Heights and Distances: SPI Mathematics

Formulas (3)

• Line of sight, elevation and depression · Depression equals elevation
  $$\text{angle of depression from the top} = \text{angle of elevation from the ground point}$$
• Height and distance through tan · Height and ground distance
  $$\tan\theta = \frac{h}{d}, \qquad h = d\tan\theta, \qquad d = \frac{h}{\tan\theta}$$
• The sun's elevation and a shadow · Sun's elevation from a shadow
  $$\tan\theta = \frac{\text{height}}{\text{shadow}}$$

Watch out for (6)

• Measuring depression from the tower
  → Line of sight, elevation and depression
• Putting the depression angle at the top of the triangle
  → Line of sight, elevation and depression
• Turning tan upside down
  → Height and distance through tan
• Dividing when you should multiply
  → Height and distance through tan
• Reading the ratio the wrong way
  → The sun's elevation and a shadow
• A high sun does not lengthen the shadow
  → The sun's elevation and a shadow

Formulas (1)

• The slanting side is the hypotenuse · Height and ground distance from a slant length
  $$h = L\sin\theta, \qquad d = L\cos\theta, \qquad L = \frac{h}{\sin\theta}$$

Watch out for (4)

• Using tan with a slant length
  → The slanting side is the hypotenuse
• Swapping sin and cos
  → The slanting side is the hypotenuse
• Using the taller pole's full height
  → Finding the triangle in a wire, a broken tree or a rectangle
• Calling the broken part the whole tree
  → Finding the triangle in a wire, a broken tree or a rectangle

Formulas (3)

• Complementary angles from two points: h = √(ab) · Height from complementary angles
  $$h = \sqrt{ab}$$
• An observer who walks, or a shadow that changes · Height from two angles and the walk between them
  $$h = \frac{d}{\cot\alpha - \cot\beta}$$
• A flagstaff on a tower, a cloud and its reflection · Reflection in still water
  $$\text{rise to the cloud} = C - h, \qquad \text{drop to its image} = C + h$$

Watch out for (7)

• Adding or multiplying the distances
  → Complementary angles from two points: h = √(ab)
• Giving the farther point the bigger angle
  → Complementary angles from two points: h = √(ab)
• Using the walk as a whole distance
  → An observer who walks, or a shadow that changes
• Subtracting tan instead of cot
  → An observer who walks, or a shadow that changes
• Leaving a speed in km per second
  → An observer who walks, or a shadow that changes
• Using the flagstaff alone for its top
  → A flagstaff on a tower, a cloud and its reflection
• Measuring the reflection from the observer
  → A flagstaff on a tower, a cloud and its reflection