If you know the sine of an interior angle (α₀) geometric shapes to calculate something is not necessary - the sine of the corresponding external angle (α₁) will have exactly the same value: sin(α₁) = sin(α₀). This is determined by the properties of the trigonometric functions sin(α₀) = sin(180°-α₀). If you wanted to find out, for example, cosine or tangent of the outer edge, this value would need to be taken with the opposite sign.
There is a theorem that in a triangle the sum of any two interior angles equal the exterior angle of a third vertex. Use it in that case, if the magnitude of the internal angle corresponding to the considered external (α₁), unknown, and the angles (β₀ and γ₀) the other two vertices are given in the conditions. Find the sine of the sum of the known angles: sin(α₁) = sin(β₀+γ₀).
The task with the same initial conditions as in the previous step, has another solution. It follows from another theorem - the sum of the interior angles of a triangle. As this amount is, according to the theorem, must be equal to 180°, the value of the unknown internal angle can be represented using two known (β₀ and γ₀) - it will be equal to 180°-β₀-γ₀. This means that you can use the formula from the first step, replacing in this the value of internal angle of this expression: sin(α₁) = sin(180°-β₀-γ₀).
In a regular polygon the value of the external angle at any vertex is equal to the size of the Central angle and, therefore, can be calculated using the same formula that he did. Therefore, if the conditions of the problem given the number of sides (n) of the polygon, when computing the sine of any angle (α₁) assume that its value is equal to the turnover divided by the number of sides. Full rotation in radians is expressed by twice the number PI, so the formula should look like this: sin(α₁) = sin(2*π/n). When calculating in degrees double PI replace 360°: sin(α₁) = sin(360°/n).
Advice 2 : How to find the cosine of the external angle
Any angle can be extended to the deployed, if to extend over the top of one of its sides. While the other side will divide a straight angle into two. The angle formed by the second party and the first called adjacent, and when we are talking about polygons, so it called external. The fact that the sum of external and internal angles, by definition, equal to the value of the maximized angle allows you to calculate trigonometric functions on the known ratios of parameters of the polygons.
Knowing the result of the computation of the cosine of the internal angle (α) you'll know the module of the cosine external (α₀). The only thing that you need to produce with this value to change its sign, i.e., multiplied by -1: cos(α₀) = -1*cos(α).
If we know the value of the internal angle (α) to calculate the cosine of an external (α₀) you can use the method described in the previous step is to find its cosine, and then to change the sign. But it can be done differently - just compute the cosine of the external angle, subtracting this value from the inner 180°: cos(α₀) = cos(180°-α). If the magnitude of the internal angle is given in radians, the formula needs to be converted to this: cos(α₀) = cos(π-α).
The regular polygon to calculate the value of the external angle (α₀) does not need to know any parameters except the number of vertices (n) of this figure. This number divide 360° find the cosine of the resulting numbers: cos(α₀) = cos(360°/n). For calculations in radians on the number of vertices you have to share twice the number PI, and the formula should acquire the following form: cos(α₀) = cos(2*π/n).
In a right triangle, the cosine of the external angle at the vertex lying opposite the hypotenuse is always equal to zero. For the other two vertices, this value can be calculated, knowing the length of the hypotenuse (c) leg (a), which form the top. No trigonometric functions to calculate required, just divide the length of the short side length of the larger, and change the sign of the result is: cos(α₀) = -a/c.
If you know the lengths of two sides (a and b), too, can do the calculations without trigonometric functions, but the formula is somewhat more complicated. Fraction, the denominator of which is the length of the side adjacent to the top outer corner, and in the numerator - the length of the other leg defines the tangent of the internal angle. Knowing the tangent, you can calculate the cosine of the internal angle: √(1/(1+a2/b2). This expression will replace the cosine in the right part of the formula from the first step: cos(α₀) = -1*√(1/(1+a2/b2).