. In the table of primitive numbers, we see that 61 is the number primitive. Incidentally, the subsequent cause of the number? 610 is unrealistic. We get the final result | AC | \u003d? 610. If the square of the hypotenuse was equal to, for example, 675, then? 675 \u003d? (3 * 25 * 9) \u003d 5 * 3 *? 3 \u003d 15 *? 3 In the event that similar accuracy is permissible, execute the return check - take the result in the square and compare with the initial value.

3. Let us famous for us one of the cathets and angle adjacent to it. For definiteness, let it be cathets | AB | and corner?. Then we can take advantage of the formula for the trigonometric function of the cosine - cosine of the angle is equal to the attitude of the adjacent catech for hypotenuse. Those. In our designations COS? \u003d | AB | / | AC | Panel Get the length of hypotenuse | AC | \u003d | AB | / COS?. If we were famous for us Kartat | BC | and an angle?, then we use the formula for calculating the sine angle - the corner sinus is equal to the attitude of the opposite category to the hypotenuse: SIN? \u003d | BC | / | AC | We get that the length of the hypotenuse is like | AC | \u003d | BC | / COS?

4. For clarity, we will see an example. Let Dana Cate Length | AB | \u003d 15. And the angle? \u003d 60 °. We get | AC | \u003d 15 / Cos 60 ° \u003d 15 / 0.5 \u003d 30. We will see how it is allowed to check your result with the pythagorette theorem. To do this, we need to count the length of the second category | BC |. Using the formula for tangent TG corner? \u003d | BC | / | AC |, Get | BC | \u003d | AB | * TG? \u003d 15 * TG 60 ° \u003d 15 *? 3. Further apply the Pythagore theorem, we obtain 15 ^ 2 + (15 *? 3) ^ 2 \u003d 30 ^ 2 \u003d\u003e 225 + 675 \u003d 900. The test is executed.

Helpful advice
Calculating the hypotenuse, execute the check - whether the obtained value of the Pythagora theorem satisfies.

Many types of triangles are famous: positive, equiced, acute and so farther. All of them are classic only for them properties and in all their rules for finding values, be it a party or angle at the base. But from each manifold of these geometric shapes, a triangle with a direct angle is allowed to select the triangle with a direct angle.

You will need

• Pure sheet, pencil and ruler for a schematic image of a triangle.

Instruction

1. The triangle is called rectangular if one of its corners is 90 degrees. It consists of 2 cathetes and hypotenuses. The hypotenuses call the major side of this triangle. It lies contrary to the direct corner. Cate, respectively, call smaller sides. They can be both equal among themselves and have a different value. Equality of cathets means that you work with an equilibried rectangular triangle. The charm of him is that it combines the properties of 2 figures: a rectangular and an isced triangle. If the cathets are not equal, then the triangle is arbitrary and obeys the basic law: the more like the corner, the more likely the opposite of him rolls him.

2. There are several methods for finding hypotenuses on cathetu and corner. But earlier than to use one of them, you should determine which catt and the angle are famous. If the angle and the adjacent catat is given, then the hypotenuse is easier to detect everything on the cosine of the angle. The cosine of an acute angle (COS a) in a rectangular triangle call the ratio of the adjacent catech for hypotenuse. The panel implies that hypotenuse (C) will be equal to the ratio of the adjacent category (b) to the cosine of the angle A (COS A). This is allowed to write this way: cos a \u003d b / c \u003d\u003e c \u003d b / cos a.

3. If you are given an angle and opposite catat, then you should work with sine. Sinus of an acute angle (SIN A) in a rectangular triangle is the ratio of an opposite category (a) to hypotenuse (C). The thesis is running here as in the previous example, only a sinus is taken inspire a kosinus function. Sin A \u003d A / C \u003d\u003e C \u003d A / SIN A.

4. Also allowed to take advantage of such a trigonometric function as Tangent. But the finding of the desired magnitude is slightly complicated. A tangent of an acute angle (TG A) in a rectangular triangle is called the ratio of an opposite category (a) to the adjacent (b). Finding both categories, apply the Pythagore's theorem (the square of the hypotenuse is equal to the sum of the squares of the cathets) and the huge side of the triangle will be detected.

The hypotenuse is called the side in a rectangular triangle, which is the opposite of an angle of 90 degrees. In order to calculate its length, it is enough to know the length of one of the cathets and the magnitude of one of the sharp corners of the triangle.

Instruction

1. When you host the nut and the acute corner of the rectangular triangle, the size of the hypotenuse can be equal to the ratio of the cote to the cosine / sinus of this angle, if this angle is to it opposite / adjacent: H \u003d C1 (either C2) / SIN ?; H \u003d C1 (or C2 ) /cos?. For example: Let the ABC rectangular triangle with a hypothenuisa AB and a direct angle of C. Let the angle B are 60 degrees, and the angle A 30 degrees of the BC Cate length 8 cm. You need to detect the length of the AB hypotenuse. To do this, it is allowed to use any of the methods proposed above: AB \u003d BC / COS60 \u003d 8 cm.ab \u003d Bc / sin30 \u003d 8 cm.

Word " cathe "Comes from the Greek words" perpendicular "or" blind "- this explains why it was exactly the way both sides of the rectangular triangle, which constitute its ninety-gradual angle. Detect the length of all of cathe OH is easy if the value of the angle adjacent to it and some of the parameters are famous, because in this case the famous values \u200b\u200bof all 3 angles will actually become.

Instruction

1. If, in addition to the magnitude of the adjacent angle (β), the second length of the second cathe A (B), then the length cathe A (a) allowed to determine as a private from dividing the length of the famous cathe And on the vested angle tangent: a \u003d b / tg (β). This follows from the definition of this trigonometric function. It is permitted to do without tangent, if you use the sinus theorem. It follows from it that the ratio of the length of the desired side to the sinus of the opposite angle is equal to the ratio of the vigorous length cathe And to the sinus of the famous angle. Antoled desired cathe The acute angle is allowed to express via the famous angle as 180 °-90 ° -β \u003d 90 ° -β, because the sum of all angles of any triangle should be 180 °, and according to the definition of a rectangular triangle, one of its corners is 90 °. So, the desired length cathe And it is allowed to calculate according to the formula a \u003d sin (90 ° -β) * b / sin (β).

2. If the magnitude of the adjacent angle (β) and the hypotenuse (C) length are carried out, then the length cathe A (A) is allowed to calculate as a product of the length of hypotenuses on the cosine of the famous angle: a \u003d C * COS (β). This follows from the definition of cosine, as a trigonometric function. But allowed to use, as in the previous step, the sinus theorem and then the length of the desired cathe And will be equal to the product of the sinus of the difference between 90 ° and the vast angle to the ratio of the length of the hypotenuse to the sinus of direct angle. And on the fact that the sinus of 90 ° is equal to one, the formula is allowed to write like this: a \u003d sin (90 ° -β) * c.

3. The actual calculations are allowed to make, say, using the Windows Calculator available in Windows. To start it, it is allowed in the main menu on the "Start" button, prefer to "execute" item, dial the CALC command and click the OK button. In the default, the simplest variant of the interface of this program, trigonometric functions are not provided, and later it is necessary to click on the section "View" section and prefer the "scientist" line or "engineering" (depends on the operating system version used).

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The word "catat" came to Russian from Greek. In the exact translation, it denotes a plumb, that is, perpendicular to the surface of the Earth. In mathematics, customs are referred to as the sides forming the straight corner of the rectangular triangle. The party opposes this corner is called hypotenuse. The term "cathe" is also used in the architecture and special technologies of welding.

Instruct the rectangular triangle of the DC. Indicate its cathets as a and b, and the hypotenuse is like with. All sides and corners of the rectangular triangle are interconnected by certain relationships. The ratio of the catech, opposing one of the sharp corners, is referred to as a sinus of this angle. In this triangle sincab \u003d A / C. The cosine is a relationship to the hypotenus of the adjacent category, that is, CoscaB \u003d b / c. Reverse relationships are referred to as secondary and costerans. This angle is obtained in the division of hypotenuses to the adjacent catat, that is, SECCAB \u003d C / B. It turns out the value, reverse cosine, that is, to express it is allowed using the SECCAB \u003d 1 / COSSAB formula. The coskanes is equal to the private from the division of hypotenuses on the opposite catat and this is a quantity, inverse sinus. It can be calculated using the COSECCAB \u003d 1 / SINCABABA product formula related to Tangent and Kotangent. In this case, the Tangent will be the ratio of the side A to the side B, that is, the opposite category to the adjacent. This ratio can be expressed by the TGCAB \u003d A / B formula. Accordingly, the backstatitude will be a catangent: CtgCab \u003d b / a. The ratio between the sizes of hypotenuses and both cathets has identified ancient Greek mathematician Pythagores. Theorem called him name, people use until now. It states that the square of the hypotenuse is equal to the sum of the squares of the cathets, that is, C2 \u003d A2 + B2. Accordingly, any catat will be equal to square root from the difference in the squares of hypotenuse and other category. This formula is allowed to write down as B \u003d? (C2-A2). The length of the category is allowed to express and through the ratios that are famous for you. According to the theorems of sinuses and cosine, roll is equal to the product of hypotenuses to one of these functions. Allowed to express it through Tangent or Kotangent. Roots and allowed to detect, say, according to the formula A \u003d B * Tan Cab. It is true in the same way, depending on the specified Tangent or Kotangent, is determined by the 2nd catat. The architecture also uses the term "catat". It is used in relation to the ionic capitals and denotes a plumb through the middle of her tail. That is, in this case, this term is denoted by the perpendicular to the specified line. In the special technologies of welding work there is a representation of the "catat of the angular seam". As in other cases, this is the shortest distance. Here we are talking about the interval between one of the welded parts to the border of the seam located on the surface of a different detail.

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Note!
Working with the Pythagora theorem, do not forget that you are dealing with the degree. Finding the sum of the squares of cathets, to purchase a final result, you should remove the square root.

Instruction

If you need to calculate on the Pythagoreo Theorem, use the following algorithm: - Determine in the triangle, which parties are categories, and - hypotenurus. Two sides forming an angle in ninety degrees and there are kartets remaining the third - hypotenuse. (cm) - take into the second degree each cattata of this triangle, that is, multiply upon yourself. Example 1. Let it be necessary to calculate the hypotenuse if one catat in the triangle is 12 cm, and the other - 5 cm. First, the squares of the cathets are equal to: 12 * 12 \u003d 144 cm and 5 * 5 \u003d 25 cm. Next, determine the sum of the squares cathets. A certain number is hypotenuses, you need to get rid of the second degree of number to find length of this side of the triangle. To do this, remove the value of the quantity of cathets from under square root. Example 1. 144 + 25 \u003d 169. Square root out of 169 will be 13. Therefore, the length of this hypotenuses equal to 13 cm.

Another way to calculate length hypotenuses lies in the terminology of sinus and corners in the triangle. By definition: Sine angle of alpha - opposite catech for hypotenuse. That is, looking at the drawing, Sin A \u003d CV / AB. Hence, hypotenuse Av \u003d sv / sin a. Example 2. Let an angle of 30 degrees, and the passing knife - 4 cm. It is necessary to find the hypotenuse. Solution: AV \u003d 4 cm / sin 30 \u003d 4 cm / 0.5 \u003d 8 cm. Reply: Length hypotenuses equal to 8 cm.

A similar way to stay hypotenuses From the definition of cosine angle. Cosine angle - the ratio of the adjacent category and hypotenuses. That is, COS A \u003d AC / AB, from here AV \u003d AC / COS a. Example 3. In the ABC triangle, AV - hypotenuse, the angle of you is 60 degrees, catat the speakers - 2 cm. Find AV.
Solution: AV \u003d AC / COS 60 \u003d 2 / 0.5 \u003d 4 cm. Answer: hypotenuse is 4 cm in length.

Helpful advice

If you find the value of the sine or cosine of the angle, use either the sinus and cosine table, or the bradys table.

Tip 2: How to find the length of hypotenuses in a rectangular triangle

The hypotenuse is called the longest out of the sides in the rectangular triangle, so it is not surprising that from the Greek language this word is translated as "stretched". This side always lies opposite the angle of 90 °, and the sides forming this angle are called customers. Knowing the lengths of these sides and the magnitudes of acute angles in different combinations of these values \u200b\u200bcan be calculated and the length of the hypotenuse.

Instruction

If the lengths of both triangles (A and B) are known, then use the lengths of hypotenuse (C) the most, perhaps known to the mathematical postulate - Pythagore's theorem. It says that the square of the length of the hypotenuses is the sum of the squares of the spells of cathets, which implies that you should calculate the root of the sum of the erected length of the two sides: C \u003d √ (a² + c²). For example, if the length of one category is 15, a - 10 centimeters, then the length of the hypotenuse will be approximately 18.0277564 centimeters, since √ (15² + 10²) \u003d √ (225 + 100) \u003d √ 325-18,0277564.

If the length of only one of the cathets (a) in a rectangular triangle is known, as well as the value of the angle lying opposite it (α), the length of the hypotenuse (C) can be using one of the trigonometric functions - sinus. To do this, divide the length of the known side to the sinus of the known angle: C \u003d a / sin (α). For example, if the length of one of the cathets is 15 centimeters, and the magnitude of the angle in the opposite vertex of the triangle is 30 °, the length of the hypotenuse will be equal to 30 centimeters, since 15 / sin (30 °) \u003d 15 / 0.5 \u003d 30.

If the value of one of the sharp angles (α) is known in the rectangular triangle and the length of the category adjacent to it (B), then another trigonometric function can be used to calculate the length of the hypotenuse (C) - cosine. You should divide the length of the known category on the cosine of the known angle: C \u003d B / COS (α). For example, if the length of this category is 15 centimeters, and the magnitude of the acute angle, to it adjacent, is 30 °, the length of the hypotenuse will be approximately 17,3205081 centimeters, since 15 / COS (30 °) \u003d 15 / (0.5 * √3) \u003d 30 / √3≈17,3205081.

It is customary to denote the distance between the two points of any segment. It can be straight, broken or closed line. Calculate the length can be quite simple, if you know some other segments.

Instruction

If you need to find the length of the sides of the square, then this will not be if it is known for its Square S. Due to the fact that all parties of the square have, calculate the value of one of them by the formula: a \u003d √s.

Geometry - Science is not simple. It can come in handy both for the school program and in real life. Knowledge of many formulas and theorems will simplify geometric calculations. One of the most simple figures in geometry is a triangle. One of the varieties of triangles, equilateral, has its own characteristics.

Features of the equilateral triangle

According to the definition, the triangle is a polyhedron, which has three angle and three sides. This is a flat two-dimensional figure, its properties are studied in high school. By the type of angle distinguish with acute-angular, stupid and rectangular triangles. The rectangular triangle is such a geometric figure, where one of the corners is 90º. Such a triangle has two categories (they create a straight angle), and one hypotenuse (it is opposite the direct angle). Depending on which values \u200b\u200bare known, there are three simple methods to calculate the hypothen of the rectangular triangle.

The first way to find the hypothen of the rectangular triangle is. Pythagorean theorem

Pythagoreo Theorem is ancient way to calculate any of the sides of the rectangular triangle. It sounds like this: "In a rectangular triangle, the square of hypotenuse is equal to the sum of the squares of the cathets." Thus, in order to calculate the hypotenuse, it is necessary to withdraw the square root of two cathets in the square. For clarity, formulas and scheme are shown.

The second way. Calculation of hypotenuse with 2 known values: Cate and adjacent angle

One of the properties of the rectangular triangle states that the ratio of the length of the catech to the length of the hypotenuse is equivalent to the cosine of the angle between these or hypotenuse. We call the corner-known angle α. Now, due to a known definition, it is easy to formulate a formula for calculating hypotenuses: hypotenuse \u003d catat / COS (α)

Third way. Calculation of hypotenuse with 2 known values: Cate and an opposing corner

If the opposite angle is known, it is possible to take advantage of the properties of the rectangular triangle again. The ratio of the length of the catech and hypotenuse is equivalent to the sinus of the opposite corner. Again we call the known angle α. Now for calculations we will apply a little different formula:
Hypotenuse \u003d catat / sin (α)

Examples that will help deal with formulas

For a deeper understanding of each of the formulas, visual examples should be considered. So, suppose there is a rectangular triangle, where there are such data:

• Carthew - 8 cm.
• The adjacent angle cosα1 - 0.8.
• The opposite corner of SINα2 - 0.8.

According to Pythagore: hypotenuse \u003d square root (36 + 64) \u003d 10 cm.
The magnitude of the category and adjacent angle: 8 / 0.8 \u003d 10 cm.
The magnitude of the category and the opposite angle: 8 / 0.8 \u003d 10 cm.

Having understood in the formula, it can be easily calculated with hypotenuse with any data.

Video: Pythagora theorem