Here's an alternative solution if you don't want to use the cosine formula: Slice a unit circle into 24 sectors each forming a 15-degree angle. Create 24 isosceles triangles inside these sectors using the two edges (of length one) with 15 degrees between them. The area size of the unit circle, i.e., pi, is greater than the area sum of 24 triangles thus created. Now, the base of each triangle is 1 and its height is sin(15), and thus the area size of the triangle is sin(15)/2. We don't know the value of sin(15) but we know that of 30 and 45 (and cosines thereof). Using the compound-angle formula: sin(15) =sin(45-30) = sin 45 cos 30 - cos 45 sin 30 = sqrt(2)/2*sqrt(3)/2 - sqrt(2)/2/2. Then the area sum of 24 triangles is 24*sin(15)/2 = sqrt(2)*(sqrt(3)-1)*3 > 3.1058. Thus pi is at least 3.105.
Пікірлер: 1 М.