Angle of elevation, angle of depression, aur real-life trigonometry problems.
🌟 Real Life Applications
🔭 Engineer, Surveyor, Pilot — sab use karte hain!
• Lighthouse operator samundar mein ships track karta hai — angle of depression
• Mountaineer pehle se height measure karta hai — angle of elevation
• Architect buildings ki height plan karta hai — trig formulas
• GPS satellites ka angle se position calculate hota hai
Yeh sab Heights & Distances chapter ka use hai!
📐 Important Terms
Angle of Elevation (Unnayan Kona):
Jab hum neeche se kisi upar ki cheez ko dekhte hain — horizontal se upar ki taraf — woh angle = angle of elevation.
tan θ = height/distance = h/d
Angle of Depression (Avnaman Kona):
Jab hum upar se kisi neeche ki cheez ko dekhte hain — horizontal se neeche ki taraf — woh angle = angle of depression. Important: Angle of depression = Angle of elevation at object (alternate angles, parallel lines)
Key Formulas
tan θ = Opposite / Adjacent = h/d
h = d × tan θ (height nikalo jab distance pata ho)
d = h / tan θ (distance nikalo jab height pata ho)
sin θ = h/slant → slant = h/sin θ
Condition
Formula Use
Height + horizontal distance pata
tan θ = h/d
Slant distance + angle pata
sin θ = h/L, cos θ = d/L
Two angles given (same base)
Do equations, eliminate h ya d
🛠️ 4-Step Problem Solving Strategy
Problem Solving Strategy
Diagram banao — careful aur labeled, right triangles identify karo
Variables define karo — height = h, distance = d, etc.
Trig equation likhein — tanθ ya sinθ ya cosθ
Solve aur answer verify karo — units check karo
📝 Examples — Step by Step
Example 1 (Basic): Tower 30m door. Angle of elevation = 30°. Height?
Diagram
Right △, base=30m, angle=30°, h=?
Equation
tan30° = h/30 → 1/√3 = h/30
Solve
h = 30/√3 = 30√3/3 = 10√3 m ≈ 17.32 m
Example 2: Kite ke saath 100m lambi string hai, 60° angle se. Kite ki height?
sin60° = h/100 → √3/2 = h/100 → h = 50√3 m ≈ 86.6 m
Example 3 (Two angles): Ek 75m unchai wale tower ke upar se ek lighthouse ka angle of elevation 30° hai, neeche se 60°. Lighthouse ki height nikalo.
Neeche se (ground): tan60° = H/d → d = H/√3 ...(1)
Tower ke top se (75m height): tan30° = (H−75)/d
1/√3 = (H−75)/d → d = √3(H−75) ...(2)
Equate: H/√3 = √3(H−75)
H = 3(H−75) = 3H−225
225 = 2H → H = 112.5 m
d = 112.5/√3 = 37.5√3 m
Example 4: River ki width nikalo — ek tree 60° angle se ek side, 30° doosri side se.
Tree ki height = h. Ek side: tan60° = h/d₁ → d₁ = h/√3
Doosri side: tan30° = h/d₂ → d₂ = h√3
Total width = d₁+d₂ = h/√3 + h√3 = h(1/√3 + √3) = h(1+3)/√3 = 4h/√3
Agar tree = 30m: Width = 120/√3 = 40√3 m ≈ 69.3 m
Example 5: Ship ki angle of depression 45° hai lighthouse se (height 120m). Ship ki distance nikalo.
Angle of depression = 45°. Alternate angle → angle of elevation from ship = 45°
tan45° = 120/d → 1 = 120/d → d = 120 m
Example 6 (Hard): 80m lambi rod vertically khadi hai. Uski shadow ki length nikalo jab sun ka elevation 30° ho.
Q1. Ek tower ki height 10m hai. Uski chhaya ki length sunset par 10√3 m hai. Suraj ka elevation angle kya hai?
tanθ = 10/(10√3) = 1/√3. θ = 30°
Q2. Building ke upar se ek flag pole ki top aur bottom ke angles of elevation 60° aur 45° hain. Building 15m high hai. Flag pole ki height nikalo.
Let d = horizontal distance, h = flag pole height, building = 15m.
tan45° = 15/d → d = 15m. tan60° = (15+h)/15 → √3 = 1+h/15 → h = 15(√3−1) ≈ 15(0.732) ≈ 10.98 m
Q3. Ek hill ke top par khade aadmi ko 200m neeche ek boat 45° angle of depression se dikhti hai. Boat ki hill se horizontal distance?
tan45° = 200/d → 1 = 200/d → d = 200 m
Q4. Do towers 100m apart hain. Pehli 20m high, doosri 30m high. Ek dusre ke tops se angle of elevation kya hoga?