Instruction

1

If in addition the values of the cosine of the angle are known the lengths of pairs of sides (b and c), which form this angle, to calculate the value of the unknown side (a) you can use the theorem of cosines. She argues that the squared length of the right side is equal to the sum of the squares of the lengths of the other two, if it is reduced by twice the product of the lengths of these sides on the well-known of the terms of the cosine of the angle between them: a2 = b2 + c2 - 2*a*b*cos(α).

2

Since the value of the angle α you do not know and calculate it is not necessary, indicate this in terms of the variable (the cosine of) any letter (e.g., f) and substitute into the formula: a2 = b2 + c2 - 2*a*b*f. Get rid of the extent in the left part of the expression to get a General view of the final calculation formula of the desired length of side: a = √(b2+c2-2*a*b*f).

3

To find the length of a side (a), provided that in addition to the value of cosine (f = cos(α)) lying opposite this side of the angle, given the value of another angle (β) and length lying on the opposite side (b), we can use the theorem of sines. According to her the ratio of desired length to the sine of the opposite angle is equal to the ratio of the length of the known side to the sine of the angle, which is also given in: a/sin(a) = b/sin(β).

4

The sum of the squares of the sine and cosine of the same angle equals one - use this identity to Express the sine in the left side of the equation using the given in terms of cosine: a/√(1-f2) = b/sin(β). Find a formula to calculate the length of the right side in General, moving the denominator from the left side of the identity in the right: a = √(1-f2)*b/sin(β).

5

In a right triangle to calculate the values of the parties is sufficient to complement the cosine of an acute angle (f = cos(α)) a single parameter - the length of any of the parties. To find the length of side (b) adjacent to the vertex, the cosine of which is known, multiply this value by the length of the hypotenuse (c): b = f*c. If you need to calculate the length of hypotenuse and length of side is known, transform this formula accordingly: c = b/f.

# Advice 2: How to calculate the length of triangle side

To calculate lengths of sides in an arbitrary triangle most often has to apply the theorem of sines and cosines. But among the whole set of arbitrary polygons of this kind there are of them "more correct" variations - equilateral, isosceles, rectangular. If the triangle is known that it belongs to one of these species, methods of calculation of its parameters is much easier. When calculating the lengths of their sides is often possible to do without trigonometric functions.

Instruction

1

**The length of the**

**side**(A) of an equilateral

*triangle*to find the radius of the inscribed circle (r). To do so, increase it to six times and divide by the square root of triples: A = r*6/√3.

2

Knowing the radius of the circumscribed circle (R), too it is possible to calculate the length of

**side**(A) right*triangle*. This radius is twice that used in the previous formula, so triple it and also divide by the square root of triples: A = R*3/√3.3

The perimeter (P) of an equilateral

*triangle*to calculate the length of its**side**(A) is even simpler, because the lengths of the sides in this figure are the same. Just divide the perimeter into three pieces: A = R/3.4

In an isosceles triangle the calculation of the length

**of the sides**of known perimeter is a bit more complicated - you need to know more and a length of at least one of the parties. If you know the length**of side**A lying in the base of the figure, the length of any side (In) find halving the difference between the perimeter (P) and the size of the base: B = (R-A)/2. And if you know the side, the length of the base define by subtracting from the perimeter to twice the length of the side: A = R-2*V.5

Knowledge area (S) occupied on the plane right triangle, is also sufficient for finding the lengths of its

**sides**(A). Extract the square root of the area ratio and the square root of three, and the result double A = 2*√(S/√3).6

In a right triangle, unlike any other, to calculate the length of one of the parties is sufficient to know the lengths of the other two. If the target side is the hypotenuse (C), to do this, find the square root of the sum of the lengths of the known sides (A and b) squared: C = √(A2+B2). But if you want to calculate the length of one of the other two sides, then the square root must be obtained from the difference of the squares of the lengths of the hypotenuse and another side: A = √(C2-B2).

# Advice 3: 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.

# Advice 4: How to find the cosine of the angle of the triangle with vertices

The cosine of an angle is the ratio adjacent to this corner of the leg to the hypotenuse. This value, like other trigonometric ratios used to solve not just right triangles, but for many other tasks.

Instruction

1

For an arbitrary triangle with vertices A, b and C the task of finding the cosine is the same for all three angles, if the triangle is acute-angled. If the triangle has an obtuse angle, the determination of its cosine should be considered separately.

2

Acute-angled triangle with vertices A, b and C find the cosine of the angle at the vertex A. Lower the height from the vertex To the side of the triangle AC. The intersection point of heights with the party as mark D and consider the right triangle АВD. In this triangle side AB of the original triangle is the hypotenuse, and the legs — the height BD of the original acute triangle and cut AD belonging to the side AC. The cosine of angle A is equal to the ratio AD/AB, because AD is the side adjacent to the angle A in a right triangle АВD. If you know in which the ratio of the altitude BD divides the side AC of the triangle, the cosine of the angle As found.

3

If the AD value is not given, but known height BD, is the cosine of the angle can be determined by its sine. The sine of angle A is equal to the ratio of the height BD of the original triangle to the side AC. Basic trigonometric identity establishes a relationship between the sine and cosine of the angle:

Sin2 A+ Cos2 A=1. To find the cosine of the angle And calculate: 1- (BD/AC)2, from the result to extract the square root. The cosine of the angle As found.

Sin2 A+ Cos2 A=1. To find the cosine of the angle And calculate: 1- (BD/AC)2, from the result to extract the square root. The cosine of the angle As found.

4

If the triangle is known to all parties, the cosine of any angle find the cosine theorem: the square of the sides of a triangle is equal to the sum of the squares of the other two parties without twice the product of these sides into the cosine of the angle between them. Then the cosine of the angle A in a triangle with sides a, b, C compute by the formula: Cos A = (A2-b2-c2)/2*b*C.

5

If the triangle you need to determine the cosine of obtuse angle, use the formula of the coercion. The obtuse angle of a triangle is more direct, but less deployed, it can be written as 180°-α, where α is an acute angle complementary to the obtuse angle of the triangle to the deployed. According to the formula cast, find the cosine: Cos (180°-α)= Cos α.