In life, we often have to face math problems: at school, at university, and then helping our child with homework. People of certain professions will encounter mathematics on a daily basis. Therefore, it is useful to memorize or recall mathematical rules. In this article, we will analyze one of them: finding the leg of a right triangle.
What is a right triangle
First, let's remember what a right triangle is. A right triangle is a geometric figure of three segments that connect points that do not lie on the same straight line, and one of the angles of this figure is 90 degrees. The sides that form a right angle are called the legs, and the side that lies opposite the right angle is called the hypotenuse.
Finding the leg of a right triangle
There are several ways to find out the length of the leg. I would like to consider them in more detail.
Pythagorean theorem to find the leg of a right triangle
If we know the hypotenuse and the leg, then we can find the length of the unknown leg using the Pythagorean theorem. It sounds like this: “The square of the hypotenuse is equal to the sum of the squares of the legs.” Formula: c²=a²+b², where c is the hypotenuse, a and b are the legs. We transform the formula and get: a²=c²-b².
Example. The hypotenuse is 5 cm, and the leg is 3 cm. We transform the formula: c²=a²+b² → a²=c²-b². Next, we decide: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).
Trigonometric relations to find the leg of a right triangle
It is also possible to find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding the leg using trigonometric functions: by sine, cosine, tangent, cotangent. To solve the problems, the table below will help us. Let's consider these options.
Find the leg of a right triangle using the sine
The sine of an angle (sin) is the ratio of the opposite leg to the hypotenuse. Formula: sin \u003d a / c, where a is the leg opposite the given angle, and c is the hypotenuse. Next, we transform the formula and get: a=sin*c.
Example. The hypotenuse is 10 cm and angle A is 30 degrees. According to the table, we calculate the sine of angle A, it is equal to 1/2. Then, using the transformed formula, we solve: a=sin∠A*c; a=1/2*10; a=5 (cm).
Find the leg of a right triangle using cosine
The cosine of an angle (cos) is the ratio of the adjacent leg to the hypotenuse. Formula: cos \u003d b / c, where b is the leg adjacent to the given angle, and c is the hypotenuse. Let's transform the formula and get: b=cos*c.
Example. Angle A is 60 degrees, the hypotenuse is 10 cm. According to the table, we calculate the cosine of angle A, it is equal to 1/2. Next, we solve: b=cos∠A*c; b=1/2*10, b=5 (cm).
Find the leg of a right triangle using the tangent
The tangent of an angle (tg) is the ratio of the opposite leg to the adjacent one. Formula: tg \u003d a / b, where a is the leg opposite to the corner, and b is adjacent. Let's transform the formula and get: a=tg*b.
Example. Angle A is 45 degrees, the hypotenuse is 10 cm. According to the table, we calculate the tangent of angle A, it is equal to Solve: a=tg∠A*b; a=1*10; a=10 (cm).
Find the leg of a right triangle using the cotangent
The cotangent of an angle (ctg) is the ratio of the adjacent leg to the opposite leg. Formula: ctg \u003d b / a, where b is the leg adjacent to the corner, and is opposite. In other words, the cotangent is the "inverted tangent". We get: b=ctg*a.
Example. Angle A is 30 degrees, the opposite leg is 5 cm. According to the table, the tangent of angle A is √3. Calculate: b=ctg∠A*a; b=√3*5; b=5√3 (cm).
So, now you know how to find the leg in a right triangle. As you can see, it is not so difficult, the main thing is to remember the formulas.
side triangles can be detected not only along the perimeter and area, but also along a given side and corners. For this, trigonometric functions are used - sinus and co sinus. Problems with their application are found in the school course of geometry, as well as in the university course of analytic geometry and linear algebra.
Instruction
1. If one of the sides of the triangle and the angle between it and its other side are famous, use the trigonometric functions - sinus om and co sinus ohm. Imagine a right triangle HBC, which has an angle? is equal to 60 degrees. The triangle HBC is shown in the figure. Because of sinus, as you know, is the ratio of the opposite leg to the hypotenuse, and to sinus- the ratio of the adjacent leg to the hypotenuse, to solve the problem, use the further relationship between these parameters: sin?=HB/BCAccordingly, if you want to know the leg of a right triangle, express it through the hypotenuse in the following way:
2. If, on the contrary, the leg of the triangle is given in the condition of the problem, find its hypotenuse, guided by the further relationship between the given values: BC \u003d HB / sin? By analogy, find the sides of the triangle and using sinus a, changing the previous expression in the following way: cos ?=HC/BC
3. In elementary mathematics there is a representation of the theorem sinus ov. Guided by the facts that this theorem describes, it is also possible to find the sides of the triangle. In addition, it allows you to find the sides of a triangle inscribed in a circle, if you know the radius of the latter. To do this, use the relation below: a/sin ?=b/sin b=c/sin y=2R .
4. Beyond the theorem sinus ov, there is also a theorem similar to it in essence sinus ov, which, like the previous one, is also applicable to triangles of all 3 varieties: right-angled, acute-angled and obtuse-angled. Guided by the facts that prove this theorem, it is allowed to find unknown quantities using the following relations between them: c^2=a^2+b^2-2ab*cos ?
A geometric figure consisting of three points that do not belong to the same line, called vertices, and three pairwise segments connecting them, called sides, is called a triangle. There are a lot of problems for finding the sides and angles of a triangle given a limited number of initial data, one of these problems is finding the side of a triangle given one of its sides and two corners .
Instruction
1. Let a triangle be built? ABC and famous - side BC and angles ?? and ??. It is known that the sum of the angles of any triangle is 180? will be equal? = 180? – (?? + ??). It is possible to find the sides AC and AB by applying the sine theorem, which reads AB / sin?? = BC/sin?? = AC/sin?? \u003d 2 * R, where R is the radius of the circle described about the triangle? ABC, then we get R \u003d BC / sin??, AB \u003d 2 * R * sin??, AC \u003d 2 * R * sin??. The sine theorem can be used for any given 2 corners and a side.
2. The sides of a given triangle can be found by calculating its area using the formula S \u003d 2 * R? *sin?? *sin?? * sin??, where R is calculated by the formula R = BC / sin??, R is the radius of the circumscribed triangle? ABC from hereThen side AB can be found by calculating the height dropped on it h = BC * sin??, from the formula S = 1/2 * h * AB we have AB = 2 * S / h side AC.
3. If the external angles of a triangle are given as angles? and??, then it is possible to detect internal angles with the support of the corresponding relations? = 180? – ??,?? = 180? – ??,?? = 180? – (?? + ??). Then we act like the first two points.
The comprehension of triangles has been carried out by mathematicians for several millennia. The science of triangles - trigonometry - uses special quantities: sine and cosine.
Right triangle
Initially, the sine and cosine appeared due to the need to calculate the quantities in right triangles. It was noticed that if the value of the degree measure of the angles in a right triangle is not changed, then the aspect ratio, no matter how these sides change in length, remains invariably identical. This is how the representations of sine and cosine were introduced. The sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse, and the cosine is the ratio of the adjacent leg to the hypotenuse.
Theorems of cosines and sines
But cosines and sines can be used not only in right triangles. In order to find out the value of an obtuse or acute angle, the side of any triangle, it is enough to apply the cosine and sine theorem. The cosine theorem is quite primitive: “The square of the side of a triangle is equal to the sum of the squares of the other 2 sides, minus the double 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: "In a triangle, the angles are proportional to the opposite sides." This theorem is often expanded due to the property of the circle circumscribed about a triangle: "In a triangle, the angles are proportional to opposite sides, and their ratio is equal to the diameter of the circumscribed circle."
Derivatives
The derivative is a mathematical tool that shows how rapidly a function changes with respect to the metamorphosis of its argument. Derivatives are used in algebra, geometry, economics and physics, and in a number of technical disciplines. When solving problems, you need to know the tabular values \u200b\u200bof the derivatives of trigonometric functions: sine and cosine. The derivative of the sine is the cosine, and the derivative of the cosine is the sine, but with a minus sign.
Application in mathematics
Especially often, sines and cosines are used in solving right-angled triangles and problems associated with them. The convenience of sines and cosines found its reflection in technology. It was primitive to estimate angles and sides using the cosine and sine theorems, breaking difficult figures and objects into “primitive” triangles. Engineers and architects, who often deal with aspect ratios and degrees, have spent a lot of time and effort calculating cosines and sines of non-table angles. Then the tables of Bradis came to the rescue, containing thousands of values of sines, cosines, tangents and cotangents of various angles. In Soviet times, some teachers forced their wards to memorize the pages of the Bradis tables.
The sine is one of the basic trigonometric functions, the application of which is not limited to geometry alone. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and the calculation of the sine is sometimes necessary to solve various problems. In general, the calculation of the sine will help to consolidate drawing skills and knowledge of trigonometric identities.
Ruler and pencil games
A simple task: how to find the sine of an angle drawn on paper? To solve, you need a regular ruler, a triangle (or a compass) and a pencil. The simplest way to calculate the sine of an angle is by dividing the far leg of a triangle with a right angle by the long side - the hypotenuse. Thus, first you need to complete the acute angle to the figure of a right triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the angle. It will be necessary to observe an angle of exactly 90 °, for which we need a clerical triangle.
Using a compass is a bit more precise, but will take longer. On one of the rays, you need to mark 2 points at a certain distance, set a radius on the compass approximately equal to the distance between the points, and draw semicircles with centers at these points until these lines intersect. By connecting the points of intersection of our circles with each other, we get a strict perpendicular to the ray of our angle, it remains only to extend the line until it intersects with another ray.
In the resulting triangle, you need to measure the side opposite the corner and the long side on one of the rays with a ruler. The ratio of the first measurement to the second will be the desired value of the sine of the acute angle.
Find the sine for an angle greater than 90°
For an obtuse angle, the task is not much more difficult. It is necessary to draw a ray from the vertex in the opposite direction using a ruler to form a straight line with one of the rays of the angle we are interested in. With the resulting acute angle, you should proceed as described above, the sines of adjacent angles, forming together a developed angle of 180 °, are equal.
Calculating the sine from other trigonometric functions
Also, the calculation of the sine is possible if the values of other trigonometric functions of the angle or at least the length of the sides of the triangle are known. Trigonometric identities will help us with this. Let's look at common examples.
How to find the sine with a known cosine of an angle? The first trigonometric identity, coming from the Pythagorean theorem, says that the sum of the squares of the sine and cosine of the same angle is equal to one.
How to find the sine with a known tangent of an angle? The tangent is obtained by dividing the far leg by the near one or by dividing the sine by the cosine. Thus, the sine will be the product of the cosine and the tangent, and the square of the sine will be the square of this product. We replace the squared cosine with the difference between unity and the square sine according to the first trigonometric identity and, through simple manipulations, we bring the equation to calculate the square sine through the tangent, respectively, to calculate the sine, you will have to extract the root from the result obtained.
How to find the sine with a known cotangent of an angle? The cotangent value can be calculated by dividing the length of the near leg from the leg angle by the length of the far one, as well as dividing the cosine by the sine, that is, the cotangent is the inverse function of the tangent with respect to the number 1. To calculate the sine, you can calculate the tangent using the formula tg α \u003d 1 / ctg α and use the formula in the second option. You can also derive a direct formula by analogy with the tangent, which will look like this.
How to find the sine of the three sides of a triangle
There is a formula for finding the length of the unknown side of any triangle, not just a right triangle, given two known sides using the trigonometric function of the cosine of the opposite angle. She looks like this.
If the problem is given the lengths of two sides of a triangle and the angle between them, then you can apply the formula for the area of \u200b\u200bthe triangle through the sine.
An example of calculating the area of a triangle using the sine. Given sides a = 3, b = 4, and angle γ= 30°. The sine of an angle of 30° is 0.5 
The area of the triangle will be 3 sq. cm.
There may also be other conditions. If the length of one side and the angles are given, then first you need to calculate the missing angle. Because the sum of all the angles of a triangle is 180°, then:
The area will be equal to half the square of the side multiplied by the fraction. In its numerator is the product of the sines of the adjacent angles, and in the denominator is the sine of the opposite angle. Now we calculate the area using the following formulas:
For example, given a triangle with side a=3 and angles γ=60°, β=60°. Calculate the third angle: 
Substituting the data into the formula
We get that the area of the triangle is 3.87 square meters. cm.
II. Area of a triangle in terms of cosine
To find the area of a triangle, you need to know the lengths of all sides. By the cosine theorem, you can find unknown sides, and only then use .
According to the law of cosines, the square of the unknown side of a triangle is equal to the sum of the squares of the remaining sides minus twice the product of these sides by the cosine of the angle between them.
From the theorem we derive formulas for finding the length of the unknown side:
Knowing how to find the missing side, having two sides and an angle between them, you can easily calculate the area. The formula for the area of a triangle in terms of cosine helps you quickly and easily find a solution to various problems.
An example of calculating the formula for the area of a triangle through cosine
Given a triangle with known sides a = 3, b = 4, and angle γ= 45°. Let's find the missing part first. With. By cosine 45°=0.7. To do this, we substitute the data into the equation derived from the cosine theorem.
Now using the formula, we find
The area of a triangle is equal to half the product of its sides and the sine of the angle between them.
Proof:
Consider an arbitrary triangle ABC. Let side BC = a in it, side CA = b and S be the area of this triangle. It is necessary to prove that S = (1/2)*a*b*sin(C).
To begin with, we introduce a rectangular coordinate system and place the origin at point C. Let's position our coordinate system so that point B lies on the positive direction of the Cx axis, and point A would have a positive ordinate.
If everything is done correctly, you should get the following figure.
The area of a given triangle can be calculated using the following formula: S = (1/2)*a*h, where h is the height of the triangle. In our case, the height of the triangle h is equal to the ordinate of point A, that is, h \u003d b * sin (C).
Given the results obtained, the formula for the area of a triangle can be rewritten as follows: S = (1/2)*a*b*sin(C). Q.E.D.
Problem solving
Task 1. Find the area of triangle ABC if a) AB = 6*√8 cm, AC = 4 cm, angle A = 60 degrees b) BC = 3 cm, AB = 18*√2 cm, angle B= 45 degrees c ) AC = 14 cm, CB = 7 cm, angle C = 48 degrees.
According to the theorem proved above, the area S of triangle ABC is equal to:
S = (1/2)*AB*AC*sin(A).
Let's do the calculations:
a) S = ((1/2) *6*√8*4*sin(60˚)) = 12*√6 cm^2.
b) S = (1/2)*BC*BA*sin(B)=((1/2)* 3*18*√2 *(√2/2)) = 27 cm^2.
c) S = (1/2)*CA*CB*sin(C) = ½*14*7*sin48˚ cm^2.
We calculate the value of the sine of the angle on a calculator or use the values from the table of values of trigonometric angles. Answer:
a) 12*√6 cm^2.
c) approximately 36.41 cm^2.
Problem 2. The area of triangle ABC is 60 cm^2. Find side AB if AC = 15 cm, angle A = 30˚.
Let S be the area of triangle ABC. By the triangle area theorem, we have:
S = (1/2)*AB*AC*sin(A).
Substitute the values we have into it:
60 = (1/2)*AB*15*sin30˚ = (1/2)*15*(1/2)*AB=(15/4)*AB.
From here we express the length of side AB: AB = (60*4)/15 = 16.