Height and Distance is one of the most important and scoring topics in Trigonometry under Quantitative Aptitude. Most questions are based on right-angled triangles formed by objects observed from the ground level. If you clearly understand three ideas:

✔ Right-angled triangle
✔ Trigonometric ratios
✔ Angle of elevation and depression

Then you can solve any Height & Distance question easily.

This guide covers all formulas, concepts, diagrams, identities, T-ratio values, and exam-ready understanding in a simple and complete manner.

Quick Overview: Height & Distance – All Formulas

1. Trigonometric Ratios (Based on Right-Angled Triangle)

In a right-angled triangle OAB, where ∠BOA = θ:

        A (Perpendicular)
        |
        |
        | AB
        |
        |_______ B (Hypotenuse = OB)
       O
      (Base = OA)

Using this diagram:

i. sinθ = Perpendicular / Hypotenuse = AB / OB

ii. cosθ = Base / Hypotenuse = OA / OB

iii. tanθ = Perpendicular / Base = AB / OA

iv. cosecθ = 1/sinθ = OB / AB

v. secθ = 1/cosθ = OB / OA

vi. cotθ = 1/tanθ = OA / AB

These ratios are the foundation for all height and distance questions because every problem forms a right-angled triangle.

2. Trigonometric Identities (Must-Know)

These identities apply to all angles used in aptitude exams:

1. sin²θ + cos²θ = 1
2. 1 + tan²θ = sec²θ
3. 1 + cot²θ = cosec²θ

These relations help simplify expressions and solve missing ratios quickly.

3. Standard Values of Trigonometric Ratios

These values must be memorized because Height & Distance problems depend on them directly.

4. Angle of Elevation (Definition + Diagram)

When an observer looks upward at an object placed above eye level, the angle formed by the line of sight with the horizontal is called the angle of elevation.

Definition

Angle of elevation = ∠AOP, where object P is above horizontal O.

          P (Object)
           *
          /|
         / |
        /  | Height
       /   |
      O----*-------------- Horizontal line
     (Observer)

Explanation

• The observer is at point O
• The object is at point P (above O)
• The line of sight OP makes an angle with the horizontal
• This angle = angle of elevation

Keywords in questions:

✔ “Looked up at”
✔ “Seen from a point below”
✔ “Angle made by the line of sight with horizontal while looking upward”

5. Angle of Depression (Definition + Diagram)

When an observer looks downward at an object placed below eye level, the angle made by the line of sight with the horizontal is called the angle of depression.

Definition

Angle of depression = ∠AOP, where object P is below horizontal O.

         O----*-------------- Horizontal line
        (Observer)
         \
          \
           \   Line of sight
            \
             * P (Object below)

Explanation

• Observer stands at a higher point
• Object lies below
• Angle formed with horizontal while looking down → angle of depression

Important: Angle of elevation = angle of depression because they form alternate interior angles.

6. How Height & Distance Problems Work (Core Concept)

Every question reduces to one right-angled triangle:

✔ Height = Perpendicular
✔ Horizontal distance = Base
✔ Line of sight = Hypotenuse
✔ Angle given = θ
✔ Use sin, cos, tan accordingly

Most questions use tanθ because:

tanθ = Perpendicular / Base

So height = distance × tanθ
or
distance = height / tanθ

Smart Tips and Practical Tricks for Solving Height and Distance Problems

Mastering Height and Distance becomes easy once you understand how right-angled triangles, trigonometric ratios, and angles work together. Most students don’t struggle because formulas are hard, they struggle because they apply them without visualizing the situation. This section simplifies every concept into clear, practical tips so you can solve questions faster, draw diagrams correctly, select the right trigonometric ratio, and avoid common mistakes in competitive exams.

1. Visualize the Right-Angled Triangle First

Every height and distance problem forms a right-angled triangle.
If you draw the triangle correctly, most of the question is already solved.

How to visualize:

• Height = perpendicular (vertical)
• Ground distance = base (horizontal)
• Line of sight = hypotenuse

Example:
A tree 15 m high is seen at 30°.
Draw perpendicular (tree), base (distance), hypotenuse (line of sight).
Then apply:
tan30° = height / distance.

2. Identify Whether the Angle is Elevation or Depression

Understanding the type of angle changes your diagram.

• Elevation → looking UP
• Depression → looking DOWN

But both angles are always measured from the horizontal.

Example:
From a lighthouse top, depression to ship = 40°.
Elevation from ship to lighthouse = 40°.

3. Use tanθ When Height and Base Are Involved

Most height–distance problems become easy by using:

tanθ = Perpendicular / Base

This is the most frequently used ratio in exams.

Example:
Height = distance × tanθ
Distance = height ÷ tanθ

4. Use sinθ or cosθ Only When Line of Sight Is Given

If hypotenuse is mentioned, never apply tanθ.

• Use sinθ = height / hypotenuse
• Use cosθ = base / hypotenuse

Example:
A 50 m kite string makes 30°.
Height = 50 × sin30 = 25 m.

5. Draw a Clear Diagram Before Calculating

A simple diagram avoids 90% of mistakes.

Include:

• Height
• Horizontal distance
• Angle at observer
• Line of sight

Why it helps:
You instantly recognize which ratio (sin, cos, tan) is needed.

6. Use Two-Observation Formulas When Angle Changes from Two Points

When angle increases as you move closer, height formula becomes:

h = (d × tanθ₁ × tanθ₂) / (tanθ₂ − tanθ₁)

This is common in advanced SSC, Banking, and CDS questions.

Example:
Tower seen at 30° from A and 60° from point 20 m closer.
Use above formula to get height = 20√3.

7. Memorize Trigonometric Values Thoroughly

Most calculations rely on:

• sin, cos, tan of 30°, 45°, 60°
• tan60° = √3
• tan30° = 1/√3

Without these, you cannot solve quickly.

8. Check Units Carefully

Heights in meters, distances in meters = fine.
Angles are always in degrees for these questions.

Never convert degrees to radians in aptitude questions.

9. Identify the Perpendicular and Base Correctly

Students mix up “height” and “distance.”
Use this rule:

• Vertical = height = perpendicular
• Ground = distance = base

If angle touches ground → base is adjacent.
If angle touches height → perpendicular is opposite.

10. Practice Common Exam Patterns

Typical repeated question types:

• Height from one angle
• Height from two angles
• Shadow length
• Angle of depression from tower
• Balloon rising vertically
• Person moving toward/away from object

Mastering these patterns guarantees full marks in Height and Distance.

FAQs About Height & Distance

Q1. Why are trigonometric ratios important in height and distance questions?

Because every height–distance question forms a right-angled triangle, and trigonometric ratios directly relate angles with heights and distances.

Q2. Why is tanθ used most frequently?

Because height = perpendicular and ground distance = base, and tanθ = P/B.

Q3. Why is angle of elevation used in finding heights?

When the observer looks upward, the height forms the opposite side, making it easy to calculate using trigonometric ratios.

Q4. Why is tan90° not defined?

Because tanθ = sinθ/cosθ, and cos90° = 0, making division undefined.

Q5. What is the difference between angle of elevation and angle of depression?

Elevation = looking upward
Depression = looking downward

But both angles are measured from the horizontal.

Q6. Why are most questions based on 30°, 45°, and 60°?

Because these angles have simple, standard trigonometric values that make calculations fast in exams.

Q7. Why do angles of elevation and depression form alternate interior angles?

Because the line of sight acts as a transversal between two horizontal lines.

Q8. How to identify perpendicular, base, and hypotenuse in a question?

Opposite to θ = perpendicular
Adjacent to θ = base
Longest side connecting them = hypotenuse

Q9. Why is cosθ rarely used in height problems?

Because cosθ relates base and hypotenuse, while height questions generally involve perpendicular values.

Q10. How can I avoid mistakes in height and distance?

Draw a neat diagram, mark the angle, identify P/B/H, and apply the correct trigonometric ratio.