Height & Distance

“Height and Distance” in trigonometry involves calculating the height of objects or distances between points using angles of elevation and depression.

Key Concepts

  • Line of Sight: The imaginary line between an observer's eye and the object.
  • Angle of Elevation: Angle between the horizontal and the line of sight when looking up.
  • Angle of Depression: Angle between the horizontal and the line of sight when looking down.
  • Trigonometric Ratios: sinθ = Perpendicular/Hypotenuse, cosθ = Base/Hypotenuse, tanθ = Perpendicular/Base.

Useful Formulas

  • tanθ = Height / Distance
  • Height = Distance × tanθ
  • Distance = Height / tanθ

Previous Year Questions with Explanations

  • SSC CGL 2020: A man observes the top of a tower at an angle of elevation of 30°. If the distance between the man and the tower is 50 m, find the height of the tower.
  • Height = 50 × tan(30°) = 50 × 1/√3 ≈ 28.87 m
  • RRB JE 2019: A man sees a balloon at an angle of elevation of 60°. The balloon is 173.2 m above the ground. What is the distance of the man from the balloon's foot?
  • Distance = 173.2 / tan(60°) = 173.2 / √3 ≈ 100 m
  • SSC CHSL 2018: From the top of a building 60 m high, the angle of depression to a car is 45°. Find the car's distance from the building.
  • Distance = 60 / tan(45°) = 60 m
  • HSSC 2021: A ladder makes a 60° angle with the ground. The base is 5 m from the wall. Find the wall's height.
  • Height = 5 × tan(60°) = 5√3 ≈ 8.66 m
  • SSC GD 2023: The angle of elevation to a 100 m high tower is 45°. What is the distance from the tower?
  • Distance = 100 / tan(45°) = 100 m
  • SSC CPO 2017: A boy flying a kite observes an angle of elevation of 60°, string length is 100 m. Find the height of the kite.
  • Height = 100 × sin(60°) = 100 × √3/2 ≈ 86.6 m
  • RRB ALP 2018: A tower's shadow is 25 m when the angle of elevation of the sun is 45°. Find the height of the tower.
  • Height = 25 × tan(45°) = 25 m
  • SSC MTS 2021: If angle of elevation is 60° and the shadow is 5√3 m, what’s the height of the tower?
  • Height = 5√3 × tan(60°) = 5√3 × √3 = 15 m
Note: Always sketch a triangle to understand the problem and then apply the appropriate trigonometric ratio.

Conclusion

Height and Distance problems are solved using trigonometry. Mastery of angle concepts and trigonometric ratios is key to success in competitive exams.

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