Trigonometry
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Transcript Trigonometry
Trigonometry
3D Trigonometry
s
p, q and r are points on level ground, [sr] is
a vertical flagpole of height h. The angles
of elevation of the top of the flagpole
from p and q are α and β, respectively.
q
h
β
30º
(i) If | α | = 60º and | β | = 30º,
express | pr | and | qr | in terms of h.
60º
α
p
r
s
s
h
OPP
60º
p
ADJ
h
tan 60
pr
h
3
pr
r
q
h
30º
60º
h
pr
3
p
r
s
h
tan 30
qr
1
h
3 qr
q
30º
ADJ
qr 3h
p
60º
h
OPP
r
(ii) Find | pq | in terms of h, if tan qrp = 8.
s
Pythagoras’ Theorem
x 2 12 ( 8 )2
1 8
9
q
30º
3h
3x
8
A
1
p
1h
cos A
3
r
60º
h
3
a2 = b2 + c2 – 2bccosA
2
h
h 1
qp 3h
2 3h
3
3
3
2
2
2
2
9
h
h
2
h
h
2
3h 2
h2
3
3 3
1
2
cos A
8h
3
3
q
r
2
qp
2
2
8h
3
8
h
3
p
h
3
a2 = b2 + c2 – 2bccosA
The great pyramid at Giza in
Egypt has a square base and
four triangular faces.
The base of the pyramid is
of side 230 metres and the
pyramid is 146 metres high.
slanted edge
The top of the pyramid is directly above the centre of the base.
(i) Calculate the length of one of the slanted edges, correct to the
nearest metre.
Pythagoras’ theorem
x 2 2302 2302
x 105800
x
2
146
162·6
x 325·269..
162·6
2
2006 Paper 2 Q5 (b)
162·6
230 m
230 m
The great pyramid at Giza in
Egypt has a square base and
four triangular faces.
The base of the pyramid is
of side 230 metres and the
pyramid is 146 metres high.
slanted edge
The top of the pyramid is directly above the centre of the base.
(i) Calculate the length of one of the slanted edges, correct to the
nearest metre.
Pythagoras’ theorem
l 2 1462 162·62
l
l
47754·76
l 218·528..
2
146
162·6
219 m
2006 Paper 2 Q5 (b)
162·6 m
146 m
(ii) Calculate, correct to
two significant numbers,
the total area of the four
triangular faces of the
pyramid (assuming they
are smooth flat surfaces)
Pythagoras’ theorem
219 h 115
2
2
2
slanted edge
219 m
h
h2 2192 1152
115 m230 m
h 2 34736 186·375.. 186·4
m
1
Area of triangle base × height
2
1
(230)(186·4)
2
2
21436
m
2006 Paper 2 Q5 (b)
(ii) Calculate, correct to
two significant numbers,
the total area of the four
triangular faces of the
pyramid (assuming they
are smooth flat surfaces)
Pythagoras’ theorem
219 h 115
2
2
2
slanted edge
219 m
h2 2192 1152
115 m
h 2 34736 186·375.. 186·4
m
Total area 21436 4
85744 m2
86000 m2
2006 Paper 2 Q5 (b)
h
s
r
h
θ
p
q
3x
2θ
p
h
tan
3x
r
t
x
h
θ
3x
h 3x tan
pqrs is a vertical wall of height h on level ground. p is a point
on the ground in front of the wall. The angles of elevation of r
from p is θ and the angle of elevation of s from p is 2θ.
| pq | = 3| pt |.
Find θ.
2005 Paper 2 Q5 (c)
q
s
s
h
p
2θ
x
2θ
t
p
h
tan 2
x
r
t
x
h
θ
3x
h x tan 2
pqrs is a vertical wall of height h on level ground. p is a point
on the ground in front of the wall. The angles of elevation of r
from p is θ and the angle of elevation of s from p is 2θ.
| pq | = 3| pt |.
Find θ.
2005 Paper 2 Q5 (c)
q
s
2 tan
tan 2
1 tan 2
2 tan
3xtan θ xtan 2θ
1 tan 2
2θ
p
3t 1 t
t 3t 0
t 1 3t
1
t
3
2
x
h
θ
q
3x
Let t = tan θ t 0
2t
3t
2
1 t
3
r
t
2
2
2t
3t 3t 2t
0
1 3t 0
1
t tan
3
2005 Paper 2 Q5 (c)
3
2
6
d
abc is an isosceles triangle on a horizontal
plane, such that |ab| |ac| 5 and |bc| 4.
m is the midpoint of [bc].
2
5
b
A
5
(i) Find | bac | to the nearest degree.
a 2 b2 c 2 2bc cos A
m4
42 52 52 2(5)(5)cos A
16 25 25 50cos A
16 50 50cos A
c
50cos A 50 16
34
cos A
50
A 47·156....
A 47
a
d
abc is an isosceles triangle on a horizontal
plane, such that |ab| |ac| 5 and |bc| 4.
m is the midpoint of [bc].
(ii) A vertical pole [ad] is erected
at a such that |ad | 2, find
|amd | to the nearest degree.
am 2 5
2
2
5
b
2
m
am 25 4
am 21
amd
2
21
a
21 5
2
2
2
tan amd
2
c
amd 23·578.. 24