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📐 Height & Distance (ऊँचाई और दूरी)

📘 Angle of Elevation

When an object is above eye level, the angle formed is called Angle of Elevation.

Angle between horizontal line and line of sight (upwards)
Example: Watching top of a tower.

📘 Angle of Depression

When an object is below eye level, the angle formed is called Angle of Depression.

Angle between horizontal line and line of sight (downwards)
Example: Looking down from building.

📘 Angle–Side Ratio

Important Trigonometric Ratios:

sinθ = Perpendicular / Hypotenuse
cosθ = Base / Hypotenuse
tanθ = Perpendicular / Base
tan30° = 1/√3
tan45° = 1
tan60° = √3

📘 Standard Angles

Common Values:

30° → 1 : √3 : 2
45° → 1 : 1 : √2
60° → √3 : 1 : 2

📘 Change of Angle (30° → 45°)

When elevation changes from 30° to 45°:

Distance changes by ( √3 − 1 ) × Height
Used in tower problems.

📘 Change of Angle (30° → 60°)

When elevation changes from 30° to 60°:

Distance = Height × 3

📘 Change of Angle (45° → 60°)

When elevation changes from 45° to 60°:

Distance = Height × (√3 − 1)

📘 Change of Angle (15° → 30°)

When elevation changes from 15° to 30°:

Distance = Height × (2 − √3)

✍️ Practice 1

A tower subtends angle 30°. Height = 10m. Find distance.

tan30° = 1/√3
Distance = 10√3 m

✍️ Practice 2

Angle of elevation = 45°. Height = 20m. Find distance.

tan45° = 1
Distance = 20m
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📐 Height & Distance – Advanced Flow Chart

📘 Change of Angle (General)

When elevation angle changes:

asinθ, acosθ, hcotθ
Distance = h(cotθ₁ − cotθ₂)

📘 From θ° to 2θ°

If angle changes from θ° to 2θ°:

h² = y² − x²
Used in double-angle problems

📘 Complementary Angles

If θ₁ + θ₂ = 90°:

h = √(xy)
Very important shortcut

📘 Distance Formula

If two angles observed:

a = h(cotθ₁ + cotθ₂)

📘 Cross Height Formula

Two towers problem:

x = √(h₁h₂)

📘 Reciprocal Height

Intersecting lines case:

1/a = 1/h₁ + 1/h₂

📘 Special Case

If AB = CD:

x = y tan((α+β)/2)

📘 General Formula

Using cot values:

x = h(cotα + cotβ)/(cotα − cotβ)

✍️ Practice 1

Height = 20m
Angles = 30°, 45°

Apply: h(cot30 − cot45)

✍️ Practice 2

Find distance between towers.

Use: x = √(h₁h₂)
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Height & Distance Special Notes