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A person stands on a building that is 12.6 tall. A phone is on the ground 58.5 meters away from the building. Find the angle of depression from your eyes to the object.

A person stands on a building that is 1212.66 tall. A phone is on the ground 5858.55 meters away from the building.\newlineFind the angle of depression from your eyes to the object.

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Full solution

Q. A person stands on a building that is 1212.66 tall. A phone is on the ground 5858.55 meters away from the building.\newlineFind the angle of depression from your eyes to the object.
  1. Identify Relationship: Identify the relationship between the angle of depression, the height of the building, and the distance from the building to the phone.\newlineThe angle of depression is the angle formed between the line of sight from the person's eyes to the phone and a line parallel to the ground. This angle is equal to the angle of elevation from the phone to the person's eyes when looking at the building.
  2. Determine Triangle: Determine the right triangle that will be used to calculate the angle of depression. The height of the building (12.612.6 meters) will be the opposite side of the right triangle, and the distance from the building to the phone (58.558.5 meters) will be the adjacent side of the right triangle.
  3. Use Tangent Function: Use the tangent function to find the angle of elevation, which is equal to the angle of depression.\newlineThe tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Therefore, we will use the formula tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}.
  4. Plug in Values: Plug in the values for the opposite and adjacent sides into the tangent function. tan⁡(θ)=12.658.5\tan(\theta) = \frac{12.6}{58.5}
  5. Calculate Tangent: Calculate the value of the tangent of the angle. tan⁡(θ)=12.658.5≈0.2153846154\tan(\theta) = \frac{12.6}{58.5} \approx 0.2153846154
  6. Use Inverse Tangent: Use the inverse tangent function to find the angle θ\theta.θ=arctan⁡(0.2153846154)\theta = \arctan(0.2153846154)
  7. Calculate Angle: Calculate the angle θ\theta using a calculator or a trigonometric table.\newlineθ≈arctan⁡(0.2153846154)≈12.3\theta \approx \arctan(0.2153846154) \approx 12.3 degrees

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