Heights and Distances

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high importance~2 Q in Tier 120 formulas⚡ 11 shortcuts5 subtopics

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Angles of elevation and depression

Tangent rule

\tan\theta = \frac{\text{height above eye}}{\text{horizontal distance}}

The angle sits at the observer. Height is opposite, distance is next to the angle.

Height and distance

h = d\tan\theta,\qquad d = h\cot\theta

Slanting length (thread, wire, ladder)

h = L\sin\theta,\qquad d = L\cos\theta

L is the slanting line of sight, the hypotenuse of the triangle.

Depression to elevation

\text{depression from top} = \text{elevation from bottom}

The two horizontal lines are parallel, so the angles are equal.

Standard angles: 30°, 45°, 60°

Tangent values

\tan 30^\circ = \frac{1}{\sqrt{3}},\quad \tan 45^\circ = 1,\quad \tan 60^\circ = \sqrt{3}

Distance from height

d = h\cot\theta

cot 30 = sqrt3, cot 45 = 1, cot 60 = 1/sqrt3.

Ladder on a wall

h = L\sin\theta,\quad d = L\cos\theta

Theta is the ladder's angle with the ground.

Fifteen and seventy-five

\tan 15^\circ = 2-\sqrt{3},\qquad \tan 75^\circ = 2+\sqrt{3}

Two observation points (two angles)

Same side (walk towards)

h = \frac{d}{\cot\alpha - \cot\beta}

d is the distance walked; beta is the nearer, bigger angle.

Opposite sides

h = \frac{d}{\cot\alpha + \cot\beta}

d is the full distance between the two observers.

Gap between two objects from a height

\text{gap} = h(\cot\alpha - \cot\beta)

Tower seen from foot and roof of a building

d = \frac{b}{\tan\beta - \tan\alpha},\quad H = d\tan\beta

b = building height; beta from the foot, alpha from the roof.

Moving observers: speed and time

Moving observer chain

v\,t = h(\cot\alpha - \cot\beta)

alpha = first (farther) angle, beta = second (nearer) angle.

Time to reach the foot

t = \frac{h\cot\beta}{v}

Use the angle at the car's current position.

Vertical rise

\text{rise} = d(\tan\beta - \tan\alpha)

d = fixed horizontal distance; angles of depression shrink as the balloon rises.

Speed conversion

1\ \text{km/h} = \frac{5}{18}\ \text{m/s}

Compound figures: buildings, pedestals, broken objects

Stacked object (statue on pedestal)

s = d(\tan\beta - \tan\alpha)

d comes from the lower triangle: d = pedestal height x cot alpha.

Tower on a building

t = d(\tan\beta - \tan\alpha),\quad d = b\cot\alpha

From a roof: depression and elevation

d = b\cot\alpha,\qquad H = b + d\tan\beta

Broken tree

\text{stump} = x\tan\theta,\quad \text{broken} = \frac{x}{\cos\theta}

x = distance from the foot to where the top touches.