So far we have used trigonometry to find unknown side lengths and angles in right-angled triangles. A common application is to compare the vertical position of objects in the real world to calculate either their difference in height, or their distance from you, the observer. These kinds of problems are described in terms of angles of elevation or angles of depression.
The angle formed between the horizontal and an observer's line of sight to an object is either called the angle of elevation, or the angle of depression, depending on the placement of the object.
Even when there is no human 'observer', you can still talk about angles of elevation and depression. For example, the top of a diving board may have a $30^\circ$30° angle of elevation from the side of a pool, and a campsite may have a $20^\circ$20° angle of depression from the top of a nearby mountain.
Using the angle of elevation or depression, we can create right-angled triangles. This will often mean that you need to draw a diagram and add vertical lines in the appropriate places. Once you can identify the right-angled triangles in the diagram, you can then use trigonometric ratios to find the missing information as in 3.02.
Exploration
Council regulations require that trees above a height of $5$5 metres cannot be cut down. Jemma is looking to build a home on her plot of land, and wants to see if she is able to remove a particular tree. She stands at a point on the ground and makes some measurements as shown below:
The distance from the base of the tree to her chosen point of observation is $4.2$4.2 m, and the angle of elevation from this point on the ground to the top of the tree is $38$38 degrees. In the right-angled triangle she forms above, the height of the tree that she wants to calculate is opposite the angle of elevation and she knows the adjacent angle, so she used $\tan$tan to find the height:
| $\tan38^\circ$tan38° | $=$= | $\frac{\text{Height of tree }}{4.2}$Height of tree 4.2 |
| $\text{Height of tree }$Height of tree | $=$= | $4.2\times\tan38^\circ$4.2×tan38° |
| $\text{Height of tree }$Height of tree | $=$= | $3.28$3.28 |
According to her calculations the height of this tree is $3.28$3.28 m, so council regulations would allow her to remove the tree.
- Surveyors routinely measure an angle and a length, then use trigonometry to get another length that would be difficult to measure directly, such as a distance across a lake.
- Architects use trigonometry to calculate structural load, roof slopes, ground surfaces, sun shading and light angles.
- Ship captains use trigonometry in navigation to find the distance of the shore from a point at sea.
- Oceanographers use trigonometry when calculating the height of tides in the ocean.
Practice questions
Question 1
Considering the diagram below, find $19$19, the angle of elevation to point $A$A from point $C$C.
Give your answer correct to two decimal places.
Question 2
Considering the diagram below, find $\theta$θ, the angle of depression from point $B$B to point $C$C.
Give your answer correct to the nearest degree.
Question 3
At a certain time of the day a light post, $6$6 m tall, has a shadow of $5.8$5.8 m. If the angle of elevation of the sun at that time is $\theta$θ, find $\theta$θ correct to the nearest degree.
Question 4
From the top of a rocky ledge $188$188 m high, the angle of depression to a boat is $13^\circ$13°. If the boat is $d$d m from the foot of the cliff find $d$d correct to $2$2 decimal places.