MathematicsSome applications of trigonometryClass 10

Heights and distances (Trigonometry applications): interactive simulation

Trigonometry enables engineers and surveyors to calculate unreachable heights and distances without direct measurement.

Useful for: CBSE Boards, ICSE Class 10

Try this

The Angle of Elevation is the angle formed by the line of sight with the horizontal when the point observed is above the horizontal level. The Angle of Depression is formed when the object observed is below horizontal level.

  1. 1

    In Scenario 1, set θ = 45°.

    What you should see

    At θ = 45°, tan(45°) = 1, so the tower height exactly equals the observer distance d!

  2. 2

    Select Scenario 2 (Lighthouse & Two Ships) and observe the depression angles.

    What you should see

    The ship closer to the lighthouse has a larger angle of depression than the farther ship.

Formulas behind it

Tangent Ratio

tan⁡θ=Opposite (Height)Adjacent (Distance)=hd\tan\theta = \frac{\text{Opposite (Height)}}{\text{Adjacent (Distance)}} = \frac{h}{d}

Tower Height

h=dtan⁡θ+hobserverh = d \tan\theta + h_{\text{observer}}

Two Ships Distance

Δx=H(cot⁡θ2−cot⁡θ1)\Delta x = H(\cot\theta_2 - \cot\theta_1)

Kite String Length

L=hsin⁡θL = \frac{h}{\sin\theta}

Common mistakes

  • Confusing Angle of Depression with angle with the vertical. Angle of depression is ALWAYS measured from the horizontal line of sight downward.
  • Forgetting to add the observer’s eye height (e.g. 1.5 m) to the final tower height when specified in board exam questions.
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FAQ

What is a Theodolite?

A surveying instrument with a rotating telescope used to measure horizontal and vertical angles with high precision.

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