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CDS Mathematics · Formula sheet

Heights and Distances formulas

5 formulas and 6 common traps for CDS Mathematics Heights and Distances, grouped by subtopic.

Full notes with worked examples

One Line of Sight

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Height, distance and the angle

Height

h=dtan⁡θh = d\tan\theta

Common traps

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.

Two Points of Observation

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Difference and sum of the distances

Same side

d=h(cot⁡α−cot⁡β)d = h(\cot\alpha - \cot\beta)

Complementary angles

Complementary elevations

h2=pqh^2 = pq

Common traps

Same side or opposite sides?

'On either side of the tower' or 'on the opposite banks' means the distances ADD. Subtracting them gives an option that is always there.

Nearer point, larger angle

The point closer to the tower sees the larger elevation. Swapping the angles makes the gap negative.

Heights multiply, distances multiply

h2=pqh^2 = pq uses the PRODUCT of the distances. With 44 and 1616, the height is 88, not 1010, their average.

Observer Above the Ground

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Elevation and depression together

Height of the object

H=h+hcot⁡βtan⁡αH = h + h\cot\beta\tan\alpha

Common traps

Add the part below eye level

The elevation gives only the part of the tower ABOVE the observer's eye. The observer's own height hh must be added to get the whole tower.

Towers on Plane Figures and Bearings

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Ground distances from the figure

Hexagon diagonals

AB:AC:AD=1:3:2AB : AC : AD = 1 : \sqrt3 : 2

Common traps

Which diagonal of the hexagon?

From AA, the next-but-one vertex CC is s3s\sqrt3 away and the opposite vertex DD is 2s2s. Using 2s2s for CC changes the ratio of the towers.

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