Trigonometry Case Study - Lighthouse Navigation
Case Study: The Lighthouse Navigation
Exploring real-world applications of trigonometric ratios in maritime navigation
The Scenario
What is a Lighthouse? A lighthouse is a tower, building, or other type of structure designed to emit light from a system of lamps and lenses and to serve as a navigational aid for maritime pilots at sea or on inland waterways.
The Problem: A guard stationed at the top of a 100 m high lighthouse observes two ships approaching it. The ships are on the same side of the lighthouse. The angles of depression of the two ships are observed to be 60° and 30° respectively.
Key Concept: Angle of Depression
๐ Definition
The angle of depression is the angle formed between the horizontal line of sight and the line of sight looking downward to an object below the horizontal.
๐ Important Property
The angle of depression from point A to point C equals the angle of elevation from point C to point A (alternate interior angles with parallel lines).
Question 1
Relate the angle of depression to the angle of elevation using the properties of parallel lines.
✓ Solution
Given: Horizontal line through A is parallel to ground line BD
Property Used: When a transversal cuts two parallel lines, alternate interior angles are equal.
∠ of Depression from A = ∠ of Elevation from ground point
(Alternate Interior Angles)
Therefore:
- Angle of depression to Ship 1 (60°) = Angle of elevation from C to A (60°)
- Angle of depression to Ship 2 (30°) = Angle of elevation from D to A (30°)
Question 2
Which ship is closer to the lighthouse: the one with the 60° angle of depression or the one with 30°? Justify.
✓ Solution
Using: tan ฮธ = Opposite / Adjacent = Height / Distance
Distance = Height / tan ฮธ
Ship 1 (60° depression)
tan 60° = √3 ≈ 1.732
Distance = 100/√3 ≈ 57.7 m
Ship 2 (30° depression)
tan 30° = 1/√3 ≈ 0.577
Distance = 100√3 ≈ 173.2 m
๐ข Ship with 60° angle of depression is CLOSER
Greater angle of depression → Closer distance
Question 3
Find the distance between the two ships.
✓ Step-by-Step Solution
Set up the diagram
Let BC = x (distance of Ship 1 from lighthouse)
Let BD = y (distance of Ship 2 from lighthouse)
Height AB = 100 m
In right triangle ABC (for Ship 1)
tan 60° = AB / BC
√3 = 100 / x
x = 100/√3 = 100√3/3 m
In right triangle ABD (for Ship 2)
tan 30° = AB / BD
1/√3 = 100 / y
y = 100√3 m
Calculate distance between ships (CD = BD - BC)
CD = y - x
CD = 100√3 - 100√3/3
CD = 100√3(1 - 1/3)
CD = 100√3 × (2/3)
CD = 200√3/3 m ≈ 115.47 m
๐ Distance between the two ships
200√3/3 m ≈ 115.47 m
Alternative Question
If a third ship is observed exactly at the foot of the lighthouse, what would be its angle of depression? Calculate the distance of the ship with the 60° angle of depression from the foot of the lighthouse.
A Angle of Depression for Ship at Foot
If a ship is exactly at the foot of the lighthouse (point B), the guard would look straight down.
The line of sight would be perpendicular to the horizontal.
Angle of Depression = 90°
B Distance of Ship with 60° Depression
Set up the triangle
In right triangle ABC:
- AB = 100 m (height)
- ∠ACB = 60° (angle of elevation = angle of depression)
- BC = ? (distance to find)
Apply trigonometric ratio
tan 60° = AB / BC
√3 = 100 / BC
BC = 100 / √3
BC = 100 / √3 × √3 / √3
BC = 100√3 / 3 m
Calculate numerical value
BC = 100 × 1.732 / 3
BC ≈ 57.74 m
๐ Distance from foot of lighthouse
100√3/3 m ≈ 57.74 m
Quick Reference: Trigonometric Values
| Angle (ฮธ) | sin ฮธ | cos ฮธ | tan ฮธ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
SOH
Sin = Opposite / Hypotenuse
CAH
Cos = Adjacent / Hypotenuse
TOA
Tan = Opposite / Adjacent
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