Overview
This lecture explains how to solve word problems involving angles of elevation and depression using right triangle trigonometry, particularly focusing on the tangent function.
Drawing Right Triangles for Word Problems
- Draw a right triangle to visualize problems dealing with elevation and depression.
- The angle of elevation is measured above the horizontal line, while the angle of depression is below it.
- Label the diagram with known distances (horizontal, vertical) and unknowns to solve for.
Trigonometric Ratios (SOH-CAH-TOA)
- Sine (sin θ) = opposite / hypotenuse.
- Cosine (cos θ) = adjacent / hypotenuse.
- Tangent (tan θ) = opposite / adjacent.
- Decide which trigonometric ratio to use based on given and required sides.
Example 1: Finding Building Height with Angle of Elevation
- Given: Distance from building = 800 ft, angle of elevation = 30°.
- Use tangent: tan(30°) = height / 800.
- Solve: height = 800 × tan(30°) ≈ 461.88 ft.
- Exact value: tan(30°) = √3/3, so height = 800√3/3.
Example 2: Finding Angle of Elevation
- Given: Height of tree = 50 ft, distance from tree = 20 ft.
- Use tangent: tan(θ) = 50 / 20 = 2.5.
- Find angle: θ = tan⁻¹(2.5) ≈ 68.2°.
- Use inverse tangent to solve for angles when two sides are known.
Example 3: Distance from Tower with Angle of Depression
- Given: Observation tower height = 100 ft, angle of depression = 10°.
- Use tangent: tan(10°) = 100 / x (x = distance from tower to boat).
- Solve: x = 100 / tan(10°) ≈ 567.1 ft.
Summary of Elevation and Depression Angles
- Angle of elevation: measured above the horizontal.
- Angle of depression: measured below the horizontal.
Key Terms & Definitions
- Angle of Elevation — The angle above the horizontal observed from a lower point.
- Angle of Depression — The angle below the horizontal observed from a higher point.
- Tangent (tan θ) — Ratio of the length of the opposite side to the adjacent side in a right triangle.
- Inverse Tangent (tan⁻¹ or arctan) — Used to find the angle when the tangent ratio is known.
Action Items / Next Steps
- Practice drawing diagrams for elevation and depression problems.
- Use tan, tan⁻¹, and the appropriate ratios to solve similar word problems.
- Ensure your calculator is set to degree mode when working with these problems.