1
CHAPTER:
THREE-DIMENSION TRIGONOMETRY
Contents
1
Introduction
2
Angle between a line and a plane
3
Angle between Two Planes
4
Angle between Two Skew Lines
5
Miscellaneous Examples
 1 Introduction
A line is said to lie in a plane if every point of the line is in the plane.
A line is said to be parallel to a plane if the line never meets the plane.
A line which is perpendicular to a plane is perpendicular to every line in the plane through the point of
intersection.
Basic procedure in handling 3-D trigonometric problems:
a) Sketch a diagram
b) Identify the angle you need to find
c) Find a triangle in the diagram which includes this angle
d) Solve for the angle using sine rule, cosine rule, trigonometric ratios, or area of triangles.
Example 1.1 (VJC Prelim 97/1/14 modified)
The diagram shows a model of a house. AE, BF, CG and DH are each 5
units in length and they stand vertically above the horizontal base,
E
ABCD, where AB = 20 units, BC = 15 units, FJ = 6 units, GJ = 10 units.
J is vertically above BC and I is vertically above AD.
Find, to the nearest degree, the acute angle
A
between BG and plane GHIJ. [34o]
Solution
I
H
J
D
G
F
C
B
 2 Angle between a line and a plane
Consider a line L passing through B and meeting a plane  at A as shown below.
L
B
A

C

If C is the foot of the perpendicular to the plane from point B, then AC is called the projection of AB on
the plane . The angle between the line L and the plane  is defined as the angle  between the line and
its projection on the plane. It is also the smallest angle between the line and the plane.
2
Example 2.1
The diagram shows a unit cube. Find the angle between the lines a) AH and b) AG with the base ABCD.
[450, 35.30 ]
Solution
E
H
F
A
G
D
B
Example 2.2 (VJC Prelim 97/1/13bi)
In the diagram, the triangle ABC lies in a horizontal plane and the
angle ACB Is 90o. The point T is 2 cm vertically above C, AC = 3 cm
and AB = 12 cm. Find, to the nearest 0.10, the angle between the line
AT and plane TCB. [56.30]
Solution
A
C
T
C
B
3
Example 2.3
The prism shown below has three identical rectangular faces.
AD = BE = CF = 12 cm and the sides of triangles ABC, DEF are each 10 cm long.
Find the angle x between the line CD and the plane ABED. [33.70 ]
Solution
F
C
E
D
B
A
 3 Angle between Two Planes
Two planes are said to be parallel if they do not intersect.
Two planes which are not parallel have a line of intersection (or common line).
Consider the diagram shown. Two planes, 1 and 2, are non-parallel.
Their common line is XY.
Line AP lies in plane 1 and line AQ lies in plane 2 .
P
If AP and AQ are both perpendicular to the common line, XY,
of the planes, then the angle between the planes is angle PAQ,
 Q
denoted by  in the diagram.
X
2
1
A
Y
4
Example 3.1
In a tetrahedron ABCD, AD = BD = CD = 6 cm and AB = BC = CA = 8 cm. Find the angle between the
plane ACD and the plane ABC. [58.90]
Solution
D
B
C
A
Example 3.2 (VJC 97/1/14 modified)
The diagram shows a rectangular box 20 by 15 by 5 units. Find the acute angle between the plane AHC and
the base ABCD, to the nearest 0.10 . [ 22.60 ]
Solution
E
H
F
A
G
D
B
C
5
Example 3.3
The base of a right pyramid with vertex V is a square ABCD of length 20 cm. The vertex V is 15 cm above
the centre of the base. VA = 5 17 cm. Find
a) the angle between the slant face ABV and the base ABCD
b) the angle between the slant faces ABV and BCV. [56.30, 107.90]
Solution
V
D
A
C
B
6
 4 Angle between Two Skew Lines
Two lines which do not lie on the same plane are called skew lines. Skew lines are never parallel, and do
not intersect. The angle between the skew lines L1 and L2 (see diagram below) is the acute angle between
L1 and L2’ where L2’ is a line parallel to L2 and meets L1. Note that the lines L1 and L2’ and lie on the same
plane.
L2
L2’
L1
Example 4.1 (SRJC Prelim 97/1/13 i)
A square PQRS of side 4 cm lies in a horizontal plane. Another identical square PQTW lies in a plane
inclined at an angle of 40o to the square PQRS. The points X and Y on PQRS are vertically below T and W
respectively. Find to the nearest 0.1o, the angle between the two skew lines QW and RS. [45o]
Solution
T
R
Q
W
X
P
Y
S
7
Example 4.2
ABCD and ADEF are two unit squares such that angle CDE = angle BAF = 90 0.
a) Find the angle between the lines BC and DE
b) Find the angle between the lines BF and CD
c) Show that the tangent of the angle between the lines BE and AF is 2
6
d) Show that the cosine of the angle between the lines AD and EG is
, where G is the centre of the
6
0
0
square ABCD. [90 , 45 ]
Solution
8
 5 Miscellaneous Examples
Example 5.1
The base ABC is horizontal and angle BAC = 900, P, Q, R are vertically above A, B, C. Given AC=5 cm,
AB = 7 cm, PA = 7 cm, BQ = 3 cm and CR = 3 cm,
a) find the angle of inclination of line PQ to the horizontal
1891
b) prove the length of the perpendicular from A to RQ is
74
c)
4 74
)
35
25
[29.70]
3034
P
prove that the inclination of the plane PRQ to the horizontal is tan-1 (
d) Show that the angle between the skew edges PR and CB is cos-1
Solution
R
Q
A
B
C
9
Example 5.2
The base of a right pyramid with vertex V is an equilateral triangle ABC. In triangle ABC, two medians
AD and BE meet at M. Given AB=4cm and VA=5cm.
a) Find the length of AD,
b) Find the height of the pyramid
4 3
c) Show that the cosine of the angle between the line VA and the plane ABC is
15
d) Show that the tangent of the angle between the planes ABC and VAC is
e)
Show that the cosine of the angle between the lines AB and VE is
Solution
59
2
21
. [2 3 cm,
21
V
59
cm]
3
C
A
B
10
Example 5.3
The diagram shows a cube with all edges of unit length. P is a point on the edge FB such that PF=a. Find,
in terms of a, the perpendicular distance from E to the line HP. If  is the obtuse angle between the planes
1
1
EHP and GHP, show that cos  = . If a = , find, to the nearest 0.10,
1+a2
3
a) the angle between the line HP and the plane ADHE
1+a2
b) the angle between the planes EHP and EHC. [
, 43.50, 26.60]
2+a2
Solution
H
G
E
F
P
D
A
C
B
11
Example 5.4
ABCD is a horizontal rectangle with AB = 4 cm, BC = 3 cm. E is a point on CD such that CE = 1 cm. P is
6
a point vertically above E with PE = 3 cm and sin PAB =
. Calculate
3
a) the sine of the angle APB
b) the tangent of the angle between the skew lines PB and AD
c) the angle between the plane PAB and the horizontal
d) the perpendicular distance from the point E to the plane PAB
e) the angle between the planes PAB and PAD
Give your answers in surd form or to the nearest 0.1 0 as appropriate.
4
6
10
3 2
[
,
, 450,
, 1200]
3
19 3
2
Solution
12
SUMMARY (Three-Dimension Trigonometry)
Basic procedure in handling 3-D trigonometric problems:
a) Sketch a diagram
b) Identify the angle you need to find
c) Find a triangle in the diagram which includes this angle
d) Solve for the angle using sine rule, cosine rule, trigonometric ratios, or area of triangles.
Angle between a line and a plane
Consider a line L passing through B and meeting a plane  at A as shown below.
L
B
A

C

If C is the foot of the perpendicular to the plane from point B, then AC is called the projection of AB on
the plane . The angle between the line L and the plane  is defined as the angle  between the line and its
projection on the plane.
Angle between Two Planes
Two planes are said to be parallel if they do not intersect.
Two planes which are not parallel have a line of intersection (or common line).
Consider the diagram shown. Two planes, 1 and 2, are non-parallel.
Their common line is XY.
Line AP lies in plane 1 and line AQ lies in plane 2 .
If AP and AQ are both perpendicular to the common line, XY,
of the planes, then the angle between the planes is angle PAQ,
denoted by  in the diagram.
X
2
P

Q
1
A
Y
Angle between Two Skew Lines
Two lines which do not lie on the same plane are called skew lines. Skew lines are never parallel, and do
not intersect. The angle between the skew lines L1 and L2 (see diagram below) is the acute angle between
L1 and L2’ where L2’ is a line parallel to L2 and meets L1. Note that the lines L1 and L2’ and lie on the same
plane.
L2’
L2
L1
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