Trigonometry
Parametric form of a function
The binomial theorem
Inverse functions
Graphical relationships
5.01 Three-dimensional trigonometry
After this lesson, you will be able to…
• interpret information about a 3-dimensional context given in diagrammatic or written form
• identify and sketch relevant 2-dimensional triangles (including right-angled triangles) within 3-dimensional figures or scenarios
• apply Pythagoras’ theorem, trigonometric ratios, sine rule, and cosine rule to solve for unknown lengths and angles in these 2D triangles
• solve problems in three dimensions, including finding the angle between a line and a plane, and problems involving bearings or angles of elevation/ depression
Three-dimensional trigonometry
In 3D space, a line and a plane can be related in four ways:
The line lies in the plane.
The line is perpendicular to the plane, intersecting at a point and forming right angles.
To find the angle between a line and a plane:
The line is parallel to the plane and does not intersect it.
The line intersects the plane at a point P and forms an angle with the plane.
• From a point A on the line (distinct from its intersection point P with the plane), drop a perpendicular to the plane, intersecting at M.
• The angle is formed between the line segments AP and PM
3D trigonometry problems can be solved by breaking them into 2D triangles. If a diagram is provided, identify the 2D components. If no diagram is provided, construct 2D diagrams from the given information to form a 3D model.
Plan views (top-down) are useful for bearings, while elevation views help with angles of elevation and depression.
Interactive exploration
Discover this concept in action online mathspace.co
Example 1
A rectangular prism has dimensions 3 cm, 3 cm, and 14 cm as shown.
a Calculate the length of diagonal HF , denoted as z cm, rounded to two decimal places.
Create a strategy
Identify the right-angled triangle FGH and use Pythagoras’ theorem: c 2 = a 2 + b2
Apply the idea
In triangle FGH, GH = 3 cm (since GH = EF ) and FG = 14 cm.
Write the formula
Substitute c = z, a = 3 and b = 14
Evaluate the squares
Evaluate the addition
Take the square root of both sides
Evaluate and round
The length of HF is approximately 14.32 cm.
b Find the size of ∠DFH, denoted as θ°, rounded to two decimal places.
Create a strategy
In right-angled triangle DHF , use the tangent ratio: tan (θ) = .
Apply the idea
In triangle DHF , the opposite side DH = 3 cm and the adjacent side HF = 14.32 cm.
Write the formula
Substitute Opposite = 3 and Adjacent = 14.32
Take the inverse tangent
Evaluate and round
The angle ∠DFH is approximately 11.83°.
Example 2
A room measures 5 m in length and 4 m in width. The angle of elevation from the bottom left corner to the top right corner is 57°.
a Find d, the distance from one corner of the floor to the opposite corner, in surd form.
Create a strategy
Construct a 2D diagram of the floor and apply Pythagoras’ theorem: c 2 = a 2 + b2 .
Apply the idea
The distance is . 4 5 d 4
Write the formula
Substitute c = d, a = 4 and b = 5
Evaluate the squares
Evaluate the addition
Take the square root of both sides
b Find h, the height of the room, rounded to two decimal places.
Create a strategy
In the right-angled triangle with the angle of elevation 57° and adjacent side , use the tangent ratio: tan (θ) = .
Apply the idea
Write the formula
Substitute θ = 57°, Opposite = h and Adjacent =
Multiply both sides by to make h the subject
Evaluate and round
The height is approximately 9.86 m.
c Find x, the angle of elevation from the bottom corner of the 5 m wall to the opposite top corner, rounded to two decimal places.
Create a strategy
In the right-angled triangle with opposite side 9.86 m and adjacent side 5 m, use the tangent ratio.
Apply the idea
Write the formula
Substitute θ = x, Opposite = 9.86 and Adjacent = 5
Take the inverse tangent
Evaluate and round
The angle of elevation is approximately 63.11°.
d Find y, the angle of depression from the top corner of the 4 m wall to the opposite bottom corner, rounded to two decimal places.
Create a strategy
In the right-angled triangle with opposite side 9.86 m and adjacent side 4 m, use the tangent ratio.
Apply the idea
Write the formula
Substitute θ = y, Opposite = 9.86 and Adjacent = 4
Take the inverse tangent
Evaluate and round
The angle of depression is approximately 67.92°.
Example 3
Three satellites, A, B and C, orbit Earth in the same plane. The distance between A and B is 8.27 km. Angles are ∠BAE = 45°, ∠ ABC = 51° and ∠ ACB = 74°. Point E is a car on Earth.
a Find x, the distance between satellites A and C, in kilometres, rounded to two decimal places.
Create a strategy
For triangle ABC, use the sine rule:
Apply the idea
Write the formula
Substitute B = 51°, AB = 8.27 and C = 74°
Multiply both sides by sin (51°)
Evaluate and round
The distance is approximately 6.69 km.
b Find y, the distance between satellites B and C, in kilometres, using the cosine rule, rounded to two decimal places.
Create a strategy
Calculate ∠ A using the fact that the sum of interior angles of a triangle is equal to 180° and use the cosine rule: y 2 = a 2 + b2 2ab cos ( A)
Apply the idea
Solving for ∠ A:
180 = ∠ A + ∠B + ∠C
180 = ∠ A + 51 + 74
180 = ∠ A + 125
∠ A = 55°
Substitute ∠ A = 55° and y:
Write the formula
Substitute ∠B = 51° and ∠C = 74°
Evaluate the addition
Subtract 125 from both sides
Write the formula
Substitute a = 8.27, b = 6.69 and A = 55°
Take the square root of both sides
Evaluate and round
The distance is approximately 7.05 km.
c Given ∠ AEB = 90°, find z, the distance from satellite A to the car E, in kilometres, rounded to two decimal places.
Create a strategy
In the right-angled triangle AEB with adjacent side z and hypotenuse 8.27 m, use the cosine ratio: cos (θ) =
4 A cone has radius 7 cm and a slant height of 13 cm:
Determine the vertical angle, θ, at the top of the cone, rounded to two decimal places.
5 A pole is seen by two people, Jenny and Matt:
a Matt is x m from the foot of the pole. Calculate x to the nearest metre.
b Calculate the height of the pole h to the nearest metre.
6 Two straight paths to the top of a cliff are inclined at angles of 24° and 21° to the horizontal:
a If path A is 115 m long, calculate the height h of the cliff, rounded to the nearest metre.
b Calculate the length x of path B, rounded to the nearest metre.
c Let the paths meet at 46° at the base of the cliff. Calculate their distance apart, y, at the top of the cliff, rounded to the nearest metre.
7 A square prism has sides of length 3 cm, 3 cm and 14 cm as shown in the diagram:
a If the diagonal HF has a length of z cm, calculate the value of z rounded to two decimal places.
b If the size of ∠DFH is θ°, calculate θ rounded to two decimal places.
8 All edges of the given cube are 7 cm long:
a Calculate the exact length: i EG ii AG
b Calculate these angles rounded to the nearest degree: i ∠EGH ii ∠EGA iii ∠ AHG iv ∠ AGH
9 Consider the cube as shown:
a Calculate the size of: i α ii β iii γ
b Calculate the exact length of x
c Calculate the exact length of y
d Calculate the angle θ rounded to the nearest degree.
10 Consider the given rectangular prism:
a Calculate the length x.
b Calculate the length of the prism’s diagonal y
c Calculate the angle θ to the nearest degree.
11 The diagram shows a triangular prism:
a Calculate the exact length of AC
b Calculate the size of ∠ ACF in degrees.
c Calculate the exact length of AF.
d Calculate the size of ∠ AFC rounded to the nearest degree.
12 A triangular prism has dimensions as shown in the diagram:
a Calculate the size of ∠ AED, rounded to two decimal places.
b Calculate the exact length of CE.
c Calculate the exact length of BE
d Calculate the size of ∠BEC, rounded to two decimal places.
e Calculate the exact length of CX.
f Calculate the size of ∠BXC, rounded to two decimal places.
13 A cockroach starts at point B and crawls towards point C at 10 cm/min on a 3.4 m high ceiling. Sally is standing at a point A:
a Calculate the distance in centimetres the cockroach will have crawled in the 3 minutes it takes to reach point C
b The cockroach is now at point C. Calculate y, the distance in centimetres Sally will be from the point directly below the cockroach rounded to two decimal places.
c Calculate θ, the angle of elevation from Sally to the cockroach at point C.
14 From the top of a vertical pole, the angle of depression to Ian standing at the foot of the pole is 43°. Liam is on the other side of the pole such that the pole is directly between him and Ian, and the angle of depression from the top of the pole to Liam is 52°. The boys are standing 58 m apart.
Calculate the height of the pole to the nearest metre.
15 The box has a triangular divider placed inside it, as shown:
a If z = AC, calculate the value of z rounded to two decimal places.
b Calculate the area of the divider rounded to two decimal places.
16 A 25 cm × 11 cm × 8 cm cardboard box contains an insert (the shaded area) made of foam:
a Calculate the exact length b of the base of the foam insert.
b Calculate the area of foam in the insert, rounded to the nearest square centimetre.
c Calculate the value of θ rounded to the nearest degree.
17 This pyramid has a square base with side length 8 cm, and all slanted edges are 12 cm in length:
a Calculate the exact length of MD.
b Calculate the size of ∠PDM, rounded to two decimal places.
c Calculate the exact height of the pyramid, MP.
d Calculate the length of MN.
e Calculate the exact length of PN.
f Calculate the size of ∠PNM, rounded to two decimal places.
g Calculate the size of ∠PDC, rounded to two decimal places.
18 This triangular prism shaped box labelled ABCDEF needs a diagonal support inserted as shown:
a Write an expression for the length of BF in terms of BD and DF.
b Calculate the length of AF in terms of AB, BD and DF.
c If AB = 19, BD = 30 and DF = 43, calculate the length of AF to two decimal places.
d Calculate the length of AF if AB, BD and DF increased by 10, rounded to two decimal places.
19 The angle of elevation to the top of a 22 m high statue is 54° from a point, A, due west of the statue. The point B is located 60 m due south of point A:
a Calculate the distance, x, from point A to the base of the statue, rounded to two decimal places.
b Calculate y, the distance from point B to the base of the statue, rounded to one decimal place.
c Calculate θ, the angle of elevation from point B to the top of the statue, rounded to the nearest degree.
20 A hot air balloon travelling at 950 m/h at a constant altitude of 3000 m is observed to have an angle of elevation of 76°. After 20 minutes, the angle of elevation is 71°:
a Determine the initial horizontal distance x from the observer to the balloon, rounded to the nearest metre.
b Determine the final horizontal distance y from the observer to the balloon, rounded to the nearest metre.
c Calculate the angle θ through which the observer has turned during the 20 minutes, rounded to the nearest degree.
21 A lighthouse is 174 m above sea level. The angle of elevation of the lighthouse from Boat 1 is 15°. The angle of elevation of the lighthouse from Boat 2 is 19°. The distance between the two boats is 1 km:
a Determine the horizontal distance x from Boat 1 to the lighthouse rounded to the nearest metre.
b Determine the horizontal distance y from Boat 2 to the lighthouse rounded to the nearest metre.
c The two boats form an angle of θ at the base of the lighthouse. Calculate θ rounded to the nearest degree.
22 The height of a lighthouse is 210 m above sea level. The angle of elevation to the top of the lighthouse from a boat P is 17°. The bearing of the lighthouse from the boat is N 44°E. The angle of elevation to the top of the lighthouse from a second boat Q is 12°. The bearing of the lighthouse from the second boat is N 32°W:
a Calculate each rounded to the nearest metre:
i The horizontal distance from the lighthouse to P.
ii The horizontal distance from the lighthouse to Q
iii The distance between the two boats.
b Calculate each rounded to the nearest degree:
i The angle made at P between Q and the base of the lighthouse.
ii The bearing of boat Q from boat P.
23 From a point 15 m due north of a tower, the angle of elevation of the tower is 32°:
a Calculate the height of the tower h, rounded to two decimal places.
b Calculate the size θ of the angle of elevation of the tower at a point 20 m due east of the tower, rounded to the nearest degree.
24 A rectangular prism has dimensions as shown:
a Calculate these lengths in surd form: i EG ii AG iii DG
b Calculate the area of △ ADG, in surd form.
c Calculate the size of ∠DAG, rounded to two decimal places.
d A triangular wedge has been inserted into the box as shown.
Calculate the size of angle x, rounded to two decimal places.
e A different triangular wedge has been inserted into the box as shown.
Calculate the size of angle y, rounded to two decimal places.
25 AC and BD are vertical posts. AC is west of BD, AC = h and BD = 1.5h P is on the ground, south of AC.
The angle of elevation of C from P is y°and the angle of elevation of D from P is z°.
The angle of elevation D from C is x°.
Prove:
26 A person walks 2000 m due north along a road from point A to point B. The point A is due east of the mountain OM, where M is the top of the mountain. The point O is directly below point M and is on the same horizontal plane as the road. The height of the mountain above point O is h metres. From point A, the angle of elevation to the top of the mountain is 15°. From point B, the angle of elevation to the top of the mountain is 13°:
a Show that if the length of AO is x, then:
b Determine the value of x in metres, rounded to one decimal place.
c Calculate the height of the mountain rounded to one decimal place.
27 There are thugs standing at point M and N. The thug at point M is 30 m away from the base of the tree, and Robin Hood, standing on a branch at point R, is looking down at him with an angle of depression of 38°. The angle of elevation from N to B is 34°.
Calculate each rounded to one decimal place:
a The height h of the tree in metres if it reaches 1.5 m above the branch.
b The distance Robin would need to shoot from point R to the man standing at point M, rounded to one decimal place.
c The distance Robin would need to shoot from point B to the man standing at point N, rounded to one decimal place.
d The distance between the man at point M and the man at point N, rounded to one decimal place.
28 From point A, 94 m due south of the base of a tower, the angle of elevation is 36° to the top of the tower. Point B is 125 m due east of the tower:
a Determine the height of the tower, rounded to the nearest metre.
b Calculate the angle θ of elevation of the top of the tower from point B, rounded to the nearest degree.
29 Two buildings AB and CD are 200 m apart. The angle of elevation of the top B of the first building from a point E, on the ground is 22°, and from the point F, also on the ground is 18°. The points E and F are 100 m apart. EF is parallel to AC. The heights of the buildings AB and CD are 100 m and 50 m respectively:
a Calculate the length of AE rounded to one decimal place.
b Calculate the length of AF rounded to one decimal place.
c Calculate the size of ∠ AFE rounded to the nearest degree.
d Calculate the length CF rounded to one decimal place.
e Calculate the length of CE rounded to one decimal place.
f Calculate the size of ∠CFE rounded to one decimal place.
g Calculate the size of the angle of elevation of the top D of the second building from the point F, rounded to the nearest degree.
h Calculate the size of the angle of elevation of the top D of the second building from the point E, rounded to the nearest degree.