Instruction

1

If the known values of two angles of an arbitrary triangle (β and

**γ**), the value of the third (α) can be determined from the theorem about sum of angles in a triangle. She says that this sum in Euclidean geometry is always 180°. That is, for finding the only unknown angle in the triangle vertices subtract from 180° the values of the two known angles: α=180°-**β**-**γ**.2

If we are talking about a right triangle, to find the values of the unknown acute angle (α) it is enough to know the value of the other acute angle (

**β**). As in this triangle the angle lying opposite the hypotenuse is always equal to 90°, then find the value of the unknown angle subtract from 90° the value of the known angle α=90°-**β**.3

Isosceles triangle is also sufficient to know the value of one of the corners to compute the other two. If you know the angle (

**γ**) between the sides of equal length, to compute both the other angles, find the half of the difference between 180° and the value of the known angle - these angles in an isosceles triangle are equal: α=**β**=(180°-**γ**)/2. This implies that if we know the value of one of the equal angles, the angle between the equal sides can be defined as the difference between 180° and the doubled value of the known angle**γ**=180°-2*α.4

If you know the lengths of three sides (A, B, C) in an arbitrary triangle, then the measure of the angle to find the cosine theorem. For example, the cosine of the angle (

**β**) lying opposite the side B, can be expressed as the sum of the squared lengths of the sides A and C, reduced by the squared length of side B divided by twice the product of the lengths of the sides A and C: cos(**β**)=(A2+C2-B2)/(2*A*C). And to find the measure of an angle, knowing something is equal to its cosine, we need to find it, the ark function, that is the arc cosine. Then**β**=arccos((A2+C2-B2)/(2*A*C)). Similarly, you can find angles lying opposite the other sides of that triangle.# 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.

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.