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CDS Mathematics · Heights and Distances

One Line of Sight

A height, a ground distance and a line of sight make one right triangle; the angle of elevation links the two legs by its tangent.

All Heights and Distances notesCDS Mathematics strategy

Why this matters

Six PYQs, one of them HARD. Draw the right triangle, mark the angle at the observer, and pick the ratio that joins what you know to what you want: tangent for height and ground distance, sine when the slant length is given (a ladder, a broken tree, a distance to a plane).

Concept 1 of 1: Height, distance and the angle

Standing at a distance dd and looking up at angle θ\theta to a top hh high, the height and the distance are the two legs of a right triangle: h=dtan⁡θh = d\tan\theta.

Definition

  • h=dtan⁡θh = d\tan\theta; d=hcot⁡θd = h\cot\theta.
  • The slant line of sight =hsin⁡θ= \dfrac{h}{\sin\theta}; use sine when a slant length is given.
  • Standard values: tan⁡30∘=13\tan 30^\circ = \dfrac1{\sqrt3}, tan⁡45∘=1\tan 45^\circ = 1, tan⁡60∘=3\tan 60^\circ = \sqrt3.
  • Shadow equal to height: the Sun is at 45∘45^\circ.
  • Two tops on one vertical seen from one point share the same ground distance.

Height

h=dtan⁡θh = d\tan\theta

Worked example

A pole casts a shadow 3\sqrt3 times its height. What is the Sun's altitude?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — Elementary Mathematics · Q82Easy

Example 1 · Heights and Distances · One Line of Sight

If the length of the shadow of a tower is equal to its height, then what is the Sun's altitude at that time ?

Is the given length slant or level?

'1010 km from the observer' can mean the straight-line distance to the plane, not the ground distance. The first needs sine, the second tangent.
Drill 5 more on height, distance and the angle

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

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