The angle of depression from the top of a 320 foot office building to the top of a 200 foot office building is 55 degrees. How far apart are the buildings?

To solve this problem, you can use right triangle trigonometry.

1. First, represent the situation with a sketch where the taller building is one vertex of a right triangle, the top of the shorter building is the second vertex, and the base is the ground distance between the two buildings. 2. The angle of depression from the top of the taller building to the top of the shorter building is given as 55 degrees. This angle of depression equals the angle of elevation from the top of the shorter building to the top of the taller building due to alternate interior angles (since the rays are parallel). 3. Label the difference in the heights of the two buildings as the opposite side of the right triangle because it's perpendicular to the ground. This difference is 320 feet - 200 feet = 120 feet. 4. The angle of elevation from the top of the shorter building to the top of the taller building (55 degrees) is adjacent to the angle you're trying to use in your trigonometric functions. This poses an issue, as the functions you are likely to use (tan, sin, or cos) will be based on the 55-degree angle of depression/elevation. 5. Use the tangent function, which relates the opposite side to the adjacent side through the angle. In this scenario, the opposite side is the height difference (120 feet), and the tangent of the angle of elevation (55 degrees) equals the opposite side divided by the distance between the buildings, which is what you're looking for.

So the equation, based on tangent function, you're going to use is:

tan(angle) = opposite/adjacent

Therefore,

tan(55 degrees) = 120 feet / distance

Now, solve for the distance.

distance = 120 feet / tan(55 degrees)

6. Calculate using a calculator ensuring that it is in degree mode:

distance ≈ 120 feet / 1.428148

distance ≈ 83.98 feet

So, the buildings are approximately 84 feet apart.

Extra: In trigonometry, the angle of depression is the angle between the horizontal line of sight and the line of sight down to an object. The angle of elevation is the angle between the horizontal and the line of sight up to an object. These two angles are congruent when the lines of sight are parallel, which typically happens when you're looking from one horizontal plane down or up to another.

The trigonometric functions used to solve these types of problems are sine (sin), cosine (cos), and tangent (tan), which relate the angles of a right triangle to the ratios of the lengths of the sides. In this case, the tangent function is especially useful because you're dealing with the opposite and adjacent sides relative to the given angle. The opposite side is the vertical distance (elevation or depression), and the adjacent side is the horizontal distance you want to find.