11 x1 t04 07 3d trigonometry (2013)
TRANSCRIPT
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3D Trigonometry
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3D TrigonometryWhen doing 3D trigonometry it is often useful to redraw all of the faces of the shape in 2D.
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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
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16 B
H
D
1500 m
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16 B
H
D
1500 m
B
H
A
1500 m23
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16 B
H
D
1500 m
B
H
A
1500 m23
N
D
B
109
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16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
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16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
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74tan1500
BD
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74tan1500
BD
74tan1500BD
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74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
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16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500
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16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500 N”
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74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
BN"||ND180,s'cointerior 180" DBNNDB
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74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
BN"||ND180,s'cointerior 180" DBNNDB
71"180"109
DBNDBN
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74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
BN"||ND180,s'cointerior 180" DBNNDB
71"180"109
DBNDBN
41" Similarly;
ABN
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74tan1500
BD
74tan1500BD 67tan1500 Similarly;
AB
BN"||ND180,s'cointerior 180" DBNNDB
71"180"109
DBNDBN
41" Similarly;
ABN
s'common "" ABNDBNABD30 ABD
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16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500 N”
30
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30cos74tan150067tan1500274tan150067tan1500 22222 b
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30cos74tan150067tan1500274tan150067tan1500 22222 b
metre)nearest (to 279996.2798
b
Anna and David are 2799 m apart.
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16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500 N”
30
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30cos74tan150067tan1500274tan150067tan1500 22222 b
metre)nearest (to 279996.2798
b
Anna and David are 2799 m apart.
bDAB
30sin
74tan1500sin
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30cos74tan150067tan1500274tan150067tan1500 22222 b
metre)nearest (to 279996.2798
b
Anna and David are 2799 m apart.
bDAB
30sin
74tan1500sin
bDAB
30sin74tan1500sin
'51110or '969 DAB
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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
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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
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16 B
H
D
1500 m
B
H
A
1500 m23
N
N’
D
B
109
A 139
b
74tan1500
67tan1500 N”
30
'51110
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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
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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
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(i) Express the length of OB in terms of h.
300
T
B
h
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(i) Express the length of OB in terms of h.
300
T
B
h
60tanh
OB
60tanhOB
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(i) Express the length of OB in terms of h.
300
T
B
h
60tanh
OB
60tanhOB
(ii) Show that 250h
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(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
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(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
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(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
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(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
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(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
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(iii) Calculate the bearing of B from the base of the tower.
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(iii) Calculate the bearing of B from the base of the tower.
A B
O
100
250
N
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(iii) Calculate the bearing of B from the base of the tower.
A B
O
100
250
N
250100tan AOB
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(iii) Calculate the bearing of B from the base of the tower.
A B
O
100
250
N
250100tan AOB
'4454AOB
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(iii) Calculate the bearing of B from the base of the tower.
A B
O
100
250
N
250100tan AOB
'4454AOB'4454180bearing
'16125
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(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