IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~6 min read

Angles of Elevation & Depression

Look up at something — that’s an angle of elevation. Look down — that’s an angle of depression. Both are measured from the horizontal. Once you draw the right triangle, it’s just SOH CAH TOA — usually tan.

📘 What you need to know

Angle of elevation: between the horizontal and a line going up to an object.
Angle of depression: between the horizontal and a line going down to an object.
Both are always measured from the horizontal — never from the vertical.
Tan is your default: most problems give you opposite (height) and adjacent (horizontal distance), so tan θ = O/A is the natural choice.
Draw a clear diagram first — every problem becomes obvious once the triangle is on paper.
Alternate angles: the angle of depression from A to B equals the angle of elevation from B to A (parallel horizontals).

Elevation vs depression

Elevation

looking UP ↗

angle from horizontal up to the object

Depression

looking DOWN ↘

angle from horizontal down to the object

The two are equal: if you’re on a cliff looking down at a boat with depression 30°, the boat looking up at you sees an elevation of 30°. Same triangle, same angle — measured between parallel horizontals.

🧭 Recipe — every elevation/depression problem

Draw a sketch. Mark the horizontal, the line of sight, and the angle between them.
Identify the right triangle. The horizontal distance and the vertical height form the two legs.
Label H, O, A relative to the angle.
Pick the trig ratio — usually tan, since you usually have/want height (O) and ground distance (A).
Solve — multiply or divide to find a length, or use tan⁻¹ to find an angle.

Worked examples

WE 1

Find a height from an angle of elevation

A person stands 30 m from the foot of a building. The angle of elevation from the person to the top of the building is 42°. Find the height of the building, correct to 3 s.f.

Step 1: Sketch — right triangle with 42° at ground level A = 30 m (horizontal); O = h (height); angle = 42° Step 2: Use tan (have A, want O) tan 42° = h/30 h = 30 × tan 42° h = 30 × 0.9004… = 27.013… height ≈ 27.0 m (3 s.f.)

WE 2

Find a distance from an angle of depression

From the top of a tower 50 m high, the angle of depression to a boat at sea is 28°. Find the horizontal distance from the boat to the foot of the tower, correct to 3 s.f.

Step 1: Sketch — depression 28° at top equals elevation 28° at boat O = 50 m (height); A = d (distance); angle = 28° Step 2: Use tan tan 28° = 50/d d = 50/tan 28° d = 50/0.5317… = 94.05… d ≈ 94.1 m (3 s.f.) depression at the top = elevation at the bottom (alternate angles)

WE 3

Find an angle of elevation

A bird is 25 m above the ground. A person stands 40 m from the point on the ground directly below the bird. Find the angle of elevation from the person to the bird, correct to 3 s.f.

Step 1: Right triangle — height 25, base 40 O = 25; A = 40 Step 2: Use inverse tan tan θ = 25/40 = 0.625 θ = tan⁻¹(0.625) = 32.005…° θ ≈ 32.0° (3 s.f.)

WE 4

Two angles from one point — find the flagpole

A flagpole stands on top of a building. From a point 80 m from the foot of the building, the angle of elevation to the top of the building is 25° and the angle of elevation to the top of the flagpole is 32°. Find the height of the flagpole, correct to 3 s.f.

Step 1: Find the building height tan 25° = h_b/80 h_b = 80 × tan 25° = 80 × 0.4663… = 37.30… Step 2: Find total height (top of flagpole) tan 32° = h_t/80 h_t = 80 × tan 32° = 80 × 0.6249… = 49.99… Step 3: Subtract to get the flagpole alone flagpole = 49.99… − 37.30… = 12.68… flagpole ≈ 12.7 m (3 s.f.) two angles from same observation point — find each height, then subtract

WE 5

Two boats from one lighthouse

An observer at the top of a lighthouse 45 m high sees two boats in line directly out at sea. The angles of depression to the boats are 27° (closer boat) and 18° (farther boat). Find the distance between the two boats, correct to 3 s.f.

Step 1: Distance to closer boat (depression 27°) tan 27° = 45/d₁ d₁ = 45/tan 27° = 45/0.5095… = 88.30… Step 2: Distance to farther boat (depression 18°) tan 18° = 45/d₂ d₂ = 45/tan 18° = 45/0.3249… = 138.49… Step 3: Distance between boats d = d₂ − d₁ = 138.49 − 88.30 = 50.19… distance ≈ 50.2 m (3 s.f.) smaller depression → farther boat (line of sight is more horizontal)

WE 6

Two observation points — find the cliff height

From point A on level ground, the angle of elevation to the top of a cliff is 35°. From point B, which is 50 m closer to the cliff than A on the same line, the angle of elevation is 50°. Find the height of the cliff, correct to 3 s.f.

Step 1: Let x = distance from B to base of cliff; h = height from A: tan 35° = h/(x + 50) → h = (x+50) tan 35° from B: tan 50° = h/x → h = x tan 50° Step 2: Set equal and solve for x x tan 50° = (x+50) tan 35° x(tan 50° − tan 35°) = 50 tan 35° x = 50 tan 35° / (tan 50° − tan 35°) x = 50 × 0.7002 / (1.1918 − 0.7002) x = 35.01 / 0.4916 = 71.21… Step 3: Find h h = 71.21 × tan 50° = 71.21 × 1.1918 = 84.86… cliff height ≈ 84.9 m (3 s.f.) two-equation simultaneous setup — both equations express h, set them equal and solve

💡 Top tips

Always sketch. Mark horizontal lines, the line of sight, and the angle between them. Bigger sketch = fewer mistakes.
Use tan most of the time — opposite (height) over adjacent (ground distance).
Depression = elevation from the other end. If stuck, swap viewpoints.
Calculator in degree mode for these problems.
For two-elevation problems: write one equation per angle, then solve simultaneously.

⚠ Common mistakes

Measuring from the vertical instead of the horizontal. Always from the horizontal.
Confusing depression and elevation. Read carefully — “from the cliff to the boat” is depression; “from the boat to the cliff” is elevation.
Putting the angle in the wrong corner of the triangle. Sketch slowly.
Forgetting tan⁻¹ when finding an angle. tan θ = 25/40 means θ = tan⁻¹(25/40), not 25/40 itself.
Mixing units (metres and kilometres). Convert first.

Next note: Bearings & Constructions — same trig, but now with compass directions. North-based, clockwise, three-digit angles. The setup changes, the maths stays the same.

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