Angles of elevation and depression

At B, where is the angle of depression towards A measured? The sketch is not to scale

A sight line gets steeper while its endpoints still use parallel horizontal references. What do you predict?

Move the slider below and above its starting position. Compare the elevation, depression and vertical-reference angles

Which relationship continues to hold away from the starting position?

Choose the horizontal reference

Identify the observer and draw a horizontal through that point. Then use the right triangle. Side solving and inverse tangent work as they did in the earlier topics

1. 1Start at the observer Elevation is measured upwards from a horizontal ray. Depression is measured downwards from a horizontal ray
2. 2Transfer between the same two points The two horizontal rays are parallel. Their alternate angles with the sight line are equal. The angle from the vertical is the complement \(\displaystyle \begin{aligned}e&=d\\v&=90^{\circ}-d\end{aligned}\)
3. 3Use the vertical difference Tangent connects the vertical difference between the endpoints to their horizontal separation x. Ground height only equals this difference when the observer is at ground level \(\displaystyle \tan e=\frac{\Delta h}{x}\)
4. 4Calculate in degree mode Choose the side or inverse calculation you learnt earlier. Keep the full calculator value and round the requested result at the end

Horizontal distance AF

\(\displaystyle 23\ \mathrm{m}\)

Elevation at A

\(\displaystyle 37^{\circ}\)

A sighting from ground point A reaches the top B of a vertical signal post. Calculate FB to one decimal place. The sketch is not to scale

Vertical height FB

\(\displaystyle 17\ \mathrm{m}\)

Depression at B

\(\displaystyle 29^{\circ}\)

From B, a sensor sights a ground marker at A. Calculate AF to one decimal place. The sketch is not to scale

Vertical difference FB

\(\displaystyle 19\ \mathrm{m}\)

Horizontal distance AF

\(\displaystyle 31\ \mathrm{m}\)

Calculate the angle of depression from B to A to one decimal place. The sketch is not to scale

An instrument is 1.6 m above level ground. A target is 12 m horizontally away, with elevation 35°. What does 12 tan 35° represent?

The small arc at the eye marks the elevation angle. Keep the instrument height and horizontal distance fixed. Compare elevation readings of 34°, 35° and 36°, then explore the wider range

What happens when the measured elevation rises from 34° to 36°?

Measure a height with a clinometer

A clinometer measures an angle of sight. Establish its zero and whether its scale is referenced to horizontal or vertical before interpreting the reading

1. 1Record the setup Measure horizontal distance to the base and the instrument height above the same level ground. Sight the target from the instrument position
2. 2Convert the reference if needed If a reading is explicitly the acute angle between the sight line and vertical, subtract it from 90°. A horizontal-reference elevation reading needs no complement
3. 3Add the instrument height The tangent calculation gives the rise above the instrument. Add its height to estimate the target height \(\displaystyle H=h_0+d\tan\alpha\)
4. 4Evaluate the model Assume level ground, a vertical target and a clear straight sight line. Repeat readings and report uncertainty; more calculator digits do not improve the measurements

Measurements

\(\displaystyle \begin{aligned}h_0&=1.6\text{ m}\\d&=12\text{ m}\\\alpha&=35^{\circ}\end{aligned}\)

The instrument is 1.6 m high on level ground, 12 m horizontally from a vertical target. Its elevation reading is 35°. Estimate the total target height to one decimal place

Measurements

\(\displaystyle v=55^{\circ}\)

A device reports that an upward sight line is 55° from vertical. What is its elevation above horizontal? Give one decimal place

The elevation reading could be anywhere from 34° to 36°. The same model estimates heights from about 9.69 m to 10.32 m. Which report is justified?

Measurements

\(\displaystyle \begin{aligned}H&=20\text{ m}\\h&=8\text{ m}\\d&=25\text{ m}\end{aligned}\)

A window is 20 m above level ground and a target is 8 m above that same ground. Their horizontal separation is 25 m. Find the angle of depression from the window to one decimal place

A tape is laid along sloping ground from an observer to a target base. Can that measured length go straight into the horizontal-distance position in H = h₀ + d tan α?

Link two right triangles

Draw both triangles and mark which lengths they share. Work towards one final result. A rounded intermediate display is not the value to reuse

1. 1Two targets on the same side Let F be directly below the observer, N the nearer target and R the farther target, in the order F, N, R. Use capital H for the shared height, alpha for the nearer depression angle and beta for the farther depression angle. Find both horizontal distances, then subtract the nearer from the farther \(\displaystyle \begin{aligned}FN&=\frac{H}{\tan\alpha}\\FR&=\frac{H}{\tan\beta}\\NR&=FR-FN\end{aligned}\)
2. 2Two sightings with a known baseline Let h be the shared height, x the nearer distance and b the baseline. Use alpha for the nearer elevation angle and beta for the farther elevation angle. Write the height in two ways, expand and collect the nearer-distance terms. The nearer angle is larger and the baseline is positive \(\displaystyle \begin{gathered}h=x\tan\alpha\\h=(x+b)\tan\beta\\x\tan\alpha=(x+b)\tan\beta\\x\tan\alpha=x\tan\beta+b\tan\beta\\x\tan\alpha-x\tan\beta=b\tan\beta\\x(\tan\alpha-\tan\beta)=b\tan\beta\\x=\frac{b\tan\beta}{\tan\alpha-\tan\beta}\end{gathered}\)

Two targets lie on level ground on opposite sides of the point directly below an observer. How do you combine their horizontal distances from that point?

Height TF

\(\displaystyle 26\ \mathrm{m}\)

Depression towards N

\(\displaystyle 48^{\circ}\)

Depression towards R

\(\displaystyle 31^{\circ}\)

A camera at T sights markers N and R on level ground, on the same side of F. Calculate NR to one decimal place. The sketch is not to scale

Baseline NR

\(\displaystyle 9\ \mathrm{m}\)

Elevation α at N

\(\displaystyle 53^{\circ}\)

Elevation β at R

\(\displaystyle 28^{\circ}\)

N and R sight the top T of a vertical structure from level ground. Calculate TF to one decimal place. The sketch is not to scale

Vertical height FB

\(\displaystyle 22\ \mathrm{m}\)

Depression at B

\(\displaystyle 36^{\circ}\)

A learner uses the given depression angle to find AF. Which step first goes wrong? The sketch is not to scale