Chapter 9: Applications of Trigonometry
Finding Heights and Distances through the Line of Sight.
📖 Essential Terminology
The line of sight is the imaginary straight line drawn from the eye of an observer to the point being viewed on an object. The horizontal level is the straight line parallel to the ground passing through the observer's eye.
To solve height/distance problems, choose a trigonometric ratio (usually sin, cos, or tan) that connects the known value with the unknown value you need to determine.
⬆️ Angle of Elevation
The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.
Case: Raising your head to look up.
⬇️ Angle of Depression
The angle formed by the line of sight with the horizontal when the point is below the horizontal level.
Case: Lowering your head to look down.
⭐ Board Exam Super-Hacks
Alternate Interior Angles Trick
When dealing with an angle of depression, always draw the horizontal line at the top. The angle of depression is equal to the angle of elevation from the object to the observer due to alternate interior angles.
The "Distance & Angle" Rule
As you walk towards a building, the angle of elevation increases (e.g., from 30° to 60°). If you walk away, it decreases. Use this to check if your diagram makes sense!
⚠️ Common Exam Traps
Trap 1: Ignoring Observer's Height
If the problem states the observer's height (e.g., "A 1.5m tall girl..."), you MUST subtract this height from the total height of the object when using trigonometric ratios, and add it back at the end!
Trap 2: The "Depression" Diagram Error
Students often mark the angle of depression inside the triangle between the line of sight and the VERTICAL wall. This is wrong. It must always be between the line of sight and the HORIZONTAL level.