Elevation and Depression Problems (AA SL)
Elevation and depression questions are ordinary right-angled trigonometry wearing a real-world costume: a flagpole, a cliff, a radio mast. The hard part usually isn't the trig, it's turning the words into a correctly-labelled triangle. This page picks out that skill with worked examples and the traps that trip people up. It's part of the broader Angles of Elevation and Depression topic.
20 questions on this sub-topic.
Setting up the triangle
Covered under IB syllabus reference SL3.3: applications of right-angled trigonometry, including angles of elevation and depression, built from labelled diagrams constructed from the written description.
Finding the angle
\(\theta = \tan^{-1}\!\left(\dfrac{\text{opposite}}{\text{adjacent}}\right)\)
Use when the height and the ground distance are both known and you need the angle itself.
Finding a height or distance
\(\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}\)
The same relation rearranged - use it when the angle is given and you need the missing side. Not in the formula booklet, so it's worth knowing by heart.
Need the full syllabus wording, non-right-angled cases, and the GDC walkthrough? See Angles of Elevation and Depression.
Worked examples
From a point 25 m from the base of a flagpole, the angle of elevation to the top is \(40^\circ\).
Find the height of the flagpole (3 significant figures).
Worked solution
The height \(h\) is opposite the \(40^\circ\) angle; 25 m is adjacent, so use tangent: \(\tan 40^\circ=\frac{h}{25}.\) M1
\(h=25\tan 40^\circ \approx 21\) m. A1
From point \(A\) the angle of elevation of a cliff top is \(20^\circ\). After walking 40 m directly towards the cliff to point \(B\), the angle of elevation is \(35^\circ\).
Find the height of the cliff.
Worked solution
With base distance \(x\) from \(B\): \(h=x\tan 35^\circ\) and \(h=(x+40)\tan 20^\circ\). M1 A1
Eliminate \(h\): \(x\tan 35^\circ=(x+40)\tan 20^\circ \Rightarrow x \approx 43.3\) m. M1 A1
So \(h\approx 43.3\tan 35^\circ\approx 30.3\) m. A1
Common mistakes
- Mixing up elevation and depression. Elevation is measured looking up from the lower point; depression is measured looking down from the higher point - sketching the horizontal line at your own eye level in the diagram 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 by alternate angles - no extra calculation is needed.
- Rounding the intermediate distance too early. In two-triangle problems like the cliff example above, rounding \(x\) before using it to find \(h\) introduces error that can shift the final answer outside the accepted range - carry full calculator accuracy through and round only the final answer.
Ready to practise properly?
20 elevation-and-depression questions, marked instantly like the real exam.
Quick answers
What is the difference between an angle of elevation and an angle of depression?
An angle of elevation is measured upward from the horizontal at the lower observer's eye level; an angle of depression is measured downward from the horizontal at the higher observer's eye level. Because the two horizontals are parallel, alternate angles make the two angles equal.
How do I find a height using an angle of elevation?
Draw a right triangle with the horizontal ground distance as the adjacent side and the height as the opposite side, then use \(\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}\) and rearrange for the unknown side.