3D trigonometry is very similar to 2D trigonometry. The key difference is being able to process a 3D problem into a 2D one to make it easier to visualise and solve.
Planes and lines in 3D
In 3D trigonometry, you need to know what a plane is and be able to find angles between lines and planes.
Definition: A plane is a flat, two dimensional surface.
Converting 3D problems to 2D problems
To convert 3D trigonometry problems to 2D ones, you have to be able to identify exactly what angles and sides you need, then draw a 2D shape that simplifies the question. It is best shown with an example.
Example 1
Give a 2D representation of the angle between the line AF and the plane ABGH in this cuboid.
First, identify the line AF and the plane ABGH. The line AF is the diagonal of the cuboid. The plane ABGH is the bottom rectangle of the cuboid. To identify the angle between the line and place, you need to draw a triangle.
One of the sides of this triangle is AF. Another one of the sides will be on the plane ABGH. To keep this two dimensional, use the line AH as it is in the same direction as AF. To complete the triangle, the third line has to be FH. Here is an illustration:
The transformation from a 3D drawing to a 2D one simplifies the problem and makes it much easier to solve.
Solving 3D trigonometry problems
To solve a 3D trigonometry problem, you first have to convert it to a 2D one and then solve like normal.
Example 2
In the cuboid from the previous example, the sides AB, BH, and FH are 7cm, 3cm, 15cm respectively. Find the angle between the line AF and the plane ABGH to 1 decimal place.
The above example already transformed the problem to a 2 dimensional one. Now, you can use SOHCAHTOA to find the missing angle.
However, there is not enough information to find the angle - at least 2 sides are needed. Therefore, use Pythagoras to find the side AH by focusing on the triangle ABH:
(AH)2=(AB)2+(BH)2
Substitute in AB=7cm,BH=3cm:
(AH)2=72+32
(AH)2=49+9
(AH)2=58
AH=58
Now, use trigonometric ratios to find the desired angle - call it x for simplicity: