Angles of Elevation & Depression (Cambridge (CIE) IGCSE Maths: Extended): Revision Note
Exam code: 0580 & 0980
Elevation & depression
What are angles of elevation and depression?
An angle of elevation or depression is the angle measured between the horizontal and the line of sight
Looking up at an object creates an angle of elevation
Looking down at an object creates an angle of depression
Right-angled trigonometry can be used to find
an angle of elevation or depression
or a missing distance
The tan ratio is often used in real-life scenarios
You may know the height of an object and want to find the distance you are from it
You may know the distance you are from an object and want to find its height
Examiner Tips and Tricks
It may be useful to draw more than one diagram if the triangles that you are interested in overlap one another.
Worked Example
A cliff is perpendicular to the sea and the top of the cliff, T, stands 24 metres above the level of the sea.
The angle of depression from the top of the cliff to a boat at sea is 35°.
At a point metres vertically up from the foot the cliff is a flag marker, M.
The angle of elevation from the boat, B, to the flag marker is 18°.
(a) Draw a diagram of the situation. Label all the angles and distances given above.
Answer:
(b) Find the distance from the boat to the foot of the cliff.
Answer:
Consider triangle TBF where F is the foot of the cliff
Angle TBF = 35º because of alternate angles
Use SOHCAHTOA to find the missing distance
We know the opposite (TF) and we want to find the adjacent (BF), so use
BF = 34.3 m (3 s.f.)
(c) Find the value of .
Answer:
Consider triangle MBF
Use SOHCAHTOA to find the missing distance
We know the adjacent (BF) and we want to find opposite (MF), so use
(3 s.f.)
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Questions students ask about Angles of Elevation & Depression
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How do you draw a diagram for angle of elevation and depression questions?
Draw the vertical object and the horizontal ground first, then add a dashed horizontal line from the observer and the line of sight — the angle always sits between the horizontal and the line of sight.
- Draw the object (cliff, tower) as a vertical line on horizontal ground or sea, with a right angle at the foot
- Label the observer and the object with the question's letters
- Draw a dashed horizontal from the observer, then the line of sight to the object
- Mark the angle between them: looking up is elevation, looking down is depression
- Add the given lengths and label the unknown
Page example: the cliff TF is 24 m high; from T the angle of depression to boat B is 35°, and from B the angle of elevation to marker M is 18°.
- The 35° goes at T, between the dashed horizontal and TB
- By alternate angles, angle TBF = 35° as well, which is easier to use inside the triangle
- The 18° goes at B, between the sea and BM
If the triangles overlap, as here, redraw each one separately: TBF and MBF.
How do you find the angle of elevation or depression given the opposite and adjacent sides?
Use tan: tan θ = opposite ÷ adjacent, then take the inverse, θ = tan⁻¹(opposite ÷ adjacent).
- Draw the right-angled triangle, with the angle between the horizontal and the line of sight
- Label the sides from that angle: opposite (O), adjacent (A), hypotenuse (H)
- You have O and A, so SOHCAHTOA gives TOA: tan θ = O/A
- Find θ with the inverse tan key (usually SHIFT then tan)
Example using the page's cliff: TF = 24 m and BF = 34.27555... m. Find the angle of elevation from the boat.
- Opposite = 24, adjacent = 34.27555...
- tan θ = 24 ÷ 34.27555...
- θ = tan⁻¹(24 ÷ 34.27555...) = 35°, which matches the page
Another example: a 12 m tree, 20 m away — θ = tan⁻¹(12 ÷ 20) = 31.0° (3 s.f.)
Tips:
- Use the unrounded length (34.27555..., not 34.3)
- Label O, A and H from the angle you actually want, or you will pick the wrong ratio
What is an angle of elevation?
An angle of elevation is the angle between the horizontal and your line of sight when you are looking up at an object.
- It is measured at the observer: start at the horizontal line through them and turn upwards to the line of sight.
- It is always measured from a horizontal, never from a vertical such as a cliff face. On a diagram the horizontal is the dashed line.
- Looking up gives an angle of elevation; looking down at something below you gives an angle of depression.
In the cliff example on this page:
- you are in the boat, B, looking up at the flag marker, M
- the horizontal is BF, the line from the boat to the foot of the cliff
- your line of sight is BM, so the angle of elevation is the 18° between BF and BM
- the 35° at the cliff top, T, is turned downwards from the horizontal there, so that one is an angle of depression
The two horizontals are parallel, so the angle of elevation of T from the boat is also 35° (alternate angles).
