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10 Further trigonometry
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Stage 5.1-3

10.11 Trigonometry in 3D

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Introduction

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.

Trigonometry in 3D

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.

Examples

Example 1

Consider the given rectangular prism:

A rectangular prism with dimensions 12, 9 and 8 centimetres. Ask your teacher for more information.
a

Find the length x.

Worked Solution
Create a strategy

Use Pythagoras' theorem c^2=a^2+b^2.

Apply the idea

The lengths x,\, 9 and 12 create a right-angled triangle on the base of the prism where x is the hypotenuse.

\displaystyle x^2\displaystyle =\displaystyle 9^2+12^2Use Pythagoras' theorem
\displaystyle x^2\displaystyle =\displaystyle 225Evaluate
\displaystyle x\displaystyle =\displaystyle 15\text{ cm}Square root both sides
b

Find the length of the prism's diagonal y.

Worked Solution
Create a strategy

Use Pythagoras' theorem c^2=a^2+b^2.

Apply the idea

The lengths x=15,\, 8 and y create a right-angled triangle where y is the hypotenuse.

\displaystyle y^2\displaystyle =\displaystyle 8^2+15^2Use Pythagoras' theorem
\displaystyle y^2\displaystyle =\displaystyle 289Evaluate
\displaystyle y\displaystyle =\displaystyle 17\text{ cm}Square root both sides
c

Find the angle \theta to the nearest degree.

Worked Solution
Create a strategy

Use a trigonometric ratio.

Apply the idea

In the triangle with angle \theta , \, x=15 is the adjacent side, and y=17 is the hypotenuse. So we can use the cosine ratio to find \theta .

\displaystyle \cos \theta\displaystyle =\displaystyle \dfrac{15}{17}Use the cosine ratio
\displaystyle \theta\displaystyle =\displaystyle \cos ^{-1}\left( \dfrac{15}{17} \right)Find the inverse cosine of both sides
\displaystyle =\displaystyle 28\degreeEvaluate and round

Example 2

Two straight paths to the top of a cliff are inclined at angles of 24 \degree and 21 \degree to the horizontal:

A triangular prism with a height of h metres and width of y metres. Ask your teacher for more information.
a

If path A is 115\text{ m} long, find the height h of the cliff, rounded to the nearest metre.

Worked Solution
Create a strategy

Use a trigonometric ratio for the triangle with angle 24\degree and hypotenuse 115\text{ m.}

Apply the idea

h is the opposite side to the angle of 24\degree , and 115 is the hypotenuse. So we can use the sine ratio.

\displaystyle \sin 24 \degree\displaystyle =\displaystyle \dfrac{h}{115}Use the sine ratio
\displaystyle h\displaystyle =\displaystyle 115 \sin 24 \degreeMultiply both sides by 115
\displaystyle =\displaystyle 47\text{ m}Evaluate and round
b

Find the length x of path B, correct to the nearest metre.

Worked Solution
Create a strategy

Use a trigonometric ratio for the triangle with angle 21\degree and hypotenuse x\text{ m.}

Apply the idea

h=47 is the opposite side to the angle of 21\degree , and x is the hypotenuse. So we can use the sine ratio.

\displaystyle \sin 21 \degree\displaystyle =\displaystyle \dfrac{47}{x}Use the sine ratio
\displaystyle x\displaystyle =\displaystyle \dfrac{47}{\sin 21 \degree}Swap x and \sin 21 \degree
\displaystyle =\displaystyle 131\text{ m}Evaluate and round
c

Let the paths meet at 46 \degree at the base of the cliff. Find their distance apart, y, at the top of the cliff, to the nearest metre.

Worked Solution
Create a strategy

Use the cosine rule.

Apply the idea

Path A (115\text{ m}), Path B (131\text{ m}), and y make a triangle with angle opposite y of 46\degree . So we have a problem involving 3 sides and one angle. So we can use cosine rule.

\displaystyle c^2\displaystyle =\displaystyle a^2+b^2-2ab \cos CUse cosine rule
\displaystyle y^2\displaystyle =\displaystyle 115^2+131^2-2\times 115 \times 131 \cos 46\degreeSubstitute values
\displaystyle y^2\displaystyle =\displaystyle 9455.9433Evaluate
\displaystyle y\displaystyle =\displaystyle 97\text{ m}Evaluate and round
Idea summary

To solve trigonometric problems in 3D we can use any of the tools that we have previously learnt:

  • Pythagoras' theorem

  • Trigonometric ratios

  • Sine rule

  • Cosine rule

  • Area rule

  • Angles of elevation and depression

  • Bearings

Outcomes

MA5.3-15MG

applies Pythagoras' theorem, trigonometric relationships, the sine rule, the cosine rule and the area rule to solve problems, including problems involving three dimensions

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