How to calculate the cosine of the angle

Cosine one of the trigonometric functions, which is used in the solution of geometric and physical problems. Actions with vectors also, rarely do without the use of a cosine. There are several ways to compute the cosine of the angle from the simplest arithmetic to the decomposition in a Taylor series. The choice of method depends on the required accuracy of values of the cosine.

Instruction

1. Any student known table Bradis. He produced many laborious calculations, but freed mathematicians from the time-consuming calculation of the values of the basic trigonometric functions for a large number of angles. Before the wide dissemination of calculators and computers, these tables were used by virtually all engineers, mathematicians, physicists and students.
2. To calculate the cosine of the angle the table very easily. It is sufficient to find the column of values of angles angle degrees and then go through the table line to the intersection with the minute angle. The figure shows a fragment of the table Bradis. It is seen that the cosine value for the angle 72°30' is 0,3007. The tables Bradis, you can find function values with accuracy to 0.0001, for most calculations, such accuracy is quite sufficient.
3. Originally the trigonometric functions were related to the right triangle and the ratio of its sides. It is possible to recall and to apply a known ratio, if the angle is acute. Construct a right triangle with the given angle. To do this, swipe two beams and lower from any one of them perpendicular to the other. Now, if we denote the intersection points of rays with the letters a, B and C, it can be argued that cos ∠BAC = CA/AB or against adjacent thereto of the AC leg to the hypotenuse AB. The accuracy of this method is low and depends heavily on the accuracy of the build.
4. For more precise calculations of trigonometric functions are laid out in Taylor series. The Taylor series for the cosine, see figure. The decomposition in a number allows you to calculate cosine with any accuracy. The higher the accuracy, the more members of the series will have to find. Bradis in its tables laid out cosine in a number and found the first few members. Modern calculators do the same thing.
5. Try to manually calculate the cosine of 72°30'. To do this, first set the angle to radians: 72°30' = 72,5°*π rad/180° = 1,2654 happy (note that the value of the number π it is necessary to take also quite accurate, in the formula used π≈3,1416). Now substitute this value into a number and calculate a few first terms of series: 1 - 1,2654^2/2 + 1,2654^4/24 - 1,2654^6/720 + 1,2654^8/40320 = 1 - 0,8006 + 0,1068 - 0,0057 + 0,0002 = 0,3006, where 720 = 6!, 40320 = 8!.
   Thus, cos 72°30' = cos 1,2654 glad ≈ 0,3006.