Angles of Elevation and Depression (AA SL)
Angles of elevation and depression turn everyday "how tall / how far" questions - towers, cliffs, planes, kites - into a single right-angled triangle you can solve with SOHCAHTOA. This topic covers the definitions of both angles, why they're often equal thanks to alternate angles, and the two situations that trip students up most: problems with two separate sightings, and problems where the observer's own height needs to be added in.
What the syllabus says
This topic maps onto one point in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| SL3.3 | Applications of right and non-right angled trigonometry, including Pythagoras's theorem. Angles of elevation and depression. Construction of labelled diagrams from written statements. Contexts may include use of bearings. |
This is a core AA SL syllabus point, examined on Paper 1 and Paper 2.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is the angle of elevation?
The angle of elevation is the angle measured upward from the horizontal to a line of sight, used when you're looking up at something higher than you. It's always measured from the horizontal, never from the vertical.
e.g. From a point 50 m from a building's base, if the angle of elevation to the top is \(35^\circ\), the height is \(50\tan35^\circ\approx35.0\) m.
What is the angle of depression?
The angle of depression is the angle measured downward from the horizontal to a line of sight, used when you're looking down at something lower than you - for example, from a clifftop down to a boat.
e.g. From the top of a 40 m cliff, if the angle of depression to a boat is \(15^\circ\), the boat's distance from the base is \(40/\tan15^\circ\approx149\) m.
Why does "alternate angles" matter here?
The horizontal line at the top of a cliff or tower is parallel to the horizontal line at the observer's level below. The line of sight crosses both, so the angle of depression from the top and the angle of elevation from the bottom are alternate angles - always equal.
e.g. From a 60 m tower, the angle of depression to a car is \(12^\circ\). By alternate angles, the car's angle of elevation to the tower top is also \(12^\circ\), so the horizontal distance is \(60/\tan12^\circ\approx282\) m.
What is a two-observation problem?
Some questions give the angle of elevation to a moving object (like a plane) at two different moments. Each sighting forms its own right-angled triangle sharing the same height, and you subtract the two horizontal distances to find how far the object moved.
e.g. A plane at 1800 m has elevation \(25^\circ\), then \(50^\circ\) a minute later: distances are \(1800/\tan25^\circ\approx3860\) m and \(1800/\tan50^\circ\approx1510\) m, so it travelled \(\approx2350\) m.
What is the "eye height" adjustment?
When a person's eyes (not the ground) are the point the angle is measured from, the right-angled triangle only gives the height above eye level - you then add the observer's eye height to get the total height of the object.
e.g. Eyes 1.5 m up, 20 m from a building, elevation \(50^\circ\): height above eye level \(=20\tan50^\circ\approx23.8\) m, so total height \(\approx25.3\) m.
Key formulas
This topic doesn't add any new formulas - it's an application of right-angled trigonometry you already know. The table and comparison below summarise how the pieces fit together.
Formula reference
Elevation and depression problems are solved with ordinary right-angled trig - none of these are separately listed in the formula booklet, since they're prior knowledge from SOHCAHTOA.
| Formula | Used for | Booklet? |
|---|---|---|
| \(\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}\) | Finding a height or distance from an angle | Not in booklet |
| \(\theta = \tan^{-1}\!\left(\dfrac{\text{opposite}}{\text{adjacent}}\right)\) | Finding an angle of elevation/depression | Not in booklet |
| \(c^2=a^2+b^2\) | Pythagoras' theorem, for the direct (hypotenuse) distance | Not in booklet - prior knowledge |
Elevation vs depression
The two angles describe the same idea from opposite ends of the line of sight.
| Feature | Angle of elevation | Angle of depression |
|---|---|---|
| Measured from | Horizontal, looking up | Horizontal, looking down |
| Observer is | Below the object | Above the object |
| Typical scenario | Looking up at a tower, kite, or plane | Looking down from a cliff or building at a boat or car |
| Link between them | Equals the angle of depression seen from the higher point (alternate angles) | Equals the angle of elevation seen from the lower point (alternate angles) |
Setting up the right triangle
Every elevation/depression problem reduces to identifying the right-angled triangle formed by the horizontal, the vertical, and the line of sight.
Vertical unknown
\[\text{opposite} = \text{adjacent}\times\tan\theta\]
Use when the height or depth is unknown and the horizontal distance is given.
Horizontal unknown
\[\text{adjacent} = \dfrac{\text{opposite}}{\tan\theta}\]
Use when the horizontal distance is unknown and the height is given.
Angle unknown
\[\theta = \tan^{-1}\!\left(\dfrac{\text{opposite}}{\text{adjacent}}\right)\]
Use when both the height and horizontal distance are given.
Two sightings and eye-height problems
These two variations account for most of the harder marks on this topic.
Two sightings, one height
\[\text{distance moved} = \dfrac{h}{\tan\theta_1} - \dfrac{h}{\tan\theta_2}\]
Solve two triangles sharing the same vertical height, then subtract.
Observer's eye height
\[\text{total height} = (\text{adjacent}\times\tan\theta) + \text{eye height}\]
Add the observer's own height once you've found the height above eye level.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
From a point 35 m from the base of a tower, the angle of elevation to the top of the tower is \(42^\circ\).
Find the height of the tower, correct to 3 significant figures.
Worked solution
\(\tan 42^\circ=\frac{h}{35}.\) M1 A1
\(h=35\tan 42^\circ\) M1 \(\approx31.5\) m. A1
From the top of a vertical cliff 65 m high, the angles of depression of two boats, \(A\) and \(B\), due south of the cliff, are \(10^\circ\) and \(22^\circ\) respectively. Boat \(A\) is further from the cliff than boat \(B\).
(a) Find the distance of boat \(A\) from the base of the cliff, correct to 3 significant figures.
(b) Find the distance of boat \(B\) from the base of the cliff, correct to 3 significant figures.
(c) Find the distance between the two boats, correct to 3 significant figures.
Worked solution
(a) By alternate angles, \(\tan 10^\circ=\frac{65}{d_A}\Rightarrow d_A=\dfrac{65}{\tan 10^\circ}\) M1
\(\approx369\text{ m.}\) A1
(b) \(d_B=\dfrac{65}{\tan 22^\circ}\) M1
\(\approx161\) m. A1
(c) Both boats lie on the same line due south, so \(AB=d_A-d_B\) M1
\(=369-161\) A1
\(\approx208\) m. A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Mixing up which angle is elevation and which is depression. Elevation is measured looking up from the lower point; depression is measured looking down from the higher point - drawing the horizontal line at your own eye level fixes this instantly.
- Re-deriving an angle instead of using alternate angles. If the angle of depression from the top is given, the angle of elevation from the bottom is the same value - no extra calculation needed.
- Forgetting to add the observer's eye height. If the angle is measured from a person's eyes rather than the ground, the tangent calculation only gives the height above eye level - the eye height must be added on afterwards.
- Skipping the diagram. The syllabus explicitly expects you to construct a labelled diagram from the written description - without one it's easy to swap which distance is opposite and which is adjacent.
Using your GDC
No calculator model needs a special sequence for this topic - it's ordinary right-angled trig evaluation.
Once you've drawn the triangle and written the tangent (or sine/cosine) equation, your GDC does the arithmetic - there's no dedicated calculator mode for this topic.
Check your calculator's angle setting is in degrees before you evaluate \(\tan\), \(\tan^{-1}\), or the other trig ratios - AA questions on this topic are almost always set in degrees. Enter the rearranged equation directly, and use the inverse trig key to recover an angle from a ratio like \(\tfrac{opposite}{adjacent}\). For two-sighting problems, work out each triangle's distance separately and store the values (or write them down) before subtracting, rather than trying to do it all in one calculator line.
Tip: If a height or distance comes out negative or absurdly large, the angle mode (degrees vs radians) is the first thing to check.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Elevation and depression questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between angle of elevation and angle of depression?
Angle of elevation is measured upward from the horizontal, when you're looking up at something above you. Angle of depression is measured downward from the horizontal, when you're looking down at something below you.
Why are the angle of elevation and angle of depression sometimes equal?
When two horizontal lines are parallel (the horizontal at the top point and the horizontal at the bottom point), the angle of depression from the top and the angle of elevation from the bottom are alternate angles on the same line of sight, so they're always equal.
How do I handle a problem with two different sightings?
Set up two separate right-angled triangles, one for each angle, sharing the same vertical height. Solve each for its horizontal distance, then add or subtract the two distances depending on what the question asks for.
Can I use my GDC for this topic?
Yes - your calculator's sin, cos, tan and inverse trig keys handle all the arithmetic once you've set up the right-angled triangle. See the GDC guide for general tips.
Sub-topics
Angles of Elevation and Depression broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AA SL syllabus unit, in case you want to keep going.