How to find the cosine of the angle of a triangle with vertex

Instruction
1

For an arbitrary triangle with vertices A, B and C, the problem of finding the cosine is the same for all three angles if triangle island.$ If there is a blunt angle in the triangle, the definition of its cosine should be considered separately.

2

In the island-angle triangle with vertices A, B and C, find the cosine of the angle at the vertex A. Lower the height from vertex B to the side of triangle AU. The point of elevation intersection with the AU side denote D and consider the rectangular triangle AVD. In this triangle, the AB side of the original triangle is hypotenuse, and the cathets are the height of the VD of the original point-angle triangle and the segment of AD belonging to the AU side. The cosine of angle A is equal to the AD/AB relation, since the cathete AD is adjacent to angle A in the rectangular triangle of AVD. If it is known in which ratio the height of VD divides the side of AS of the triangle, the cosine of angle A is found.

3

If the magnitude of AD is not given, but the height of VD is known, the cosine of the angle can be determined through its sine. The sine of angle A is equal to the ratio of the VD height of the original triangle to the AU side. The basic trigonometric identity establishes the relationship between the sine and the cosine of the angle:

Sin² A+ Cos² A=1. To find the cosine of angle A, calculate: 1- (HD/AC) ², from the result you need to extract the square root. Cosine of angle A found.

4

If all sides are known in a triangle, then the cosine of any angle find by the cosines theorem: the square of the side of the triangle is equal to the sum of the squares of two other parties without double the work of these sides on the cosine of the angle between them. Then the cosine of angle A in the triangle with sides a, b, with the formula: Cos A = (a²-b²-c²) /2*b*s.

5

If you need to define the cosine of the blunt angle in a triangle, use the cast formula. The blunt angle of the triangle is larger than the straight but smaller than the unfolded, it can be written as 180°-α, where α is the sharp angle complementing the blunt angle of the triangle to the unfolded. By the cast formula, find the cosine: Cos (180°-α) = Cos α.

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