Advice 1: As the cosine to find a side of a triangle

Knowledge of the values of the cosine of the angle at the vertex of an arbitrary triangle allows you to find the value of this angle. But with a single parameter it is impossible to know the length of a side of such figure, need any more related quantities. If they are given in terms of the choice of the calculation formula will depend on what parameters are chosen as additions to the cosine of the angle.
As the cosine to find a side of a triangle
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 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.
How to find the cosine of the angle of the triangle with vertices
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.
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.
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