A triangle is a primitive polygon bounded on a plane by three points and three segments connecting these points in pairs. The angles in a triangle are acute, obtuse and right. The sum of the angles in a triangle is continuous and equal to 180 degrees.

You will need

• Basic knowledge of geometry and trigonometry.

Instructions

1. Let us denote the lengths of the sides of the triangle as a=2, b=3, c=4, and its angles as u, v, w, each of which lies opposite to one of the sides. According to the cosine theorem, the square of the length of a side of a triangle is equal to the sum of the squares of the lengths of the other 2 sides minus twice the product of these sides and the cosine of the angle between them. That is, a^2 = b^2 + c^2 – 2bc*cos(u). Let's substitute the lengths of the sides into this expression and get: 4 = 9 + 16 – 24cos(u).

2. Let us express cos(u) from the resulting equality. We get the following: cos(u) = 7/8. Next we will find the actual angle u. To do this, let's calculate arccos(7/8). That is, angle u = arccos(7/8).

3. Similarly, expressing the other sides in terms of the others, we find the remaining angles.

Note!
The value of one angle cannot exceed 180 degrees. The sign arccos() cannot contain a number larger than 1 and smaller than -1.

Helpful advice
In order to detect all three angles, it is not necessary to express all three sides, it is allowed to detect only 2 angles, and the 3rd is obtained by subtracting the value of the remaining 2 from 180 degrees. This follows from the fact that the sum of all the angles of a triangle is continuous and equal to 180 degrees.

In geometry, an angle is a figure formed by two rays emanating from one point (the vertex of the angle). Angles are most often measured in degrees, with a complete angle, or revolution, being 360 degrees. You can calculate the angle of a polygon if you know the type of polygon and the magnitude of its other angles or, in the case of a right triangle, the length of two of its sides.

Steps

Calculating Polygon Angles

Count the number of angles in the polygon.

Find the sum of all the angles of the polygon. The formula for finding the sum of all interior angles of a polygon is (n - 2) x 180, where n is the number of sides as well as angles of the polygon. Here are the angle sums of some commonly encountered polygons:

• The sum of the angles of a triangle (three-sided polygon) is 180 degrees.
• The sum of the angles of a quadrilateral (four-sided polygon) is 360 degrees.
• The sum of the angles of a pentagon (five-sided polygon) is 540 degrees.
• The sum of the angles of a hexagon (six-sided polygon) is 720 degrees.
• The sum of the angles of an octagon (eight-sided polygon) is 1080 degrees.

1. Determine whether the polygon is regular. A regular polygon is one in which all sides and all angles are equal. Examples of regular polygons include an equilateral triangle and a square, while the Pentagon in Washington is built in the shape of a regular pentagon, and a stop sign is shaped like a regular octagon.

   Add up the known angles of a polygon, and then subtract this sum from the total sum of all its angles. Most geometry problems of this kind deal with triangles or quadrilaterals, since they require less input data, so we will do the same.

   • If two angles of a triangle are equal to 60 degrees and 80 degrees, respectively, add these numbers. The result will be 140 degrees. Then subtract this amount from the total sum of all angles of the triangle, that is, from 180 degrees: 180 - 140 = 40 degrees. (A triangle whose angles are all unequal is called equilateral.)
   • You can write this solution as the formula a = 180 - (b + c), where a is the angle whose value needs to be found, b and c are the values ​​of the known angles. For polygons with more than three sides, replace 180 with the sum of the angles of the polygon of that type and add one term to the sum in parentheses for each known angle.
   • Some polygons have their own "tricks" that will help you calculate an unknown angle. For example, an isosceles triangle is a triangle with two equal sides and two equal angles. A parallelogram is a quadrilateral whose opposite sides and opposite angles are equal.

   Calculating the angles of a right triangle

   1. Determine what data you know. A right triangle is so called because one of its angles is right. You can find the magnitude of one of the two remaining angles if you know one of the following:

      Determine which trigonometric function to use. Trigonometric functions express the relationships between two of the three sides of a triangle. There are six trigonometric functions, but the most commonly used are:

In life, we will often have to deal with mathematical problems: at school, at university, and then helping our child with homework. People in certain professions will encounter mathematics on a daily basis. Therefore, it is useful to memorize or recall mathematical rules. In this article we will look at one of them: finding the side 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 forming a right angle are called 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 side 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 solve: a²=5²-3²; a²=25-9; a²=16; a=√16; a=4 (cm).

Trigonometric ratios to find the leg of a right triangle

You can also find an unknown leg if any other side and any acute angle of a right triangle are known. There are four options for finding a leg using trigonometric functions: sine, cosine, tangent, cotangent. The table below will help us solve problems. Let's consider these options.

Find the leg of a right triangle using sine

The sine of an angle (sin) is the ratio of the opposite side to the hypotenuse. Formula: sin=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, angle A is 30 degrees. Using 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=b/c, where b is the leg adjacent to a given angle, and c is the hypotenuse. Let's transform the formula and get: b=cos*c.

Example. Angle A is equal to 60 degrees, the hypotenuse is equal to 10 cm. Using 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 tangent

Tangent of an angle (tg) is the ratio of the opposite side to the adjacent side. Formula: tg=a/b, where a is the side opposite to the angle, and b is the adjacent side. Let's transform the formula and get: a=tg*b.

Example. Angle A is equal to 45 degrees, the hypotenuse is equal to 10 cm. Using 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 cotangent

Angle cotangent (ctg) is the ratio of the adjacent side to the opposite side. Formula: ctg=b/a, where b is the leg adjacent to the angle, and is the opposite leg. In other words, cotangent is an “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. We calculate: b=ctg∠A*a; b=√3*5; b=5√3 (cm).

So now you know how to find a leg in a right triangle. As you can see, it’s not that difficult, the main thing is to remember the formulas.

In geometry there are often problems related to the sides of triangles. For example, it is often necessary to find a side of a triangle if the other two are known.

Triangles are isosceles, equilateral and unequal. From all the variety, for the first example we will choose a rectangular one (in such a triangle, one of the angles is 90°, the sides adjacent to it are called legs, and the third is the hypotenuse).

Quick navigation through the article

Length of the sides of a right triangle

The solution to the problem follows from the theorem of the great mathematician Pythagoras. It says that the sum of the squares of the legs of a right triangle is equal to the square of its hypotenuse: a²+b²=c²

• Find the square of the leg length a;
• Find the square of leg b;
• We put them together;
• From the obtained result we extract the second root.

Example: a=4, b=3, c=?

• a²=4²=16;
• b² =3²=9;
• 16+9=25;
• √25=5. That is, the length of the hypotenuse of this triangle is 5.

If the triangle does not have a right angle, then the lengths of the two sides are not enough. For this, a third parameter is needed: this can be an angle, the height of the triangle, the radius of the circle inscribed in it, etc.

If the perimeter is known

In this case, the task is even simpler. The perimeter (P) is the sum of all sides of the triangle: P=a+b+c. Thus, by solving a simple mathematical equation we get the result.

Example: P=18, a=7, b=6, c=?

1) We solve the equation by moving all known parameters to one side of the equal sign:

2) Substitute the values ​​instead of them and calculate the third side:

c=18-7-6=5, total: the third side of the triangle is 5.

If the angle is known

To calculate the third side of a triangle given an angle and two other sides, the solution comes down to calculating the trigonometric equation. Knowing the relationship between the sides of the triangle and the sine of the angle, it is easy to calculate the third side. To do this, you need to square both sides and add their results together. Then subtract from the resulting product the product of the sides multiplied by the cosine of the angle: C=√(a²+b²-a*b*cosα)

If the area is known

In this case, one formula will not do.

1) First, calculate sin γ, expressing it from the formula for the area of ​​a triangle:

sin γ= 2S/(a*b)

2) Using the following formula, we calculate the cosine of the same angle:

sin² α + cos² α=1

cos α=√(1 — sin² α)=√(1- (2S/(a*b))²)

3) And again we use the theorem of sines:

C=√((a²+b²)-a*b*cosα)

C=√((a²+b²)-a*b*√(1- (S/(a*b))²))

Substituting the values ​​of the variables into this equation, we obtain the answer to the problem.

The lengths of the sides (a, b, c) are known, use the cosine theorem. It states that the square of the length of any of the sides is equal to the sum of the squares of the lengths of the other two, from which twice the product of the lengths of the same two sides by the cosine of the angle between them is subtracted. You can use this theorem to calculate the angle at any of the vertices; it is important to know only its location relative to the sides. For example, to find the angle α that lies between sides b and c, the theorem must be written as follows: a² = b² + c² - 2*b*c*cos(α).

Express the cosine of the desired angle from the formula: cos(α) = (b²+c²-a²)/(2*b*c). To both sides of the equality, apply the inverse function of cosine - arc cosine. It allows you to restore the angle in degrees using the cosine value: arccos(cos(α)) = arccos((b²+c²-a²)/(2*b*c)). The left side can be simplified and the calculation of the angle between sides b and c will take the final form: α = arccos((b²+c²-a²)/2*b*c).

When finding the values ​​of acute angles in a right triangle, knowing the lengths of all sides is not necessary; two of them are sufficient. If these two sides are legs (a and b), divide the length of the one opposite the desired angle (α) by the length of the other. This way you will get the tangent value of the desired angle tg(α) = a/b, and by applying the inverse function - arctangent - to both sides of the equality and simplifying the left side, as in the previous step, derive the final formula: α = arctan(a/b ).

If the known sides are the leg (a) and the hypotenuse (c), to calculate the angle (β) formed by these sides, use the cosine function and its inverse - arc cosine. The cosine is determined by the ratio of the length of the leg to the hypotenuse, and the formula in its final form can be written as follows: β = arccos(a/c). To calculate from the same initial acute angle (α) lying opposite the known leg, use the same relationship, replacing arccosine with arcsine: α = arcsin(a/c).

Sources:

• triangle formula with 2 sides

Tip 2: How to find the angles of a triangle by the lengths of its sides

There are several options for finding the values ​​of all angles in a triangle if the lengths of its three are known parties. One way is to use two different formulas for calculating area triangle. To simplify calculations, you can also apply the sine theorem and the sum of angles theorem triangle.

Instructions

Use, for example, two formulas for calculating area triangle, one of which involves only three of his known parties s (Heron), and in the other - two parties s and the sine of the angle between them. Using different pairs in the second formula parties, you can determine the magnitude of each of the angles triangle.

Solve the problem in general form. Heron's formula determines the area triangle, as the square root of the product of the semi-perimeter (half of all parties) on the difference between the semi-perimeter and each of parties. If we replace with the sum parties, then the formula can be written as follows: S=0.25∗√(a+b+c)∗(b+c-a)∗(a+c-b)∗(a+b-c).C other parties s area triangle can be expressed as half the product of its two parties by the sine of the angle between them. For example, for parties a and b with an angle γ between them, this formula can be written as follows: S=a∗b∗sin(γ). Replace the left side of the equality with Heron's formula: 0.25∗√(a+b+c)∗(b+c-a)∗(a+c-b)∗(a+b-c)=a∗b∗sin(γ). Derive from this equality the formula for