Heights and Distances: AFCAT Numerical Ability

Formulas (3)

• Finding the angle from a value · Reading the table backwards
  $$\sin\theta = \tfrac{1}{\sqrt2} \;\Rightarrow\; \theta = 45^\circ$$
• Complementary angles and the square identity · Complement and square identity
  $$\sin(90^\circ - \theta) = \cos\theta, \qquad \sin^2\theta + \cos^2\theta = 1$$
• All the ratios from one given ratio · The third side
  $$\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2$$

Reference tables (2)

Watch out for (10)

• √3/2 belongs to 60° for sine
  → Sine, cosine and tangent at the standard angles
• Square the value, not the angle
  → Sine, cosine and tangent at the standard angles
• Cosec goes with sine, sec with cosine
  → Cosecant, secant and cotangent
• Reciprocal is not negative
  → Cosecant, secant and cotangent
• Strip the number in front first
  → Finding the angle from a value
• Flip a reciprocal before looking it up
  → Finding the angle from a value
• Complementary means adding to 90°
  → Complementary angles and the square identity
• The identity needs the same angle
  → Complementary angles and the square identity
• Put the numbers on the right sides
  → All the ratios from one given ratio
• The hypotenuse is the longest side
  → All the ratios from one given ratio

Formulas (4)

• Height and horizontal distance: the tangent · Height from distance
  $$h = d\tan\theta, \qquad d = h\cot\theta$$
• Ladders, ropes, kites and ramps: sine and cosine · A slanting length
  $$h = L\sin\theta, \qquad x = L\cos\theta$$
• The angle of depression · Depression equals elevation
  $$\tan\theta = \dfrac{\text{height of observer}}{\text{horizontal distance}}$$
• Finding the angle of elevation · The angle from the sides
  $$\theta = \tan^{-1}\dfrac{\text{rise}}{\text{run}}$$

Watch out for (8)

• Tangent uses the horizontal distance
  → Height and horizontal distance: the tangent
• Height over distance, not the reverse
  → Height and horizontal distance: the tangent
• A length is the hypotenuse
  → Ladders, ropes, kites and ramps: sine and cosine
• A broken tree is two pieces
  → Ladders, ropes, kites and ramps: sine and cosine
• Measure from the horizontal
  → The angle of depression
• Move the angle to the ground
  → The angle of depression
• Use the difference of the heights
  → Finding the angle of elevation
• Not every angle is standard
  → Finding the angle of elevation

Formulas (4)

• Two points on the same side · Height from the distance walked
  $$h = \dfrac{d}{\cot\alpha - \cot\beta}$$
• The same set-up with letters · The cotangent difference
  $$\cot\alpha - \cot\beta = \dfrac{d}{h}$$
• Two points on opposite sides · Opposite sides add
  $$D = h(\cot\alpha + \cot\beta)$$
• Speed from two sightings · Speed
  $$v = \dfrac{h(\cot\alpha - \cot\beta)}{t}$$

Watch out for (8)

• The far point has the smaller angle
  → Two points on the same side
• Do not stop at the first distance
  → Two points on the same side
• Distances use cotangent, not tangent
  → The same set-up with letters
• Convert to the options' form
  → The same set-up with letters
• Opposite sides add, same side subtracts
  → Two points on opposite sides
• Both depressions are measured from one top
  → Two points on opposite sides
• Use the change in horizontal distance
  → Speed from two sightings
• Check the units of the options
  → Speed from two sightings

Formulas (3)

• A flagstaff on a tower, seen from one point · Two elevations, one distance
  $$H = d\tan\alpha, \qquad H + h = d\tan\beta$$
• Two buildings across a gap · Height seen from a window
  $$H = w + w\cot\phi\tan\theta$$
• A cloud and its reflection, a balloon and its size · Cloud and reflection
  $$\dfrac{H + h}{H - h} = \dfrac{\tan(\text{depression})}{\tan(\text{elevation})}$$

Watch out for (6)

• The flagstaff is a difference
  → A flagstaff on a tower, seen from one point
• One distance for both angles
  → A flagstaff on a tower, seen from one point
• Add the observer's height back
  → Two buildings across a gap
• The depression finds the gap
  → Two buildings across a gap
• The reflection is below the water, not the eye
  → A cloud and its reflection, a balloon and its size
• Halve the subtended angle
  → A cloud and its reflection, a balloon and its size