9.09 Further trigonometric problems in 3D
Much like how we can use Pythagoras' theorem in 3D space, we can also use trigonometry if we have right-angled triangles and the required starting information.
Trigonometry in 3D space works the same way as in 2D space, with the same trigonometric ratios and relations.
The key difference between 3D and 2D space when using trigonometry is that there are a lot more triangles we can find. Using this abundance of triangles, we can use multiple applications of trigonometry to find previously unknown side lengths and angles in a given problem.
We can see how multiple applications of trigonometry work together in 3D space to find missing values in the example below.
Worked example
Nathan is standing $15$15 m due North of a building with a height of $h$h. Erin is standing $20$20 m due East of the building. If the angle of elevation from Nathan to the top of the building is $32^\circ$32°, what is Erin's angle of elevation to the same point?
Think: To find the angle of elevation from Erin to the top of the building, we need to know two side lengths of the triangle formed by Erin and the building. We can find the height of the building using trigonometry in the triangle formed by Nathan and the building.
Do: Using trigonometry in the triangle formed by Nathan and the building, we can find the height $h$h of the building to be:
$h=15\tan32^\circ$h=15tan32°
Using this value, we can then find the missing angle $\theta$θ to be:
$\theta$θ$=$=$\tan^{-1}\left(\frac{h}{20}\right)$tan−1(h20)$=$=$\tan^{-1}\left(\frac{15\tan32^\circ}{20}\right)$tan−1(15tan32°20)
Rounding to two decimal places, find that:
$\theta=25.11^\circ$θ=25.11°
Reflect: As shown, using trigonometry in 3D space can help us to solve problems by using given values to find unknown side lengths which can then be used to find unknown angles.
It is also important when applying trigonometry to 3D problems is to work backwards from the information we are trying to find, since it will tell us what other values are required. Repeating this until the value required are already known will give us a starting place and method for solving the overall problem.
Practice questions
Question 1
The diagram shows a rectangular prism.
Use Pythagoras' theorem to find the length $x$x.
Hence find the length of the prism diagonal $y$y.
Find the angle $\theta$θ.
Round your answer to the nearest degree.
Question 2
Question 3
Two straight paths to the top of a cliff are inclined at angles of $24^\circ$24° and $21^\circ$21° to the horizontal.
If path $A$A is $115$115 m long, find the height $h$h of the cliff.
Round your answer to the nearest metre.
Find the length $x$x of path $B$B.
Round your answer to the nearest metre.
If the paths meet at $46^\circ$46° at the base of the cliff, find their distance apart, $y$y, at the top of the cliff.
Round your answer to the nearest metre.