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.

Ideas

Trigonometry in 3D

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.

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

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

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\degree	Evaluate 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.

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 \degree	Multiply both sides by 115
	\displaystyle =	\displaystyle 47\text{ m}	Evaluate and round

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

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 C	Use cosine rule
\displaystyle y^2	\displaystyle =	\displaystyle 115^2+131^2-2\times 115 \times 131 \cos 46\degree	Substitute values
\displaystyle y^2	\displaystyle =	\displaystyle 9455.9433	Evaluate
\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

ACMMG245

Solve right-angled triangle problems including those involving direction and angles of elevation and depression

ACMMG276 (10a)

Apply Pythagoras’ Theorem and trigonometry to solving three-dimensional problems in right-angled triangles

7.10 Trigonometry in 3D

Introduction

Ideas

Trigonometry in 3D

Examples

Example 1

Example 2

Outcomes

ACMMG245

ACMMG276 (10a)

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