Learning Objectives
- Understand the concepts of angle of elevation and angle of depression
- Use trigonometric ratios to find heights and distances
- Solve problems involving two observation points
- Draw and interpret diagrams for height and distance problems
Key Concepts
Line of Sight
The line drawn from the eye of an observer to the point being viewed is called the line of sight.
Angle of Elevation
The angle formed between the horizontal line (at eye level) and the line of sight when an observer looks upward at an object. The angle of elevation increases as the object appears higher.
Angle of Depression
The angle formed between the horizontal line (at eye level) and the line of sight when an observer looks downward at an object. The angle of depression from point A to point B equals the angle of elevation from B to A (alternate interior angles).
Problem-Solving Strategy
- Step 1: Draw a clear diagram showing the right triangle(s).
- Step 2: Mark the known and unknown quantities.
- Step 3: Identify which trigonometric ratio connects the known and unknown sides/angles.
- Step 4: Solve the equation and express the answer in required form.
Common Problem Types
Finding height of a tower/building: Use tan θ = height / distance from base.
Finding distance: Use the known height and angle to compute horizontal distance.
Two-position problems: Observe the same object from two different points; form two equations and solve simultaneously.
Summary
Trigonometry is applied to find heights and distances that are difficult to measure directly. By measuring angles of elevation or depression and one known distance, unknown heights or distances can be calculated. Drawing accurate diagrams and choosing the right trigonometric ratio are key to solving these problems.
Important Terms
- Angle of Elevation
- The angle between the horizontal and the line of sight when looking upward
- Angle of Depression
- The angle between the horizontal and the line of sight when looking downward
- Line of Sight
- The straight line from the observer's eye to the object being observed
Quick Revision
- Angle of elevation is measured from the horizontal upward
- Angle of depression is measured from the horizontal downward
- Angle of depression from A to B = Angle of elevation from B to A
- tan θ = Perpendicular/Base is the most commonly used ratio in height-distance problems
- Always draw a diagram before solving; label all given information clearly