3D Trigonometry
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3D Trigonometry
3D TrigonometryWhen doing 3D trigonometry it is often useful to redraw all of the faces of the shape in 2D.
3D TrigonometryWhen doing 3D trigonometry it is often useful to redraw all of the faces of the shape in 2D.
David is in a life raft and Anna is in a cabin cruiser searching for him. They are in contact by mobile phone. David tells Ana that he can see Mt Hope. From David’s position the mountain has a bearing of , and the angle of elevation to the top of the mountain is .
Anna can also see Mt Hope. From her position it has a bearing of , and and the top of the mountain has an angle of elevation of .
The top of Mt Hope is 1500 m above sea level.
Find the distance and bearing of the life raft from Anna’s position.
2003 Extension 1 HSC Q7a)
10916
13923
16 B
H
D
1500 m
16 B
H
D
1500 m
B
H
A
1500 m23
16 B
H
D
1500 m
B
H
A
1500 m23
N
D
B
109
16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
BD
74tan1500
BD
74tan1500BD
74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500
16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500 N”
74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
BN"||ND180,s'cointerior 180" DBNNDB
74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
BN"||ND180,s'cointerior 180" DBNNDB
71"180"109
DBNDBN
74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
BN"||ND180,s'cointerior 180" DBNNDB
71"180"109
DBNDBN
41" Similarly;
ABN
74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
BN"||ND180,s'cointerior 180" DBNNDB
71"180"109
DBNDBN
41" Similarly;
ABN
s'common "" ABNDBNABD30 ABD
16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500 N”
30
30cos74tan150067tan1500274tan150067tan1500 22222 b
30cos74tan150067tan1500274tan150067tan1500 22222 b
metre)nearest (to 279996.2798
b
Anna and David are 2799 m apart.
16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500 N”
30
30cos74tan150067tan1500274tan150067tan1500 22222 b
metre)nearest (to 279996.2798
b
Anna and David are 2799 m apart.
bDAB
30sin
74tan1500sin
30cos74tan150067tan1500274tan150067tan1500 22222 b
metre)nearest (to 279996.2798
b
Anna and David are 2799 m apart.
bDAB
30sin
74tan1500sin
bDAB
30sin74tan1500sin
'51110or '969 DAB
30cos74tan150067tan1500274tan150067tan1500 22222 b
metre)nearest (to 279996.2798
b
Anna and David are 2799 m apart.
bDAB
30sin
74tan1500sin
bDAB
30sin74tan1500sin
'51110or '969 DAB
'5180then '969 If
BDADAB
30cos74tan150067tan1500274tan150067tan1500 22222 b
metre)nearest (to 279996.2798
b
Anna and David are 2799 m apart.
bDAB
30sin
74tan1500sin
bDAB
30sin74tan1500sin
'51110or '969 DAB
'5180then '969 If
BDADAB
BDADAB But '51110 BDA
16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500 N”
30
'51110
30cos74tan150067tan1500274tan150067tan1500 22222 b
metre)nearest (to 279996.2798
b
Anna and David are 2799 m apart.
bDAB
30sin
74tan1500sin
bDAB
30sin74tan1500sin
'51110or '969 DAB
'5180then '969 If
BDADAB
BDADAB But '51110 BDA '51249 is Anna from David of bearing The
2000 Extension 1 HSC Q3c)A surveyor stands at point A, which is due south of a tower OT of height h m. The angle of elevation of the top of the tower from A is 45
The surveyor then walks 100 m due east to point B, from where she measures the angle of elevation of the top of the tower to be 30
(i) Express the length of OB in terms of h.
300
T
B
h
(i) Express the length of OB in terms of h.
300
T
B
h
60tanh
OB
60tanhOB
(i) Express the length of OB in terms of h.
300
T
B
h
60tanh
OB
60tanhOB
(ii) Show that 250h
(i) Express the length of OB in terms of h.
300
T
B
h
60tanh
OB
60tanhOB
(ii) Show that 250h
45 0
T
A
h
(i) Express the length of OB in terms of h.
300
T
B
h
60tanh
OB
60tanhOB
(ii) Show that 250h
45 0
T
A
h
hAOATO
isosceles is
(i) Express the length of OB in terms of h.
300
T
B
h
60tanh
OB
60tanhOB
(ii) Show that 250h
45 0
T
A
h
hAOATO
isosceles is
A B
O
100
tan 60h h
(i) Express the length of OB in terms of h.
300
T
B
h
60tanh
OB
60tanhOB
(ii) Show that 250h
45 0
T
A
h
hAOATO
isosceles is
A B
O
100
tan 60h h
60tan100 2222 hh
(i) Express the length of OB in terms of h.
300
T
B
h
60tanh
OB
60tanhOB
(ii) Show that 250h
45 0
T
A
h
hAOATO
isosceles is
A B
O
100
tan 60h h
60tan100 2222 hh 222 3100 hh
22 1002 h
21002
2 h
2100
h
250h
(iii) Calculate the bearing of B from the base of the tower.
(iii) Calculate the bearing of B from the base of the tower.
A B
O
100
250
N
(iii) Calculate the bearing of B from the base of the tower.
A B
O
100
250
N
250100tan AOB
(iii) Calculate the bearing of B from the base of the tower.
A B
O
100
250
N
250100tan AOB
'4454AOB
(iii) Calculate the bearing of B from the base of the tower.
A B
O
100
250
N
250100tan AOB
'4454AOB'4454180bearing
'16125
(iii) Calculate the bearing of B from the base of the tower.
A B
O
100
250
N
250100tan AOB
'4454AOB'4454180bearing
'16125
Book 2: Exercise 2G odds
Exercise 2H evens
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