Line/plane relationships
We will now apply what we know about solving 2D problems involving trigonometry and Pythagoras' theorem to three dimensions. 3D problems can seem difficult, but can be separated into one or more 2D triangles. With a careful, systematic approach successful solutions can be attained.
To create 2D diagrams an understanding of how lines and planes are related in 3D space is required.
A plane and line can be related in four ways in 3D space as follows:
- The line lies on or in the plane
- The line and plane are parallel and never meet
- The line is perpendicular to the plane and intersects it at a point creating right angles.
- The line intersects the plane at the point $P$P and subtends an angle with the plane.
To construct the angle:
- Drop a perpendicular line from a point $A$A on the line to intersect the plane at $M$M.
- The angle is the one formed between the existing line and line $PM$PM
Problem Solving Strategy
The types of problems associated with 3D applications of trigonometry can be divided into two main types:
- Diagram provided
- No diagram provided
The strategy to solve them both is similar. When a diagram is provided it is useful to break the problem up into its 2D components. When there isn't a diagram provided it is useful to start by creating 2D diagrams of the information provided in order to build a more complex 3D diagram. Plan views (views looking from above) can be useful when dealing with bearings and elevation views can be useful when dealing with angles of elevation and depression.
Practice questions
Question 1
A square prism has sides of length $3$3cm, $3$3cm and $14$14cm as shown.
If the diagonal $HF$HF has a length of $z$z cm, calculate $z$z to two decimal places.
If the size of $\angle DFH$∠DFH is $\theta$θ°, find theta to two decimal places.
Question 2
A room measures $9$9 metres in length and $6$6 metres in width. The angle of elevation from the bottom left corner to the top right corner of the room is $49$49°.
Find $d$d, the distance from one corner of the floor to the opposite corner of the floor. Leave your answer in surd form.
Find $h$h, the height of the room. Give your answer to two decimal places.
Find $x$x, the angle of elevation from the bottom corner of the $9$9m long wall to the opposite top corner of the wall, correct to two decimal places.
Find $y$y, the angle of depression from the top corner of the $6$6m long wall to the opposite bottom corner of the wall, correct to two decimal places.
Question 3
Three satellites, $A$A, $B$B, and $C$C, used for GPS navigation are orbiting the earth. The distance between satellites $A$A and $B$B is $8.27$8.27 km.
$A$A, $B$B, and $C$C are in the same plane, and $E$E is a car travelling on a road on earth. The angle $\angle BAE=45^\circ$∠BAE=45°.
Determine the distance $x$x between satellite $A$A and satellite $C$C in kilometres.
Round your answer to two decimal places.
Determine the distance $y$y between satellites $B$B and $C$C in kilometres, using the cosine rule.
Round your answer to two decimal places.
$\angle AEB$∠AEB is a right-angle. Find the distance $z$z from satellite $A$A to the car $E$E in kilometres.
Round your answer to two decimal places.
Question 4
David walks along a straight road. At one point he notices a tower on a bearing of $054^\circ$054° with an angle of elevation of $20^\circ$20° After David walks $220$220 m, the tower is on a bearing of $341^\circ$341° with an angle of elevation of $25^\circ$25°. Let $h$h be the height of the tower.
Determine the angle $\theta$θ that David has walked through with respect to the tower.
Round your answer to the nearest degree.
Find the height of the tower.
Round your answer to the nearest metre.