Angles of Elevation & Depression
Learn angles of elevation and depression for NSW Year 12 Mathematics Standard 2. An angle of elevation is measured up from the horizontal to an object above you, and an angle of depression is measured down from the horizontal to an object below you. The angle of elevation from one point always equals the angle of depression from the other, because they are equal alternate angles.
You will draw each situation as a right-angled triangle and use right-angled trigonometry (tangent, sine and cosine) to find heights, horizontal distances and angles, including answers in degrees and in degrees and minutes. These height-and-distance skills are a core part of the Standard 2 Measurement topic and appear throughout surveying, navigation and everyday problems.
Theory
Angles of elevation and depression describe looking up or down from the horizontal at an object. This Year 12 Standard 2 (NSW) guide shows how the angle of elevation from one point equals the angle of depression from the other, and how to use right-angled trigonometry to find heights, horizontal distances and angles in practical problems.
An angle of elevation is the angle measured upward from the horizontal to your line of sight when you look at an object above you. An angle of depression is the angle measured downward from the horizontal when you look at an object below you. Both are always measured from the horizontal, not from the vertical.
A key fact ties the two together: the angle of elevation from \(A\) to \(B\) equals the angle of depression from \(B\) to \(A\). The horizontal lines at \(A\) and \(B\) are parallel and the line of sight is a transversal, so the two angles are equal alternate angles. This lets you slide a depression angle into the triangle as an equal elevation angle.
Each problem becomes a right-angled triangle in a vertical plane: the height is one leg, the horizontal distance is the other leg, and the line of sight is the hypotenuse. Label the sides relative to the marked angle and choose \(\tan\), \(\sin\) or \(\cos\) to find an unknown side or angle. This is core Year 12 Mathematics Standard 2 (NSW) trigonometry.
In the right-angled triangle, label the sides relative to the marked angle \(\theta\) and choose the matching ratio (SOH CAH TOA):
The elevation and depression between the same two points are equal:
How to solve an elevation or depression problem
- Draw a right-angled triangle in the vertical plane, marking the horizontal, the height and the line of sight.
- Mark the angle at the observer. For a depression, use the equal angle of elevation from the object so the angle sits inside the triangle.
- Label the sides as opposite, adjacent and hypotenuse relative to that angle.
- Choose \(\tan\), \(\sin\) or \(\cos\) so the ratio uses the side you know and the side you want.
- Solve and round as asked (a length, an angle to the nearest degree, or degrees and minutes).
The height is opposite the angle and \(35\text{ m}\) is adjacent, so use \(\tan\).
| \(\tan 52^\circ\) | \(=\) | \(\dfrac{h}{35}\) |
| \(h\) | \(=\) | \(35\times\tan 52^\circ\) |
| \(h\) | \(=\) | \(35\times 1.2799\) |
| \(h\) | \(\approx\) | \(44.8\text{ m}\) |
The depression equals the elevation from the yacht; the height is opposite and the distance is adjacent, so use \(\tan\).
| \(\tan 28^\circ\) | \(=\) | \(\dfrac{45}{d}\) |
| \(d\) | \(=\) | \(\dfrac{45}{\tan 28^\circ}\) |
| \(d\) | \(=\) | \(\dfrac{45}{0.5317}\) |
| \(d\) | \(\approx\) | \(85\text{ m}\) |
The run is the hypotenuse and the drop is opposite, so use \(\sin\), then convert to minutes.
| \(\sin\theta\) | \(=\) | \(\dfrac{90}{320}=0.28125\) |
| \(\theta\) | \(=\) | \(\sin^{-1}(0.28125)\) |
| \(\theta\) | \(=\) | \(16.335^\circ\) |
| \(0.335^\circ\) | \(=\) | \(0.335\times 60'\approx 20'\) |
| \(\theta\) | \(\approx\) | \(16^\circ 20'\) |
Write each horizontal distance in terms of the height \(h\); the points are \(15\text{ m}\) apart, so subtract.
| \(\dfrac{h}{\tan 32^\circ}-\dfrac{h}{\tan 47^\circ}\) | \(=\) | \(15\) |
| \(1.6003h-0.9325h\) | \(=\) | \(15\) |
| \(0.6678h\) | \(=\) | \(15\) |
| \(h\) | \(\approx\) | \(22.5\text{ m}\) |
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 above you, and an angle of depression is measured downward from the horizontal when you look down at something below you. Both are measured from a level line, not from the vertical.
Why does the angle of elevation equal the angle of depression?
The horizontal line at your eye and the horizontal line at the object are parallel, and the line of sight between them acts as a transversal. The angle of elevation and the angle of depression are equal alternate angles formed by that transversal, so they are always equal.
Which trig ratio should I use for elevation and depression problems?
Label the sides relative to the angle, then use SOH CAH TOA. Use tangent when you know or want the two legs (height and horizontal distance), and sine or cosine when the hypotenuse, such as a slope or line of sight, is involved.
How do I give an angle in degrees and minutes?
Work out the angle as a decimal number of degrees. Keep the whole number of degrees, then multiply the decimal part by 60 to get the minutes, since one degree equals 60 minutes. For example, 16.335 degrees becomes 16 degrees and about 20 minutes.
How do I find a height using an angle of depression?
The angle of depression from the top equals the angle of elevation from the point below, so mark that equal angle inside the right-angled triangle. Then use tangent with the horizontal distance and the height, or sine or cosine if the line of sight is known.
Do I measure the angle of depression from the vertical?
No. Both the angle of elevation and the angle of depression are always measured from the horizontal. Measuring from the vertical is the most common mistake and gives the complementary angle, which is 90 degrees minus the correct one.