SPI Mathematics · Heights and Distances

Two Angles of Elevation

When a height is seen at two angles, from two points or as two heights stacked on each other, write one tan equation for each triangle and solve the two together.

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Why this matters

Fifteen questions in the bank and the hardest page in the chapter: ten MODERATE and five HARD, and the 2022 paper's question is here. Seven use complementary angles from two points, six have an observer who walks or a shadow that changes, and two stack one height on another.

Concept 1 of 3: Complementary angles from two points: h = √(ab)

Two points on one line with the foot of a tower see its top at angles that add to

90^\circ

. One triangle gives

\tan\theta

, the other gives

\cot\theta

, and since

\tan\theta \times \cot\theta = 1

, multiplying the two equations removes the angle altogether.

Definition

From distance $a$ : $\tan\theta = \dfrac{h}{a}$ . From distance $b$ : $\tan(90^\circ - \theta) = \cot\theta = \dfrac{h}{b}$ .
Multiply: $1 = \dfrac{h^2}{ab}$ , so $h = \sqrt{ab}$ .
$30^\circ$ and $60^\circ$ are complementary, so the same result holds for them.
With two different heights seen from one point, the same multiplication works: $\dfrac{h_1}{d_1} \times \dfrac{h_2}{d_2} = 1$ .
The nearer point sees the larger angle.

Height from complementary angles

h = \sqrt{ab}

a, bdistances of the two points from the foot of the tower
hheight of the tower

Worked example

From two points 4 m and 9 m from the foot of a tower, in one straight line with it, the angles of elevation of its top are complementary. Find the height of the tower.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q75Moderate

Example 1 · Heights and Distances · Two Angles of Elevation

Angles of elevation of the top of a tower from two points at distance 'a' and distance 'b' from base and in same straight line with the tower, are complementary. The height of the tower is ____.

Adding or multiplying the distances

The height is the square root of the product of the two distances, not their sum and not their product. With distances 4 m and 9 m it is 6 m, not 13 m or 36 m.

Giving the farther point the bigger angle

The closer you stand to a tower, the steeper you look up. The point at the smaller distance sees the larger angle, so match

60^\circ

with the nearer point.

Drill 6 more on complementary angles from two points: h = √(ab)

Concept 2 of 3: An observer who walks, or a shadow that changes

One height, two triangles that share it. The observer walks a known distance d between the two sightings, or the sun moves and the shadow shortens by d. Write the ground distance from each point in terms of h; the difference of the two is d.

Definition

From each point, ground distance $= h\cot(\text{angle})$ .
The walk is the difference of the two distances: $d = h\cot\alpha - h\cot\beta$ , with $\beta$ the larger angle.
So $h = \dfrac{d}{\cot\alpha - \cot\beta}$ .
A shadow that changes is the same picture with the sun moving instead of the observer.
Speed: distance moved over time. A distance in km over a time in seconds becomes km/h when multiplied by 3600.

Height from two angles and the walk between them

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

ddistance walked towards the foot (or change in shadow length)
αthe smaller angle, from the farther point
βthe larger angle, from the nearer point

Worked example

A man sees the top of a tower at an angle of elevation of

30^\circ

. He walks 20 m straight towards it and now sees the top at

60^\circ

. Find the height of the tower.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Heights and Distances · Two Angles of Elevation

From the top of a hill, the angles of depression of two consecutive km stones due east are found to be

30^\circ

and

45^\circ

. The height of the hill is

Using the walk as a whole distance

The distance walked is the gap between the two points, not either point's distance from the foot. Give each point its own distance, x and

x + d

Subtracting tan instead of cot

The ground distances are

h\cot\alpha

and

h\cot\beta

, so the formula has

\cot\alpha - \cot\beta

below the line. A denominator of

\tan\beta - \tan\alpha

is a printed wrong option.

Leaving a speed in km per second

2 km in 10 seconds is

2 \times \frac{3600}{10} = 720

km/h. Multiply by 3600 and divide by the seconds; dividing by 3600 gives a speed far too small.

Drill 5 more on an observer who walks, or a shadow that changes

Concept 3 of 3: A flagstaff on a tower, a cloud and its reflection

When one height sits on top of another, both tops are seen from the same ground point, so both triangles share the same ground distance. Find that distance from the triangle you can solve, then use it in the other.

Definition

Flagstaff on a tower: the top of the flagstaff is at height tower $+$ flagstaff above the ground. Use that total in the second tan equation.
The shared ground distance comes from the first sighting: $d = \dfrac{\text{tower}}{\tan(\text{first angle})}$ .
Reflection in a lake: a still lake works as a mirror, so the image of a cloud C metres above the water is C metres below it. From a point h metres above the lake, the cloud is $C - h$ above eye level and its image is $C + h$ below.

Reflection in still water

\text{rise to the cloud} = C - h, \qquad \text{drop to its image} = C + h

Worked example

A flagstaff stands on a building 20 m high. From a point on the ground, the angles of elevation of the bottom and the top of the flagstaff are

45^\circ

and

60^\circ

. Find the length of the flagstaff.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Heights and Distances · Two Angles of Elevation

If the height of a tower is half the height of the flagstaff on it and the angle of elevation of the top of the tower as seen from a point on the ground is

30^\circ

, then the angle of elevation of the top of the flagstaff as seen from the same point is

Using the flagstaff alone for its top

The top of a flagstaff on a tower is the tower plus the flagstaff above the ground. Putting only the flagstaff's length over the distance gives an angle that is far too small.

Measuring the reflection from the observer

The image is as far below the water as the cloud is above it. From a point above the lake, the drop to the image is the cloud's height plus the observer's height, not minus.

Drill 1 more on a flagstaff on a tower, a cloud and its reflection

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

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

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