A vertical pole stands at the center of a regular hexagonal field. The angle of elevation of the top of the pole from any vertex of the hexagon is , where = 2 . If the angle of elevation of the top of the pole from the midpoint of any side of the hexagon is , then the value of 3 ^2 is equal to

Step-by-step solution

Let the side length of the regular hexagon be a and the height of the vertical pole be h . The distance from the center of a regular hexagon to any of its vertices is equal to its side length a . Given that the angle of elevation from a vertex is , we have: = h a = 2 h = 2a The distance from the center of the hexagon to the midpoint of any side is the apothem, which is a (30^ ) = a 3 2 . The angle of elevation from the midpoint of a side is , so: = h a 3 2 = 2h a 3 Substitute h = 2a into the equation: = 2(2a) a 3 =