Word problem using the law of sines

Question

The law of sines. A 150-ft antenna is on top of a building. From a point on the ground, the angle of elevation to the top of the antenna is 28.5 degrees, while the angle of elevation to the bottom of the antenna from the same point is 23.5 degrees. How tall is the building?

Answer 1

Let h be the height of the building and let P be the point from which the measurements are taken.

Draw a right triangle with vertical side h + 150 and angle with vertex at P with measure 28.5°. The other acute angle of the triangle has measure 90°-28.5° = 61.5°. Draw a line segment from P to the bottom of the antenna to construct another right triangle. The acute angle of this triangle with vertex at point P has measure 23.5° and let x be the hypotenuse. A third triangle is formed with one side as the antenna and the two other sides are the hypotenuses of the two right triangles.

Using the Law of Sines on the third triangle, sin5°/150 = sin61.5°/x. So x = 1512.49

Using the right triangle with acute angle of measure 23.5°, sin23.5° = h/1512.49. So h = 603.1 ft

Answer 2

(150 + h)/tan28.5° = h/tan23.5°; h(1/tan23.5° - 1/tan28.5°) = 150/tan28.5°;

h = 150cot28.5°/(cot23.5° - cot28.5°) = 603.11 ft