Heights and Distances
What is Heights and Distances?
The angle formed between the horizontal line from the observer's eye and the line of sight to an object above the horizontal.
Key formula / rule: Tangent Ratio
Key points
- To be able to identify and draw diagrams for heights and distances problems.
- To apply trigonometric ratios correctly to find unknown heights and distances.
- To understand and utilize the concepts of angle of elevation and angle of depression.
- To solve real-world problems involving heights and distances.
Common exam trap
Confusing angle of elevation with angle of depression.
Definitions
- Term
Angle of Elevation
- Meaning
The angle formed between the horizontal line from the observer's eye and the line of sight to an object above the horizontal.
- Term
Angle of Depression
- Meaning
The angle formed between the horizontal line from the observer's eye and the line of sight to an object below the horizontal.
- Term
Hypotenuse
- Meaning
The side opposite the right angle in a right-angled triangle; the longest side.
- Term
Adjacent Side
- Meaning
The side next to the angle in question, not the hypotenuse.
- Term
Opposite Side
- Meaning
The side across from the angle in question.
Learning objectives
To be able to identify and draw diagrams for heights and distances problems.
To apply trigonometric ratios correctly to find unknown heights and distances.
To understand and utilize the concepts of angle of elevation and angle of depression.
To solve real-world problems involving heights and distances.
Formulae
- Name
Tangent Ratio
- Note
Used when the height (opposite) and distance (adjacent) are involved.
- Expression
tan(θ) = Opposite / Adjacent
- Name
Sine Ratio
- Note
Used when the height (opposite) and the line of sight distance (hypotenuse) are involved.
- Expression
sin(θ) = Opposite / Hypotenuse
- Name
Cosine Ratio
- Note
Used when the horizontal distance (adjacent) and the line of sight distance (hypotenuse) are involved.
- Expression
cos(θ) = Adjacent / Hypotenuse
- Name
Height from Angle of Elevation
- Note
Assumes observer is at ground level.
- Expression
Height = Distance × tan(Angle of Elevation)
- Name
Distance from Angle of Elevation
- Note
Assumes observer is at ground level.
- Expression
Distance = Height / tan(Angle of Elevation)
- Name
Height from Angle of Depression
- Note
From the top of an object, looking down.
- Expression
Height = Distance × tan(Angle of Depression)
- Name
Distance from Angle of Depression
- Note
From the top of an object, looking down.
- Expression
Distance = Height / tan(Angle of Depression)
Prerequisites
Basic understanding of geometry (lines, angles, triangles).
Knowledge of trigonometric ratios (sine, cosine, tangent) and their values for standard angles (30°, 45°, 60°).
Ability to form and interpret right-angled triangles.
Common mistakes
Confusing angle of elevation with angle of depression.
Incorrectly applying trigonometric ratios (e.g., using sin instead of tan when opposite and adjacent sides are involved).
Calculation errors with trigonometric values or arithmetic.
Not drawing a correct diagram or misinterpreting the given information.
Assuming the observer's height is negligible when it might be significant.
Keywords
Trigonometry
Heights
Distances
Angle of Elevation
Angle of Depression
Right-angled Triangle
Sine
Cosine
Tangent
SSC CGL
Quantitative Aptitude
Practice preview
From the top of a lighthouse 75 m high, the angle of depression of a ship is 45 degrees. The distance of the ship from the lighthouse is:…
easy
A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall.…
easy
The angle of elevation of the top of a tower from a point on the ground, which is 30 meters away from the foot of the tower, is 30 degrees. Find the height of the tower.…
easy