Height & Distance — NDA Maths

Formulas (11)

• The Right Triangle of Sight · Tangent of the angle of elevation
  $$\tan\theta = \frac{h}{d} = \frac{\text{height}}{\text{horizontal distance}}$$
• A Single Observation · One triangle, three ratios
  $$\tan\theta = \frac{h}{d}, \qquad \sin\theta = \frac{h}{\ell}, \qquad \cos\theta = \frac{d}{\ell}$$
• Slant Distances and Half-Angle Heights · Height from slant + half-angle value
  $$h = \ell\sin\theta, \qquad \cos\tfrac{A}{2} = \sqrt{\tfrac{1+\cos A}{2}}$$
• Two Observations at Different Heights · Same base, two heights
  $$\tan\beta = \frac{H}{d}, \qquad \tan\alpha = \frac{H-p}{d}$$
• Tower Carrying a Flagstaff · Stacked heights, one base
  $$\tan\theta = \frac{T}{d}, \qquad \tan\phi = \frac{T+f}{d}$$
• Angle Subtended by a Raised Segment · Subtended angle (tangent subtraction)
  $$\tan\alpha = \frac{(h_2-h_1)\,x}{x^2 + h_1 h_2}$$
• Ladders — Elevation Meets Pythagoras · Trig + length together
  $$H = x\tan\theta, \qquad x^2 + (H-k)^2 = L^2$$
• Three Collinear Observation Points · Distance from foot at elevation θ
  $$d = h\cot\theta, \qquad \text{gap} = h(\cot\theta_i - \cot\theta_j)$$
• Observation Points in Different Directions · Perpendicular observers (ground Pythagoras)
  $$z^2 = h^2(\cot^2 y - \cot^2 x)$$
• A Cloud and Its Reflection in a Lake · Cloud above, image below
  $$\tan\alpha = \frac{H-p}{d}, \qquad \tan\beta = \frac{H+p}{d}$$
• A Round Object Subtending an Angle · Subtended sphere
  $$R = \frac{r}{\sin(\alpha/2)}, \qquad h = \frac{r\sin\beta}{\sin(\alpha/2)}$$

Watch out for (12)

• Depression equals the elevation back
  → The Right Triangle of Sight
• Tangent, not sine, links height to ground distance
  → The Right Triangle of Sight
• Keep tan⁻¹ as a ratio
  → A Single Observation
• Slant uses sine, ground uses tangent
  → Slant Distances and Half-Angle Heights
• Same base, different opposite side
  → Two Observations at Different Heights
• The lower angle goes with the lower height
  → Tower Carrying a Flagstaff
• A subtended angle is a difference, not a single elevation
  → Angle Subtended by a Raised Segment
• The ladder top is not the flagstaff top
  → Ladders — Elevation Meets Pythagoras
• Bigger angle ⇒ nearer point ⇒ smaller cotangent
  → Three Collinear Observation Points
• The right angle sits at the middle observer
  → Observation Points in Different Directions
• Image depth is H + observer height, not H
  → A Cloud and Its Reflection in a Lake
• Use half the subtended angle
  → A Round Object Subtending an Angle

Formulas (5)

• Shadows and the Sun's Elevation · Shadow of a vertical object
  $$s = h\cot\theta, \qquad \Delta s = h(\cot\theta_2 - \cot\theta_1)$$
• Finding the New Sun Angle from a Shadow Change · New tangent, then bracket
  $$\tan\theta = \frac{h}{s_1 + x}$$
• Leaning Towers — Separating Height from Lean · Two readings on a leaning tower
  $$\tan\alpha = \frac{h}{p-\delta}, \qquad \tan\beta = \frac{h}{q-\delta}$$
• Chord Length of a Circle · Chord subtending central angle θ
  $$\text{chord} = 2r\sin\frac{\theta}{2}$$
• Arc Length and the Equilateral-Chord Clue · Arc length
  $$\text{arc} = r\theta \quad (\theta \text{ in radians})$$

Watch out for (5)

• Lower sun, longer shadow
  → Shadows and the Sun's Elevation
• The new shadow is old + increase, not just the increase
  → Finding the New Sun Angle from a Shadow Change
• A leaning tower has two unknowns
  → Leaning Towers — Separating Height from Lean
• Half the angle, not the whole angle
  → Chord Length of a Circle
• Angle must be in radians for r·θ
  → Arc Length and the Equilateral-Chord Clue