How to find the cosine of the angle of the external
Any plane angle can be extended to full-scale if extended over the top of one of its sides. At the same time the other side is divided into two straight angle. The angle formed by the second side and the first extension is called contiguous, and in the case of polygons, it is also called external. The fact that the sum of the external and internal corners by definition equal to the straight angle allows trigonnometry polygons on known relations parameters.
Instruction how to find the cosine of the angle of the external
Knowing the result of the calculation of the cosine of the internal angle (α), you will know the cosine of an external module (α₀). The only operation that you need to make this value - to change its sign, that is, multiplied by -1: cos (α₀) = -1 * cos (α).
If the known value of the internal angle (α), to calculate the cosine of the external (α₀) you can use the method described in the previous step - find the cosine and then change sign. But you can do in a different way - just to calculate the cosine of the outer corner, subtracting this value from the internal 180 °: cos (α₀) = cos (180 ° -α). If the value of the internal angle is given in radians, the formula to convert to a form: cos (α₀) = cos (π-α).
In a regular polygon to calculate the magnitude of the external angle (α₀) do not need to know any parameters except for the number of vertices (n) of this figure. On this number, divide 360 ° and find the cosine of the number obtained: cos (α₀) = cos (360 ° / n). For the calculations in radians on the number of vertices you have to share twice the number Pi, and the formula should buy this kind: cos (α₀) = cos (2 * π / n).
In a right-angled triangle, the cosine of the angle at the top of the outer lying opposite the hypotenuse is always zero. For the other two vertices of this value can be calculated by knowing the length of hypotenuse (c) and leg (a), which form the top. No trigonometric functions without the need to calculate, simply divide the length of the lower side to the length of more and change the sign of the result: cos (α₀) = -a / c.
If you know the lengths of two of the legs (a and b), also can do calculations without the trigonometric functions, but the formula will be somewhat more complicated. The fraction, the denominator of which is the length of side adjacent to the top outer corner, and in the numerator - the length of the other leg, determines the tangent of the inner corner. Knowing the tangent can calculate the cosine of the angle of internal:. √ (1 / (1 + a² / b²) This expression replace the cosine of the right side of the first step: cos (α₀) = -1 * √ (1 / (1 + a² / b² ).