Trigonometry Case Study - Drone Surveillance
Case Study 3: Drone Surveillance
Explore how drones use angles of depression to measure distances and heights in real-world security applications
๐ฃ️ Road Monitoring | ๐ Dual Object Tracking | ๐ Height & Distance Calculations
The Scenario
๐ Setting: Drone Surveillance System
A sophisticated drone is deployed for security surveillance over a straight highway. The drone maintains a stationary hovering position at a fixed altitude. Two vehicles (cars) are parked on the road on opposite sides of the point directly below the drone.
๐ Car 1:
Angle of depression = 30°
๐ Car 2:
Angle of depression = 45°
๐ Total Distance:
100(√3 + 1) m
Note: The two cars are on opposite sides of the point directly below the drone (opposite sides of the road).
๐ Car 1 (Left)
Further away → Smaller angle = 30°
๐ Drone
Hovering stationary at height H meters
๐ Car 2 (Right)
Closer distance → Larger angle = 45°
Question 1: Understanding Angle Changes
If the angle of depression increases, does it mean the object is moving closer to or further away from the point directly below the observer?
✓ Solution with Explanation
✅ The object is moving CLOSER
Increased angle of depression → Decreased horizontal distance
๐ Mathematical Reasoning
❌ Small Angle (Object Far)
When angle of depression is small (e.g., 30°):
tan 30° = H / d₁
d₁ is LARGE
✅ Large Angle (Object Close)
When angle of depression is large (e.g., 45°):
tan 45° = H / d₂
d₂ is SMALL
๐ Visual Representation
๐ Key Relationship
For fixed height H:
tan ฮธ = H / d
When ฮธ increases → tan ฮธ increases → d decreases
When ฮธ decreases → tan ฮธ decreases → d increases
๐ก Real-World Example (From This Problem)
• Car 1 at 30° angle → distance is 100√3 ≈ 173.2 m (FARTHER)
• Car 2 at 45° angle → distance is 100 m (CLOSER)
Larger angle (45°) = Closer distance! ✓
Question 2: Trigonometric Ratio
Write the trigonometric ratio used to relate the height of the drone and the horizontal distance to the cars.
✓ Solution with Explanation
๐ The Trigonometric Ratio
tan ฮธ = Opposite / Adjacent = H / d
Where: ฮธ = angle of depression, H = height, d = horizontal distance
๐ Understanding the Components
๐ Height (H)
The vertical distance from the drone to the road (perpendicular distance)
In our problem: This is what we need to find!
↔️ Horizontal Distance (d)
The straight-line distance along the road from the point below the drone to the car
Also unknown, but will be expressed in terms of H
∠ Angle of Depression (ฮธ)
The angle between the horizontal and the line of sight downward to the car (GIVEN)
In our problem: 30° and 45°
๐ Right Triangle Formation
๐ฏ Application in This Problem
For Car 1 (30° depression):
tan 30° = H / d₁
For Car 2 (45° depression):
tan 45° = H / d₂
๐ง Memory Aid: SOH CAH TOA
sin = Opp/Hyp
cos = Adj/Hyp
tan = Opp/Adj ← We use this!
Question 3: Calculate Drone Height
Find the height (H) at which the drone is hovering.
✓ Step-by-Step Solution
Write Equations for Both Cars
For Car 1 (angle = 30°):
tan 30° = H / d₁
1/√3 = H / d₁
d₁ = H√3 ... (1)
For Car 2 (angle = 45°):
tan 45° = H / d₂
1 = H / d₂
d₂ = H ... (2)
Use the Total Distance Condition
Given: Total distance = d₁ + d₂ = 100(√3 + 1) m
Substituting from equations (1) and (2):
H√3 + H = 100(√3 + 1)
H(√3 + 1) = 100(√3 + 1)
Solve for H
H(√3 + 1) = 100(√3 + 1)
Dividing both sides by (√3 + 1):
H = 100 m
Verification
When H = 100 m:
d₁ = H√3 = 100√3 m ✓
d₂ = H = 100 m ✓
Total = 100√3 + 100 = 100(√3 + 1) m ✓
Numerical check:
100 × 1.732 + 100 = 173.2 + 100 = 273.2 m ✓
๐ฏ Height of the Drone
H = 100 m
The drone is hovering at a height of 100 meters above the road
๐ก Why This Works (Alternative Insight)
The ratio tan(45°)/tan(30°) tells us that the distance ratio is:
d₂/d₁ = tan(45°)/tan(30°) = 1/(1/√3) = √3
Since d₁ + d₂ = 100(√3+1) and d₁ = √3 × d₂, we can solve for both and verify H = 100 m.
Alternative Question: Drone Descent
If the drone descends vertically by 50 m, what will be the new angle of depression of the car that was originally at 45°? (Assume the car has not moved).
✓ Step-by-Step Solution
Recall Initial Conditions
From Question 3, we found:
• Initial drone height: H = 100 m
• Distance to Car 2 (45° car): d₂ = 100 m
• Original angle of depression: 45°
The car remains stationary at 100 m from point O.
Calculate New Drone Height
Descent distance = 50 m
New height: H' = 100 - 50 = 50 m
Calculate New Angle of Depression
Using tan ฮธ' = H' / d₂
tan ฮธ' = 50 / 100
tan ฮธ' = 1/2 = 0.5
Taking inverse tangent:
ฮธ' = tan⁻¹(0.5) ≈ 26.57°
Compare Before and After
๐ก Before Descent
Height = 100 m
Distance = 100 m
Angle = 45°
๐ก After Descent (50 m)
Height = 50 m
Distance = 100 m (same)
Angle ≈ 26.57°
๐ฏ New Angle of Depression
ฮธ' ≈ 26.57°
or
tan⁻¹(1/2)
The angle decreased from 45° to ~26.57° after the drone descended
๐ Why Did the Angle Decrease?
When the drone descends, the height H decreases while the horizontal distance d remains the same.
Since tan ฮธ = H/d, and H decreased with d constant, tan ฮธ must also decrease.
Smaller angle = Drone is now relatively closer to the car (less steep view)
๐ฏ Exact vs Approximate Answer
Exact: ฮธ' = tan⁻¹(1/2) or arctan(0.5)
Approximate: ฮธ' ≈ 26.57° or ≈ 26°34'
In decimal degrees: 26.565051177...°
Complete Summary & Reference
๐ Key Formula Used
tan ฮธ = H / d
Angle of depression relates vertical height to horizontal distance
✓ Main Finding
H = 100 m
The drone hovers at exactly 100 meters above the road
| Question | Key Point | Answer |
|---|---|---|
| Q1: Angle Increase | Larger angle = Closer | Moves CLOSER |
| Q2: Trig Ratio | Relates H and d | tan ฮธ = H/d |
| Q3: Height H | Given total distance | H = 100 m |
| Q4: New Angle | After 50 m descent | ≈ 26.57° |
๐ Standard Trigonometric Values
30°
sin = 1/2
cos = √3/2
tan = 1/√3
45°
sin = 1/√2
cos = 1/√2
tan = 1
60°
sin = √3/2
cos = 1/2
tan = √3
๐ก Key Concepts Learned
- ✓ Angle of depression depends on observer position
- ✓ Larger angle → Closer object (inverse relationship)
- ✓ tan ratio connects height to horizontal distance
- ✓ System of equations solves multi-object problems
- ✓ Drone height can be found from distance constraints
๐ Real-World Applications
- ✓ Drone surveillance for security
- ✓ Height determination of aircraft
- ✓ Distance measurement in surveying
- ✓ Object tracking systems
- ✓ Navigation and positioning
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