2.4.7 · D5Trigonometry — Foundation
Question bank — Applications — heights and distances problems
This page hunts the thinking mistakes, not the arithmetic ones. Every item below is a one-line reveal: read the prompt, answer out loud, then check yourself. Real reasoning only — no "yes/no" left naked.
Before you touch a single trap, make sure the words and symbols are locked in. This whole bank stabs at four things: the two named angles, and the two named lengths.
Look at the labelled picture below before reading on — the whole bank refers to it. The pale-yellow angle is elevation, the chalk-blue angle is depression, is drawn vertical, horizontal.
From 2.4.01-Trigonometric-ratios, recall the two words that decide which ratio you use: opposite (the side across from the angle) and adjacent (the side touching the angle, that isn't the hypotenuse). If those feel shaky, patch them first.
Recall One-line refresh before the bank
, where is the angle of elevation (or depression), and "opposite/adjacent" are decided by where the angle sits, never by where you stand.
Why elevation and depression are equal (in-page, no off-page hunt needed): look at the second figure. The top horizontal and the bottom horizontal are parallel; the line of sight is a single straight transversal crossing both. The blue depression angle up top and the yellow elevation angle down below are alternate interior angles of parallel lines — so they are equal. (This is a parallel-lines fact, distinct from complementary angles.)
True or false — justify
The angle of depression from a cliff-top to a boat equals the angle of elevation from the boat to the cliff-top.
True — as the figure above shows, the two horizontal lines are parallel and the line of sight is a transversal, so these are equal alternate interior angles of parallel lines (a parallel-line theorem, not the complementary-angle rule).
The line of sight is the vertical height of the object.
False — the line of sight is the hypotenuse (eye to object). The vertical height is a leg of the triangle, not the slanted line you look along.
If you double the angle of elevation, the computed height doubles.
False — is nonlinear, so height does not scale with angle. is about triple, not double, .
For any real tower and any real ground distance, the angle of elevation can equal exactly .
False — would mean you stand at the tower's base (), and is undefined (vertical asymptote). At there is no triangle at all.
Angle of elevation and angle of depression can both appear in the same two-position problem.
True — e.g. a lighthouse keeper looking down at two boats uses depression, while each boat looking up uses the equal elevation; you may set the equation up from whichever viewpoint is cleaner.
Two towers of different heights, seen from the same spot at the same angle of elevation, must be at different distances.
True — same fixes the ratio , so a taller forces a proportionally larger ; the triangles are similar (see 3.2.04-Similar-triangles — third figure unpacks this).
If the angle of elevation is exactly , the height equals the horizontal distance.
True — , so . This is a handy sanity anchor: at , "as tall as it is far".
Increasing the horizontal distance (same tower) increases the angle of elevation.
False — walking away makes the tower look shorter, so decreases; shrinks as grows.
For a bird flying horizontally away from a tower, the change in its two horizontal ground-distances exactly equals the horizontal flight distance.
True — constant height means the flight path is horizontal, so the ground displacement is ; this is precisely what makes the classic "bird" problem well-posed and solvable.
Spot the error
"The tower is right next to me, so it is the adjacent side; I'll use ."
The tower is opposite the angle at your feet, and uses the hypotenuse — but is a leg. The right ratio is ; "adjacent/opposite" is set by the angle, not by what's physically nearby.
", so m."
A trig ratio is a number between the sides, not the angle itself. , giving m — a sane tower, not a mile-high spike.
"Depression angle is , height is , so ."
The height is opposite the depression angle and the distance is adjacent, so , i.e. . The student divided the wrong way.
"A bird flying horizontally sees the depression to a fixed tower-top drop ; the height is measured from the flight-line, so I'll just add the two distances."
You must subtract, not add. With constant height : at , ; at , . The flown distance is ; if the bird flew m then m — a perfectly consistent, well-posed problem. The trap is the arithmetic operation, not the geometry.
"Both angles are from different points, so I can just add the two heights I get."
Angles from different observation points reference different distances; you must introduce a variable per distance and use the given displacement to link them, not add naively.
"The angle of depression is measured from the ground up to the cliff-top."
Depression is measured downward from the horizontal at the top. Measuring it from the ground would actually be the angle of elevation (equal in size, opposite viewpoint) — exactly the parallel-line pairing in the second figure.
Why questions
Why do we pick rather than or in the basic tower problem?
Because we know one leg (ground distance) and want the other leg (height); relates the two legs, while and drag in the hypotenuse (line of sight) that we neither know nor need.
Why does knowing one angle plus one side pin down the whole triangle?
The angle fixes the triangle's shape — as the third figure shows, all triangles with that angle are nested similar copies — and one measured side fixes the scale, so every other side is forced. The trig ratio is exactly that scale factor. (See 3.2.04-Similar-triangles.)
Why is the growth of height near not exponential?
It's the specific blow-up of toward its vertical asymptote at , driven by in the denominator — a different mechanism from , which grows smoothly forever without an asymptote.
Why must the horizontal distance be perpendicular to the vertical height?
Because "height" means the vertical segment and "distance" the ground-level one; only when they meet at do we have a right triangle where 1.3.07-Pythagoras-theorem holds () and the trig ratios apply.
Why can a "moving observer" problem give the height even though we never know the actual distance?
Writing at each position gives two equations in and ; subtracting eliminates the unknown , so the walked displacement alone is enough to solve for .
Why do surveyors bother with this instead of just measuring directly?
Because the target is unreachable (a cliff-top, a far bank, a star), so they convert a measurable angle plus a measurable ground length into the unreachable length — the whole point of the applications in 2.5.02-Surveying-and-navigation.
Why does the "eye-level" (instrument height) sometimes have to be added back at the end?
The angle is measured from the instrument's horizontal, so the triangle gives height above eye level; the true object height adds the observer's eye height to that result.
Edge cases
What is the angle of elevation to the top of a tower when you stand directly beneath it?
It is and is undefined (), so the model degenerates — there's no triangle, you'd have to measure straight up.
What happens to the computed height as your ground distance with the tower fixed?
The angle and , so stays finite and equal to the real height — a far-away tower looks flat but is still its true height.
If the angle of elevation is reported as , what does that mean physically?
Your line of sight is exactly horizontal, so the object's top is at your eye level — computed height above the eye is .
Can the angle of elevation exceed in a standard heights problem?
No — beyond you'd be looking behind yourself; standard problems keep so the right triangle stays valid.
Why do we never write a negative angle for depression, even though it points downward?
By convention elevation and depression are both reported as positive magnitudes measured away from the horizontal (up for elevation, down for depression); the "down-ness" is carried by the word "depression", not by a minus sign. A signed angle would only appear if you insisted on one horizontal reference axis and let below-axis directions go negative — but standard heights work keeps both angles in and lets the labelled diagram fix the direction.
If two positions give the same angle of elevation, what does that tell you?
You are the same distance from the tower's base both times (same with the same ), so you either didn't move horizontally or moved on a circle around the base.
What if the "object" is below your eye level but you still call the angle "elevation"?
You've mislabelled it — a downward line of sight is an angle of depression; the arithmetic is the same size but the roles of which side is opposite/adjacent must be re-checked against the correct triangle.