A vertical pole stands at a corner O of a horizontal rhombus OABC . The angles of elevation of the top of the pole from the adjacent corner A and the opposite corner B are 45^ and 30^ respectively. The measure of the acute angle of the rhombus is

Step-by-step solution

Let the height of the vertical pole at O be h . The horizontal distance to the adjacent corner A is: OA = h 45^ = h Since OABC is a rhombus, all its sides are equal. Thus, the side length of the rhombus is a = OA = h . The horizontal distance to the opposite corner B (which is the length of the diagonal OB ) is: OB = h 30^ = h 3 In OAB , the lengths of the sides are OA = a , AB = a , and OB = a 3 . Using the cosine rule in OAB to find OAB : ( OAB) = OA^2 + AB^2 - OB^2 2 OA AB ( OAB) = a^2 + a^2 - (a 3 )^2 2 a a ( O