Both the angle of elevation and the angle of depression are measured from the horizontal, never from the vertical wall or cliff face. The angle of depression seen from a high point equals the angle of elevation seen from the low point looking back up, because the two horizontal lines are parallel and the line of sight is a transversal (alternate angles). Sketch the horizontal first, mark the angle against it, then apply SOH CAH TOA.
The idea itself is simple, but two setup mistakes account for almost every wrong answer. The first is measuring the angle from the vertical line (the wall, the cliff, the flagpole) instead of from the horizontal. The second is placing the angle at the wrong vertex, for example drawing the angle of depression at the bottom of the diagram instead of at the observer's eye level at the top.
Both mistakes come from skipping the sketch. Once you draw the horizontal dashed line through the observer and mark the angle correctly against it, the triangle becomes an ordinary right-angled triangle and the rest is routine trigonometry.
Step 1
Sketch the right-angled triangle
Step 2
Draw the horizontal at eye level, not the ground
Step 3
Place the angle against that horizontal, at the correct vertex
Step 4
Label opposite, adjacent, hypotenuse, then pick SOH, CAH or TOA
A useful shortcut: the horizontal at the top and the horizontal at the bottom are parallel lines, and the line of sight joining the two points is a transversal cutting both. That makes the angle of elevation from the bottom and the angle of depression from the top alternate angles, so they are always equal. You only ever need to work out one of them.
A student stands 28 m from the base of a building, on level ground. The angle of elevation from the student to the top of the building is 40°. Find the height of the building, to 3 significant figures.
Solution
TOA: tan θ = opposite ÷ adjacent.tan 40° = height ÷ 28height = 28 × tan 40°tan 40° = 0.8391, so height = 28 × 0.8391 = 23.4948…, which rounds to 23.5 m (3 s.f.).Quick check: a building roughly the same height as its distance from the observer would need an angle near 45°. 40° is a little under 45°, so a height a little under 28 m (23.5 m) is exactly the shape of answer you should expect.
A lighthouse keeper stands at the top of a cliff 85 m above sea level. The angle of depression from the keeper to a ship out at sea is 15°. Find the horizontal distance from the base of the cliff to the ship, to 3 significant figures.
Solution
TOA: tan 15° = 85 ÷ distance.distance = 85 ÷ tan 15°tan 15° = 0.2679, so distance = 85 ÷ 0.2679 = 317.28…, which rounds to 317 m (3 s.f.).The step students get wrong
Do not mark the 15° angle of depression at the base of the cliff, and do not measure it from the vertical cliff face. The angle of depression belongs at the top, against the horizontal through the observer's eye level. If you are unsure which vertex to use, redraw the triangle using the equal alternate angle at the other end, at sea level in this case, where the angle sits naturally between the sea-level horizontal and the line of sight upward.
Before you trust a tan, sin, or cos value, confirm the calculator is in degree mode, since O-Level E-Maths angles are always in degrees unless the question states radians.
On the fx-97SG CW and fx-97SG X
tan(40) and press =. You should see 0.8391 (4 s.f.). If instead you see a value near −1.117, the calculator was in radian mode.Is the angle of depression measured from looking straight down?
No. It is measured from the horizontal at the observer's eye level down to the line of sight, not from a vertical line. A small angle of depression means the line of sight is close to horizontal; a large angle of depression means the line of sight is closer to straight down.
Why are the angle of elevation and angle of depression always equal?
The horizontal line at the top and the horizontal line at the bottom are parallel, since both are horizontal. The line of sight connecting the two points is a transversal cutting both parallel lines, so the angle of depression at the top and the angle of elevation at the bottom are alternate angles, and alternate angles on parallel lines are always equal.
Does the observer's height above the ground matter?
Yes, if the question gives it. Some problems place the observer's eye a metre or two above the ground, or a boat's deck a few metres above sea level. In those cases, work out the vertical height using the trig ratio first, then add or subtract the observer's own height to get the true height or distance the question is asking for. Harder versions combine this with a bearing for the horizontal direction, or place the two points on different faces of a solid, which turns it into a 3D trigonometry question.
Which ratio should I use if I am not given the horizontal distance?
If the diagram gives you the hypotenuse (the direct line-of-sight distance) instead of the horizontal distance, use SOH CAH TOA with the opposite side, or the adjacent side, depending on which one you need to find. Relabel the triangle at the correct vertex before choosing the ratio, the same way as in both worked examples.
— Mr Gan Math Tuition
Mr. Gan works through the sketch-first method until placing the angle at the right vertex becomes automatic, not guesswork.
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