Instruction

1

If you know that the triangle is rectangular, it gives you knowledge of the value of one of the corners, i.e., missing for calculation of the third parameter. The required side (C) hypotenuse - the side lying opposite the right angle. Then to calculate, extract square root and squared and folded the lengths of the other two sides (A and B) of this figure: C=√(A2+B2). If the desired direction is the leg, remove the square root from the difference between the squares of the lengths of the longer (hypotenuse) and lower (second leg) sides: C=√(A2-B2). These formulas are derived from the Pythagorean theorem.

2

Knowledge as a third parameter to the perimeter

**of the triangle**(P) reduces the problem of computing the length of a missing**side**(S) to a simple subtraction operation - subtract from perimeter the length of both (A and B) famous faces: C=P-A-B. This formula follows from the definition of the perimeter which is the length of the polygonal line bounding the area of the shape.3

The presence of baseline angle (γ) between the sides (A and B) of known length will require to find the length of the third (C) computation of trigonometric functions. Lift both the lengths of the sides in the square and add the results. Then, from this value subtract the product of their lengths into the cosine of the known angle, and in the end remove from the obtained values of the square root: C = √(A2+B2-A*B*cos(γ)). The theorem that you used in the calculations, is called the theorem of sines.

4

A well-known area

**of the triangle**(S) will require the use of three formulas. The first defines the area as half of the work length of the known sides (A and B) the sine of the angle between them. Express from it the sine of an angle and you will get the expression 2*S/(A*B). The second formula allows to Express the cosine of the same angle as the sum of the squares of the sine and cosine of the same angle equals one, the cosine is the square root of the difference between the unit and the square received the previous expression: √(1-(2*S/(A*B))2). The third formula is a theorem, the law of cosines was used in the previous step, replace in it is the cosine of the resulting expression and you will have the following formula for calculation: C = √(A2+B2-A*B*√(1-(2*S/(A*B))2)).# Advice 2 : How to calculate angle in triangle

From the school course of plane geometry known definition: a triangle is a geometrical figure consisting of three points not lying on one straight line and three segments that connect pairs of these points. Points are called vertices, and line segments – the sides of the triangle. Share the following types of triangles: acute-angled, obtuse-angled and rectangular. Also klassificeret triangles by sides: isosceles, equilateral and scalene.

Depending on the form of a triangle, there are several ways to measure its angles, sometimes it is enough to know only the shape of a triangle.

Depending on the form of a triangle, there are several ways to measure its angles, sometimes it is enough to know only the shape of a triangle.

Instruction

1

The triangle is rectangular if it has the right angle. In the measurement of its angles, you can use trigonometric calculations.

In this triangle the angle ∠C = 90º, as a direct, knowing the lengths of the sides of a triangle, the angles ∠A and ∠B are computed by the formulas: cos∠A = AC/AB cos∠B = BC/AB. Degree measures of the angles can be found by referring to the table of cosines.

In this triangle the angle ∠C = 90º, as a direct, knowing the lengths of the sides of a triangle, the angles ∠A and ∠B are computed by the formulas: cos∠A = AC/AB cos∠B = BC/AB. Degree measures of the angles can be found by referring to the table of cosines.

2

The triangle is called equilateral if all the sides are equal.

In an equilateral triangle all the angles are equal to 60º.

In an equilateral triangle all the angles are equal to 60º.

3

In the General case, for finding the angles in an arbitrary triangle we can use the theorem of cosines

cos∠α = (b2 + c2 - a2) / 2 • b • c

Degree measure of an angle can be found by referring to the table of cosines.

cos∠α = (b2 + c2 - a2) / 2 • b • c

Degree measure of an angle can be found by referring to the table of cosines.

4

A triangle is called isosceles if two sides are equal, the third side is called the base of the triangle.

In an isosceles triangle the angles at the base are equal, i.e. ∠A = ∠B. One of the properties of a triangle is that the sum of its angles always equals 180º, so calculating the cosine theorem, the angle ∠C, the angles ∠A and ∠B we can calculate: ∠A = ∠B = 180º - ∠C)/2

In an isosceles triangle the angles at the base are equal, i.e. ∠A = ∠B. One of the properties of a triangle is that the sum of its angles always equals 180º, so calculating the cosine theorem, the angle ∠C, the angles ∠A and ∠B we can calculate: ∠A = ∠B = 180º - ∠C)/2

# Advice 3 : How to find the hypotenuse on the side and corners

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

Instruction

1

At a known side and an acute angle of a right triangle, the hypotenuse can be equal to the ratio of the leg to the cosine/sine of that angle if the given angle is opposite him/yard:

h = C1(or C2)/sinα;

h = C1(or C2)/cosα.

Example: suppose that we are given right triangle ABC with hypotenuse AB and a right angle C. Let angle B equal 60 degrees, and angle a of 30 degrees Length of side BC is 8 cm Need to find the length of the hypotenuse AB. For this you can use any of the above methods:

AB = BC/cos60 = 8 cm.

AB = BC/sin30 = 8 cm.

h = C1(or C2)/sinα;

h = C1(or C2)/cosα.

Example: suppose that we are given right triangle ABC with hypotenuse AB and a right angle C. Let angle B equal 60 degrees, and angle a of 30 degrees Length of side BC is 8 cm Need to find the length of the hypotenuse AB. For this you can use any of the above methods:

AB = BC/cos60 = 8 cm.

AB = BC/sin30 = 8 cm.

# Advice 4 : How to find the angle of a triangle if you know two sides?

In a right triangle, you simply find the angle if you know two sides of him. One of the angles is equal to 90 degrees, the other two are always sharp. These angles and will need to find. In order to find an acute angle in a right triangle, you must know the values of all three sides. Depending on which side you are aware, the sinuses acute angles can be found using formulas for trigonometric functions. To find the angle values for the sine uses a four digit mathematical table.

You will need

- - Pythagorean Theorem;
- - trigonometric function sin;
- - four-digit mathematical tables Bradis.

Instruction

1

Use the following notation for ease of preparation of the formulas necessary for calculations: c is the hypotenuse of a right triangle; a, b are the legs that form a right angle, A is the acute angle opposite sides of b; B is the acute angle opposite of side a.

2

Calculate what is the length of the unknown side of the triangle. Apply for computing the Pythagorean theorem. Calculate the side a, if the known values of the hypotenuse c and leg b. To do this, subtract the square of the hypotenuse c, the square of side b and then calculate the square root of the result.

3

Calculate side b, if the known values of the hypotenuse c and a leg. To do this, subtract the square of the hypotenuse c, the square of side a, and then calculate the square root of the result.

4

Calculate the value of the hypotenuse c, if there are two sides. This will get the sum of the squares of the legs a and b and then calculate the square root of the result and, if necessary, rounded to four decimal places.

5

Calculate the sine of the angle A according to the formula sinA = a/c. Use for calculations the calculator. Rounded, if necessary, the value of the sine of the angle A to four decimal places.

6

Calculate the sine of the angle B by the formula sinB = b/c. Use for calculations the calculator. Rounded, if necessary, the value of the sine of the angle B to four decimal places.

7

Find the angles A and B by the values of their sines. Use to determine the values of the angles of table VIII four-digit mathematical tables Bradis. Find in this table the values of sine. Move from values to the left and the first column "And" take a degree. Move from the found values up and from the top row "And" take minutes. For example, if sin(A) = 0,8949, the angle A is 63 degrees 30 minutes.

# Advice 5 : How to find sides of a triangle if all the angles of this

In order to learn all sides of the triangle need to know the measure of the angle and two adjacent sides or measure of two angles and the side between them. If you are aware of all the angles of this triangle, find the length of all sides of the triangle is impossible, but you can find the ratio of the sides of the triangle.

Instruction

1

In the first case known to such data in the triangle, as the angle and length of sides that form this angle. The side opposite the known angle, you need to find the cosine theorem, according to which you want the length of the known sides squared up and folded, then subtract from the obtained sum the product of these sides multiplied by two and the cosine of known angle.

The formula for this calculation looks like the following:

h = √(e2+f2 – 2ef*cosA) where:

e and f are the lengths of the known sides;

h – the unknown leg (or side);

A – the angle formed by the known sides.

The formula for this calculation looks like the following:

h = √(e2+f2 – 2ef*cosA) where:

e and f are the lengths of the known sides;

h – the unknown leg (or side);

A – the angle formed by the known sides.

2

In the second case, when we know two angles and the side between them of the triangle to use the theorem of sines. According to this theorem, if you divide the sine of the angle to the length of the opposite leg, we get a ratio equal to any other in this triangle. Also, if you do not know the right side, you can easily find knowing the fact that the sum of the angles of a triangle is equal to hundred eighty degrees.

This statement can be represented in a formula as follows:

SinD/d = a sinF/f = sinE/e, where:

D, F, E – values of the opposite angles;

d, f, e are the legs, the opposite corresponding angles.

This statement can be represented in a formula as follows:

SinD/d = a sinF/f = sinE/e, where:

D, F, E – values of the opposite angles;

d, f, e are the legs, the opposite corresponding angles.

3

In the third case, known only to the corners of this triangle, so it is impossible to know the length of all sides of this triangle. But you can find the attitude of these parties and the method of selection is to find a similar triangle. The aspect ratio of this triangle is found using the formulation of the system of three equations with three unknowns.

Here is the formula for compiling:

d/sinD

f/sinF

e/sinE, where:

d, f, e are the unknown sides of the triangle;

D, F, E – the angle opposite the unknown sides.

Here is the formula for compiling:

d/sinD

f/sinF

e/sinE, where:

d, f, e are the unknown sides of the triangle;

D, F, E – the angle opposite the unknown sides.

4

This equation is solved as follows:

d/sinD = f/sinF = e/sinE

(d*sinF*sinE-f* sinD* sinE-e* sinD* sinF)/ sinD* sinE* sinF.

d/sinD = f/sinF = e/sinE

(d*sinF*sinE-f* sinD* sinE-e* sinD* sinF)/ sinD* sinE* sinF.

# Advice 6 : How to find third side of triangle

A closed geometric figure with three angles of non-zero magnitude is called a triangle. Knowledge of the sizes of the two sides is not enough to calculate the length of the third side, we should know the value of at least one of the corners. Depending on the mutual arrangement of the known sides and angle for calculations should apply different methods.

Instruction

1

If the conditions of the problem except for the lengths of two sides (A and C) in an arbitrary triangle is known and the angle between them (β), then apply to find the length of the third side (B) the theorem of cosines. First, lift the lengths of the sides into a square and fold the values obtained. From this value subtract twice the product of the lengths of these sides into the cosine of the known angle, and from what remains, remove the square root. In General, the formula can be written as: B=√(A2+C2-2*A*C*cos(β)).

2

If given the measure of an angle (α), which lies opposite the longer of (A) two known sides, then start with calculating the angle opposite the other known side (B). If we start from the theorem of sines, its value must be equal to arcsin(sin(α)*B/A), this means that the magnitude of the angle lying opposite the unknown side will be 180°-α-arcsin(sin(α)*B/A). Following to find the expected length of the same theorem of sines, multiply the length of the longest side to the sine of the angle is found and divide by the sine-known because of the problem of the angle: C=A*sin(α-arcsin(sin(α)*B/A))*sin(α).

3

If given the measure of an angle (α) adjacent to the unknown side length (C), and the other two parties have identical and known according to the problem dimensions (A), the formula will be much easier. Find twice the product of known length into the cosine of the known angle: C=2*A*cos(α).

4

If we consider a rectangular triangle and is known the length of its two sides (A and b), then find the length of the hypotenuse (C) use the Pythagorean theorem. Extract the square root of the sum of squared lengths of the known sides: C=√(A2+B2).

5

If in a right triangle are known the length of one of the legs (B) and the hypotenuse (C), to calculate the length of the other leg comes from the same theorem. Extract the square root of the difference between the built in the squares of the lengths of the hypotenuse and the known leg: C=√(C2-B2).

# Advice 7 : How to find the side of a triangle

Side

**of the triangle**is a direct, limited its vertices. All of them have figures of three, this number determines the number of almost all graphics characteristics: angle, midpoint, bisectors, etc. to find the**side****of the triangle**, you should carefully examine the initial conditions of the problem and determine which of them may be basic or intermediate values to calculate.Instruction

1

The sides

**of the triangle**, like other polygons have their own names: the sides, base, and hypotenuse and legs of the figure with a right angle. This facilitates the calculations and formulas, making them more obvious even if the triangle is arbitrary. The figure of graphics, so it is always possible to arrange so as to make the solution of the problem more visible.2

Sides of any

**triangle**are connected and the other characteristics of the various ratios that help calculate the required value in one or more actions. Thus the more complex the task, the longer the sequence of steps.3

The solution is simplified if the triangle is standard: the words "rectangular", "isosceles", "equilateral" immediately allocate a certain relationship between its sides and angles.

4

The lengths of the sides in a right triangle are connected by Pythagorean theorem: the sum of the squares of the legs equals the square of the hypotenuse. And the angles, in turn, are associated with the parties to the theorem of sines. It affirms the equality relations between the lengths of the sides and the trigonometric function sine of the opposite angle. However, this is true for any

**triangle**.5

Two sides of an isosceles

**triangle**are equal. If their length is known, it is enough only one value to find the third. For example, suppose we know the height held to it. This cut divides the third**side**into two equal parts, and allocates two rectangular**triangle**H. Considered one of them, by the Pythagorean theorem find the leg and multiply it by 2. This will be the length of an unknown side.6

Side

**of the triangle**can be found through other sides, corners, length, altitude, median, bisector, perimeter size, area, inradius, etc. If you can't apply the same formula to produce a series of intermediate calculations.7

Consider an example: find

**side of**an arbitrary**triangle**, knowing the median ma=5, held for her, and the lengths of the other two medians mb=7 and mc=8.8

Resented involves the use of formulas for the median. You need to find the

**way**. Obviously, there should be three equations with three unknowns.9

Write down the formulae for the medians:ma = 1/2•√(2•(b2 + c2) – a2) = 5;mb = 1/2•√(2•(a2 + c2) – b2) = 7;mc = 1/2•√(2•(a2 + b2) – c2) = 8.

10

Express c2 from the third equation and substitute it into the second:c2 = 256 – 2•a2 – 2•b2 b2 = 20 → c2 = 216 – a2.

11

Lift both sides of the first equation in the square and find a by entering explicit values:25 = 1/4•(2•20 + 2•(216 – a2) – a2) → a ≈ 11,1.