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3D Trigonometry - Higher

3D Trigonometry - Higher

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Tutor: Bilal


3D Trigonometry

In a nutshell

33​D trigonometry is very similar to 22​D trigonometry. The key difference is being able to process a 33​D problem into a 22​D one to make it easier to visualise and solve.

Planes and lines in 33​D

In 33D 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 33​D problems to 22​D problems

To convert 33​D trigonometry problems to 22​D ones, you have to be able to identify exactly what angles and sides you need, then draw a 22​D shape that simplifies the question. It is best shown with an example.

Example 1

Give a 22​D representation of the angle between the line AFAF​ and the plane ABGHABGH​ in this cuboid.

Maths; Trigonometry; KS4 Year 10; 3D Trigonometry - Higher

First, identify the line AFAF​ and the plane ABGHABGH​. The line AFAF​ is the diagonal of the cuboid. The plane ABGHABGH​ 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 AFAF​. Another one of the sides will be on the plane ABGHABGH​. To keep this two dimensional, use the line AHAH​ as it is in the same direction as AFAF​. To complete the triangle, the third line has to be FHFH​. Here is an illustration:

Maths; Trigonometry; KS4 Year 10; 3D Trigonometry - Higher

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 ABAB​, BHBH​, and FHFH​ are 7cm7cm, 3cm3cm, 15cm15cm respectively. Find the angle between the line AFAF​ and the plane ABGHABGH to 11​ decimal place.

Maths; Trigonometry; KS4 Year 10; 3D Trigonometry - Higher

The above example already transformed the problem to a 2 dimensional one. Now, you can use SOHCAHTOA to find the missing angle.

Maths; Trigonometry; KS4 Year 10; 3D Trigonometry - Higher

However, there is not enough information to find the angle - at least 2 sides are needed. Therefore, use Pythagoras to find the side AHAH​ by focusing on the triangle ABHABH​:


Substitute in AB=7cm, BH=3cmAB = 7cm,\, BH =3cm:





Now, use trigonometric ratios to find the desired angle - call it xx for simplicity:




x=63.1° (1 d.p.)\underline{x=63.1 \degree \ (1 \ d.p.)}

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FAQs - Frequently Asked Questions

How do we transform a 3D trigonometric problem into a 2D one?

How do you solve a 3D trigonometry problem?


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