Angles of Elevation & Depression
Master angles of elevation and depression for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-04). Elevation is measured up from the horizontal and depression is measured down from it, and both turn a real scene into a right-angled triangle.
You will sketch and label the triangle, use sine, cosine or tangent to find a missing height, distance or angle, and see why the angle of depression equals the angle of elevation (alternate angles) — handy for finding how tall a tree, building or cliff is without climbing it.
Theory
An angle of elevation is measured up from the horizontal and an angle of depression is measured down from the horizontal. This Year 12 Standard 1 (NSW) guide shows how to draw the right-angled triangle and use SOH CAH TOA to find heights, distances and angles.
The line of sight is the straight line from an observer's eye to the object. When the observer looks up, the angle between the horizontal and the line of sight is the angle of elevation. When the observer looks down, it is the angle of depression. Both are always measured from the horizontal.
The angle of depression from \(O\) to \(B\) equals the angle of elevation from \(B\) to \(O\). The two horizontal lines are parallel, so these are equal alternate angles. This lets you put the angle inside the right-angled triangle.
Draw the triangle, label the sides from the angle, then use \(\sin\), \(\cos\) or \(\tan\). An angle such as \(40^\circ 30'\) is entered as \(40.5^\circ\), or with the degrees-minutes-seconds key.
Label the sides from the marked angle \(\theta\) and choose the matching ratio (SOH CAH TOA):
The elevation and depression between the same two points are equal:
Changing minutes to a decimal:
How to solve an elevation or depression problem
- Draw a right-angled triangle with the horizontal, the height and the line of sight.
- Mark the angle from the horizontal. For a depression, use the equal angle of elevation at the other end so it sits inside the triangle.
- Label the sides opposite, adjacent and hypotenuse from that angle.
- Choose \(\sin\), \(\cos\) or \(\tan\), solve, and round as asked.
Sketch the triangle. \(h\) is opposite the \(38^\circ\) angle and \(24\text{ m}\) is adjacent, so use tangent.
| \(\tan 38^\circ\) | \(=\) | \(\dfrac{h}{24}\) |
| \(h\) | \(=\) | \(24\times\tan 38^\circ\) |
| \(h\) | \(=\) | \(18.75\ldots\) |
| \(h\) | \(\approx\) | \(18.8\text{ m}\) |
The building is about \(18.8\text{ m}\) tall.
Sketch the triangle. The angle of elevation from the surfer is also \(22^\circ\). The cliff (\(55\text{ m}\)) is opposite and \(d\) is adjacent.
| \(\tan 22^\circ\) | \(=\) | \(\dfrac{55}{d}\) |
| \(d\) | \(=\) | \(\dfrac{55}{\tan 22^\circ}\) |
| \(d\) | \(=\) | \(136.12\ldots\) |
| \(d\) | \(\approx\) | \(136\text{ m}\) |
(i) The surfer is about \(136\text{ m}\) from the cliff.
(ii) \(L\) is the hypotenuse and the \(55\text{ m}\) cliff is opposite, so use sine. \(L\) is on the bottom, so divide.
| \(\sin 22^\circ\) | \(=\) | \(\dfrac{55}{L}\) |
| \(L\) | \(=\) | \(\dfrac{55}{\sin 22^\circ}\) |
| \(L\) | \(=\) | \(146.82\ldots\) |
| \(L\) | \(\approx\) | \(147\text{ m}\) |
The line of sight is about \(147\text{ m}\).
Sketch the triangle. The cable is the hypotenuse and the \(11\text{ m}\) drop is opposite the angle, so use sine and its inverse.
| \(\sin\theta\) | \(=\) | \(\dfrac{11}{45}\) |
| \(\theta\) | \(=\) | \(\sin^{-1}\!\left(\dfrac{11}{45}\right)=14.149\ldots^\circ\) |
| \(0.149\ldots^\circ\) | \(=\) | \(0.149\ldots\times 60' = 8.94\ldots'\approx 9'\) |
| \(\theta\) | \(\approx\) | \(14^\circ 9'\) |
The angle of depression is \(14^\circ 9'\).
Sketch the triangle. Change the angle to a decimal, then use tangent (height opposite, \(50\text{ m}\) adjacent).
| \(27^\circ 15'\) | \(=\) | \(27+\dfrac{15}{60}=27.25^\circ\) |
| \(\tan 27.25^\circ\) | \(=\) | \(\dfrac{h}{50}\) |
| \(h\) | \(=\) | \(50\times\tan 27.25^\circ\) |
| \(h\) | \(=\) | \(25.75\ldots\) |
| \(h\) | \(\approx\) | \(25.8\text{ m}\) |
The mast is about \(25.8\text{ m}\) tall.
Common pitfalls
Frequently asked questions
What is the difference between an angle of elevation and an angle of depression?
An angle of elevation is measured upward from the horizontal when you look up at something. An angle of depression is measured downward from the horizontal when you look down at something. Both are measured from a level line.
Why is the angle of depression equal to the angle of elevation?
The horizontal line at the observer and the horizontal line at the object are parallel, and the line of sight crosses both. The angle of depression and the angle of elevation are equal alternate angles.
How do you find the height of a building using an angle of elevation?
Draw a right-angled triangle with the height opposite the angle and the distance along the ground adjacent to it. Use tangent: height equals distance times tan of the angle.
Do I measure the angle of depression from the vertical?
No. The angle of depression is always measured down from the horizontal. Measuring from the vertical gives 90 degrees minus the correct angle.
How do I enter an angle in degrees and minutes on my calculator?
Either use the degrees-minutes-seconds key, or change the minutes to a decimal by dividing by 60. For example, 27 degrees 15 minutes is 27 plus 15 divided by 60, which is 27.25 degrees.