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3D Trigonometry - Higher

Tutor: Bilal

# 3D Trigonometry

## In a nutshell

$3$​D trigonometry is very similar to $2$​D trigonometry. The key difference is being able to process a $3$​D problem into a $2$​D one to make it easier to visualise and solve.

### Planes and lines in $3$​D

In $3$D 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 $3$​D problems to $2$​D problems

To convert $3$​D trigonometry problems to $2$​D ones, you have to be able to identify exactly what angles and sides you need, then draw a $2$​D shape that simplifies the question. It is best shown with an example.

##### Example 1

Give a $2$​D 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 $AB = 7cm,\, BH =3cm$:

$(AH)^2=7^2+3^2$​​

$(AH)^2=49+9$​​

$(AH)^2=58$​​

$AH=\sqrt{58}$

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

$\tan(x)=\frac{opp}{adj}=\frac{FH}{AH}$​​

$\tan(x)=\frac{15}{\sqrt{58}}$​​

$x=\tan^{-1}(\frac{15}{\sqrt{58}})=63.0822655...$​​

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

## FAQs - Frequently Asked Questions

### How do you solve a 3D trigonometry problem?

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