How to find the cosine of a triangle

How to find the cosine of a triangle?

Cosine is a well-known trigonometricfunction, which is also one of the basic functions of trigonometry. The cosine of an angle in a triangle of rectangular type is the ratio of the adjacent triangle to the hypotenuse of the triangle. Most often, the definition of the cosine is associated with a triangle of a rectangular type. But it also happens that the angle for which it is necessary to calculate the cosine in a triangle of a rectangular type is not located in this very triangle of a rectangular type. What, then, should be done? How to find the cosine of the angle of a triangle?

If you want to calculate the cosine of the angle in exactlytriangle of a rectangular type, then everything is very simple. It is only necessary to recall the definition of the cosine, in which the solution of this problem lies. It is simply necessary to find the very relation between the adjacent leg and the hypotenuse of the triangle. Indeed, here it is not difficult to express the cosine of the angle. The formula is as follows: - cosα = a / c, here "a" is the length of the leg, and the side "c", respectively, the length of the hypotenuse. For example, the cosine of the acute angle of a right triangle can be found from this formula.

If you are interested in what the cosine of the angle inan arbitrary triangle, then the cosine theorem comes to the aid, which should be used in such cases. The cosine theorem states that the square of the side of the triangle is a priori equal to the sum of the squares of the remaining sides of the same triangle, but without the doubled product of these sides by the cosine of the angle that lies between them.

  1. If you need to find the cosine of an acute angle in a triangle, then you need to use the following formula: cosα = (a2+ b2- c2) / (2ab).
  2. If in the triangle it is necessary to find the cosine of the obtuse angle, then we need to use the following formula: cosα = (with2- a2- b2) / (2ab). The notation in the formula - a and b - is the length of the sides that are adjacent to the desired angle, c is the length of the side that is opposite the desired angle.

Also, the cosine of the angle can be calculated using thesine theorem. It says that all sides of the triangle are proportional to the sinuses of the corners, which are opposite. Using the sine theorem, we can calculate the remaining elements of a triangle, having information only on two sides and an angle that is opposite to one side, or at two angles and one side. Consider the example. Conditions of the problem: a = 1; b = 2; c = 3. The angle that is opposite to the "A" side is denoted by α, then, according to the formulas, we have: cosa = (b² + c²-a²) / (2 * b * c) = (2² + 3²-1²) / (2 * 2 * 3) = (4 + 9-1) / 12 = 12/12 = 1. Answer: 1.

If the cosine of the angle is to be calculated not intriangle, and in some other arbitrary geometric figure, then everything becomes a little more complicated. The value of the angle must first be determined in radians or degrees, and then calculate the cosine by this value. The cosine of a numerical value is determined using Bradis tables, engineering calculators or special mathematical applications.

Special mathematical applications can havesuch functions as automatic calculation of cosines of corners in this or that figure. The beauty of such applications is that they give the right answer, and the user does not spend his time on solving sometimes rather complicated tasks. On the other hand, with the constant use of only applications for solving problems, all the skills to work with solving mathematical problems on finding cosines of angles in triangles, as well as other arbitrary figures, are lost.

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