Lecture Notes: Solving Angle of Elevation and Depression Word Problems

Key Concepts

• Angle of Elevation: The angle above the horizontal line.
• Angle of Depression: The angle below the horizontal line.
• Right Triangle: Typically drawn to solve these problems.
• SOHCAHTOA: Mnemonic for trigonometric ratios:
  • Sine (SOH): ( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} )
  • Cosine (CAH): ( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} )
  • Tangent (TOA): ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} )

Solving Word Problems

Example 1: Building Height

• Problem: A man measures the angle of elevation between the ground and a building 800 feet away to be 30 degrees. Find the building height.
• Solution:
  1. Draw a right triangle with the building, ground, and line of sight.
  2. Use the tangent ratio:
     • ( \tan(30^\circ) = \frac{h}{800} )
  3. Solve for ( h ):
     • ( h = 800 \times \tan(30^\circ) = 800 \times \frac{\sqrt{3}}{3} \approx 461.88 \text{ feet} )
  4. Note: Use a calculator in degree mode.

Example 2: Angle of Elevation

• Problem: Calculate the angle of elevation from a point to a 50-foot tree 20 feet away.
• Solution:
  1. Draw the right triangle.
  2. Use the tangent ratio:
     • ( \tan(\theta) = \frac{50}{20} = 2.5 )
  3. Find ( \theta ):
     • ( \theta = \tan^{-1}(2.5) \approx 68.2^\circ )

Example 3: Distance to Boat

• Problem: A man in a 100-foot observation tower sees a boat with an angle of depression of 10 degrees. Find the distance to the boat.
• Solution:
  1. Draw a right triangle with the tower height and horizontal distance.
  2. Use the tangent ratio:
     • ( \tan(10^\circ) = \frac{100}{x} )
  3. Solve for ( x ):
     • ( x = \frac{100}{\tan(10^\circ)} \approx 567.1 \text{ feet} )

Summary

• Angle of Elevation: Measured above the horizontal.
• Angle of Depression: Measured below the horizontal.
• Use trigonometric ratios (primarily tangent for these problems) to solve for unknowns in right triangles.