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Some Applications of Trigonometry

NCERT Class 10 · Mathematics Based on NCERT Class 10 Mathematics textbook · Free CBSE study kit

Chapter Notes

**APPLICATIONS OF TRIGONOMETRY: HEIGHTS AND DISTANCES**

**KEY DEFINITIONS**

• Line of Sight: The straight line drawn from the eye of an observer to the point being viewed on an object

• Angle of Elevation: The angle formed by the line of sight with the horizontal when the point being viewed is ABOVE the horizontal level → You raise your head to look up at the object

• Angle of Depression: The angle formed by the line of sight with the horizontal when the point being viewed is BELOW the horizontal level → You lower your head to look down at the object

• Important Relationship: The angle of elevation from point A to point B equals the angle of depression from point B to point A (alternate angles)

**GENERAL APPROACH TO SOLVING HEIGHT AND DISTANCE PROBLEMS**

  • Step 1: Draw a clear, labeled diagram showing:
  • The observer's position
  • The object being viewed
  • The horizontal reference line
  • The angle (elevation or depression)
  • Known distances
  • Step 2: Identify the right-angled triangle(s) formed
  • Step 3: Determine which trigonometric ratio to use:
  • If you know opposite and need adjacent (or vice versa) → Use tan or cot
  • If you know hypotenuse and need opposite (or vice versa) → Use sin or cosec
  • If you know hypotenuse and need adjacent (or vice versa) → Use cos or sec
  • Step 4: Set up the trigonometric equation and solve for the unknown
  • Step 5: If observer's height is given, add it to find total height of object

    • **COMMON TRIGONOMETRIC RATIOS USED**

    • tan θ = Opposite/Adjacent

    • cot θ = Adjacent/Opposite

    • sin θ = Opposite/Hypotenuse

    • cos θ = Adjacent/Hypotenuse

    Special Angles:

  • tan 30° = 1/√3, tan 45° = 1, tan 60° = √3
  • cot 30° = √3, cot 45° = 1, cot 60° = 1/√3
  • sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2
  • cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2

    • **INFORMATION NEEDED TO FIND HEIGHT OF AN OBJECT**

    You must know:

    1. The horizontal distance from the observer to the base of the object

    2. The angle of elevation from the observer's eye level

    3. The height of the observer's eye from the ground

    Formula: Height of object = Height found from triangle + Observer's height

    **TYPES OF PROBLEMS AND SOLUTION STRATEGIES**

    **Type 1: Single Object, Single Observer Position**

    Example: Tower height when distance and angle of elevation are known

    • Draw right triangle with tower as perpendicular

    • Use tan(angle) = height/distance

    • Solve for height

    • Add observer's height if needed

    **Type 2: Single Object, Multiple Observation Points**

    Example: Two different angles from different distances

    • Create two separate right triangles

    • Set up two equations with the same unknown (height)

    • Solve simultaneously to find both height and distance

    **Type 3: Multiple Objects (Shadow Problems)**

    Example: Tower's height using shadow lengths at different sun altitudes

    • Create two triangles with same height but different base angles

    • Use tan for both angles

    • Set height equal in both equations

    • Solve by substitution to find height

    **Type 4: Flagstaff Problems**

    Example: Two angles of elevation to top of building and top of flagstaff

    • Building height is known

    • Create two triangles: one to building top, one to flagstaff top

    • Use tan 30° with building height to find distance

    • Use tan 45° with distance to find total height

    • Flagstaff length = Total height - Building height

    **Type 5: Angle of Depression Problems**

    Example: Angles of depression from top of building to another building

    • Angle of depression from A to B = angle of elevation from B to A

    • Draw horizontal line from observer

    • Mark angle below horizontal

    • Create right triangle with height difference as opposite side

    • Use appropriate trigonometric ratio

    • Remember: angle of depression is measured downward from horizontal

    **CRITICAL STEPS IN SOLVING**

  • Always identify which side is opposite, adjacent, and hypotenuse with respect to the given angle
  • If observer's height is given (like 1.5 m for person), add it to the height calculated from the triangle
  • For shadow problems, the angle is at the tip of shadow (on ground), with tower as opposite side
  • For depression problems, drop a perpendicular from observer to the horizontal line passing through the object
  • In multi-object problems, label unknowns carefully and create separate equations

    • **COMMON MISTAKES TO AVOID**

    • Mistake 1: Forgetting to add observer's height when calculating total height of object

  • Always check: Is height of observer given separately?
  • Total Height = Height from triangle + Observer's height

    • • Mistake 2: Confusing angle of elevation with angle of depression

  • Elevation = looking UP (angle above horizontal)
  • Depression = looking DOWN (angle below horizontal)

    • • Mistake 3: Using wrong trigonometric ratio

  • Check: What do I know? What do I need?
  • Opposite-Adjacent relationship → tan/cot
  • Hypotenuse involved → sin/cos

    • • Mistake 4: In shadow problems, placing the angle at wrong location

  • Angle is ALWAYS at the tip of shadow on ground
  • It's the angle of elevation from shadow tip to object top

    • • Mistake 5: Not simplifying or rationalizing answers

  • √3 ≈ 1.732 (usually given)
  • 1/√3 = √3/3
  • Always rationalize denominators

    • • Mistake 6: Forgetting right angle placement

  • In height problems, right angle is ALWAYS where vertical meets horizontal
  • Building is perpendicular to ground (vertical)
  • Shadow falls on ground (horizontal)

    • **PROBLEM-SOLVING CHECKLIST**

    ✓ Draw diagram with all given information

    ✓ Mark right angles clearly

    ✓ Label all known values

    ✓ Identify what needs to be found

    ✓ Choose correct trigonometric ratio

    ✓ Write equation correctly

    ✓ Solve step by step

    ✓ Check if observer's height needs to be added

    ✓ Verify answer makes physical sense

    ✓ Use given approximations (√3 = 1.732, etc.)

    ✓ Round appropriately for final answer

    MCQs — 10 Questions with Answers

    Q1. A surveyor needs to find the height of a building but can only measure the horizontal distance and observe angles. She stands at ground level 50 m away from the building and measures the angle of elevation to the top. Which trigonometric ratio should she use to find the height directly?

    • A. tan(angle of elevation) = height / horizontal distance ✓
    • B. sin(angle of elevation) = horizontal distance / height
    • C. cos(angle of elevation) = height / slant distance
    • D. cot(angle of elevation) = height / horizontal distance

    Answer: A — tan(θ) = opposite/adjacent, where opposite is the height and adjacent is the horizontal distance; option B inverts the ratio and misidentifies the trigonometric function.

    Q2. A person observes a bird flying at a height above them. The angle of elevation from their eye level to the bird is 35°. Later, the bird descends to a position below the level of the observer. Which angle should the observer now measure, and why?

    • A. Angle of elevation, because the bird is still visible from the same position
    • B. Angle of depression, because the bird is now below the horizontal level through the observer's eye ✓
    • C. The same angle of 35°, because the bird's actual distance has not changed
    • D. Angle of elevation measured downward, which is a special case of depression

    Answer: B — Angle of depression is defined as the angle below the horizontal when looking down at an object; option A confuses position with the type of angle, and option C ignores the definition based on relative height.

    Q3. Assertion (A): If the angle of elevation of the top of a tower from a point on the ground doubles, the height of the tower also doubles. Reason (R): The angle of elevation and height of an object are directly proportional because both are measured from the same reference point. Choose the correct option:

    • A. Both A and R are true and R is the correct explanation of A
    • B. Both A and R are true but R is not the correct explanation of A
    • C. A is true but R is false
    • D. A is false but R is true ✓

    Answer: D — A is false: doubling the angle (e.g., from 30° to 60°) does not double the height due to the nonlinear nature of trigonometric functions; R is true but not a valid explanation since angle and height are not directly proportional.

    Q4. A ladder is placed against a vertical wall. An observer measures the angle of elevation from the foot of the ladder to the top of the ladder against the sky. Which statement about this angle is correct?

    • A. It is the angle between the ladder and the ground, which equals the angle of elevation ✓
    • B. It is not an angle of elevation because the top of the ladder is not higher than the observer's eye level
    • C. It is the angle between the ladder and the vertical wall, not an angle of elevation
    • D. It is the angle of elevation only if the observer stands at the foot of the ladder

    Answer: A — The angle between the ladder and the ground is measured from the horizontal ground to the ladder, which is the definition of angle of elevation; option C confuses it with the angle between the ladder and wall.

    Q5. Assertion (A): In a right triangle formed by angle of elevation problems, the horizontal distance is always the adjacent side to the angle of elevation. Reason (R): The angle of elevation is always measured at the observer's position on the ground against the horizontal line. Choose the correct option:

    • A. Both A and R are true and R is the correct explanation of A ✓
    • B. Both A and R are true but R is not the correct explanation of A
    • C. A is true but R is false
    • D. A is false but R is true

    Answer: A — Both statements are true and R correctly explains why the horizontal distance must be adjacent to the angle at the observer's position; the horizontal reference is fundamental to defining which side is adjacent.

    Q6. A student claims: 'If I know the angle of elevation and the height of an object, I can always find the horizontal distance using trigonometry.' Under what condition is this claim false?

    • A. When the angle of elevation is 45°, because tan(45°) = 1 is a special case
    • B. When the height given is the vertical distance from the observer's eye level, not from the ground
    • C. The claim is always true; knowing angle and height is sufficient to find horizontal distance using tan(θ) = height/distance ✓
    • D. When the observer is not standing on level ground at the base of the object

    Answer: C — The claim is always true mathematically; option D is a practical limitation but not a mathematical one, and option A incorrectly treats 45° as a special case for the method itself.

    Q7. Assertion (A): The angle of elevation from point P to the top of a tower and the angle of depression from the top of the tower to point P are always equal. Reason (R): These angles are alternate interior angles formed by the line of sight and a horizontal line. Choose the correct option:

    • A. Both A and R are true and R is the correct explanation of A ✓
    • B. Both A and R are true but R is not the correct explanation of A
    • C. A is true but R is false
    • D. A is false but R is true

    Answer: A — Both are true: the angles are equal as alternate interior angles formed when a transversal (line of sight) cuts two parallel horizontal lines (one at P, one at the tower top); R correctly explains the geometric reason.

    Q8. A surveyor standing at point A on a flat field observes two vertical poles. The angle of elevation to the top of pole B is 30°, and the horizontal distance is 60 m. For pole C at the same location but with angle of elevation 60° from A, the surveyor concludes pole C is twice the height of pole B. Is this reasoning correct?

    • A. Yes, because 60° is double 30°, so the height must double
    • B. No, because tan(60°) ÷ tan(30°) = √3, not 2, so pole C is √3 times taller ✓
    • C. Yes, the relationship between angle and height is always proportional
    • D. No, the heights cannot be compared without knowing the horizontal distances to each pole

    Answer: B — Using tan(θ) = height/distance with the same distance: height ratio = tan(60°)/tan(30°) = √3/（1/√3) = 3, not 2; option A incorrectly assumes linear proportionality between angle and height.

    Q9. Assertion (A): When finding the height of a building using angle of elevation, the observer's height must be added to the calculated height from the horizontal line through the observer's eyes. Reason (R): The angle of elevation is measured from the observer's eye level, not from the ground level. Choose the correct option:

    • A. Both A and R are true and R is the correct explanation of A ✓
    • B. Both A and R are true but R is not the correct explanation of A
    • C. A is true but R is false
    • D. A is false but R is true

    Answer: A — Both are true and R directly explains why A is correct: since the angle is measured from eye level, the perpendicular distance found represents height above eye level, requiring addition of observer's height to get total building height.

    Q10. Two observers at different horizontal distances from a tower measure the same angle of elevation to its top. Which conclusion is correct?

    • A. They must be at the same point because the same angle means the same height and distance
    • B. The observer closer to the tower must be standing at a higher elevation than the other
    • C. This is impossible; different distances cannot yield the same angle of elevation
    • D. They cannot be at the same height, because different distances with the same angle require different eye levels ✓

    Answer: D — For the same tower height and angle, tan(θ) = h/d means if distances differ, the heights above eye level must differ proportionally, so observers must be at different elevations; option B is correct in conclusion but option D is the most precisely reasoned answer.

    Flashcards

    What is the angle of elevation?

    The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.

    What is the angle of depression?

    The angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level.

    Formula: tan(angle) in height-distance problems

    tan(angle) = height/horizontal distance (opposite/adjacent in right triangle).

    When observer height is given, how do we find total height of object?

    Total height = calculated height from eye level + observer's eye height.

    In a shadow problem with two sun angles, what do we set up?

    Two equations using tan(30°) and tan(60°) with two different shadow lengths differing by given amount.

    How to find ladder length when angle of inclination is given?

    Use sin(angle) = height to reach / ladder length, then solve for ladder length (hypotenuse).

    In depression angle problems between two buildings, what triangles form?

    Two right triangles sharing a common vertical height, one for top point and one for bottom point of the second building.

    Why do we use tan ratio in most height-distance problems?

    Because tan relates the two quantities we typically know or need: height (opposite) and horizontal distance (adjacent).

    What does 'line of sight' mean in trigonometry applications?

    The straight line drawn from the observer's eye to the point on the object being viewed.

    In flagstaff problems, why are two separate right triangles used?

    One triangle uses the building height to find distance; second triangle uses building plus flagstaff height to find flagstaff length.

    Important Board Questions

    Define angle of elevation and angle of depression with one example each. [2 marks]

    State that angle of elevation is measured upward from horizontal when object is above eye level; angle of depression is measured downward when object is below eye level. Give one real-life example for each (e.g., looking up at plane for elevation, looking down from balcony for depression).

    A ladder 10 m long leans against a wall making an angle of 30° with the ground. Find: (i) height reached on the wall (ii) distance of the ladder's foot from the wall. [3 marks]

    Use sin 30° = height/10 to find height reached; use cos 30° = distance/10 to find distance from wall. Substitute sin 30° = 1/2 and cos 30° = √3/2 to get numerical answers.

    From a point P on ground, the angle of elevation to the top of a 15 m tall building is 30°. A flag is hoisted on the building such that the angle of elevation to the flag from P becomes 45°. Find: (i) distance of building from P (ii) length of the flag. (Take √3 = 1.732) [5 marks]

    First use tan 30° = 15/PA to find PA = 15√3 m. Then use tan 45° = (15 + flag)/PA to get flag = 15(√3 - 1) m. Show all algebraic steps and numerical substitution clearly.

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