Formulas & Concepts
Heights and Distances · Study material
Heights & Distances
Heights & Distances is one of the most important applications of Trigonometry in Quantitative Aptitude. This chapter helps us calculate the height of objects and their distances from a point when direct measurement is not possible.
Questions from this chapter are frequently asked in SSC, Banking, Railway, Insurance, Defence, CAT, NDA, CDS, and various competitive examinations.
This chapter mainly deals with:
- Line of Sight
- Horizontal Line
- Angle of Elevation
- Angle of Depression
- Trigonometric Ratios
- Height and Distance Calculations
Understanding trigonometric concepts and formulas helps candidates solve aptitude questions quickly and accurately.
What is Heights & Distances?
Heights and Distances is an application of Trigonometry used to determine the height or distance of an object using angles and trigonometric ratios.
If the angle of elevation or angle of depression from a point is known, then the required height or distance can be calculated easily.
Important Concept:
Most Heights & Distances problems are solved using:
tan θ = Height / Distance
Line of Sight
The line drawn from the eye of an observer to the object being viewed is called the Line of Sight.
Observer → Object
This imaginary line is called the line of sight.
Horizontal Line
The line parallel to the ground level passing through the observer’s eye is called the Horizontal Line.
Angle of Elevation
When the observer looks upward towards an object above the horizontal level, the angle formed between the line of sight and the horizontal line is called the Angle of Elevation.
Formula:
tan θ = Height / Distance
This formula is widely used in aptitude problems involving towers, buildings, trees, poles, and ladders.
Angle of Depression
When the observer looks downward towards an object below the horizontal level, the angle formed between the line of sight and the horizontal line is called the Angle of Depression.
Formula:
tan θ = Height / Distance
The same trigonometric formula is used for both elevation and depression problems.
Right-Angled Triangle
Most Heights & Distances problems are based on right-angled triangles.
Perpendicular = Height
Base = Distance
Hypotenuse = Line of Sight
Trigonometric Ratios
In a right-angled triangle:
1. Sine Ratio
sin θ = Perpendicular / Hypotenuse
2. Cosine Ratio
cos θ = Base / Hypotenuse
3. Tangent Ratio
tan θ = Perpendicular / Base
4. Cosecant Ratio
cosec θ = Hypotenuse / Perpendicular
5. Secant Ratio
sec θ = Hypotenuse / Base
6. Cotangent Ratio
cot θ = Base / Perpendicular
Important Trigonometric Identities
1. First Identity
sin²θ + cos²θ = 1
2. Second Identity
1 + tan²θ = sec²θ
3. Third Identity
1 + cot²θ = cosec²θ
Important Trigonometric Values
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
| cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | ∞ |
| cot θ | ∞ | √3 | 1 | 1/√3 | 0 |
Most Important Formula in Heights & Distances
tan θ = Height / Distance
This formula is used in almost every Heights & Distances problem.
Applications of Heights & Distances
- Finding height of buildings
- Distance of ships from lighthouse
- Height of towers and poles
- Aircraft and mountain problems
- River and bridge calculations
- Engineering and surveying applications
Important Concepts for Exams
1. Angle of Elevation is Above Horizontal
Observer looks upward.
2. Angle of Depression is Below Horizontal
Observer looks downward.
3. Same Formula Used
For both elevation and depression:
tan θ = Height / Distance
Quick Revision Formula Table
| Concept | Formula |
|---|---|
| sin θ | Perpendicular/Hypotenuse |
| cos θ | Base/Hypotenuse |
| tan θ | Perpendicular/Base |
| cosec θ | Hypotenuse/Perpendicular |
| sec θ | Hypotenuse/Base |
| cot θ | Base/Perpendicular |
Common Mistakes to Avoid
- Confusing angle of elevation and depression.
- Using wrong trigonometric ratio.
- Ignoring unit conversions.
- Calculation mistakes in square roots.
- Using incorrect triangle sides.
Important Exam Tips
- Memorize important trigonometric values.
- Practice tan θ based problems regularly.
- Draw diagrams for every problem.
- Identify height and distance clearly.
- Remember all trigonometric identities.
- Practice previous year aptitude questions.
- Improve simplification speed.
Heights & Distances is an important aptitude topic based on Trigonometry. Strong understanding of formulas, angles, and trigonometric ratios helps candidates solve competitive examination questions quickly and accurately.