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Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry
?
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry
30o
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry
35o
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry
40o
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry
45o
?What’s he
going to do next?
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry
45o
324 m
?What’s he
going to do next?
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.
Trigonometry
45o
324 m
324 m
Trigonometry
324 m
Eiffel Tower Facts:
•Designed by Gustave Eiffel.
•Completed in 1889 to celebrate the centenary of the French Revolution.
•Intended to have been dismantled after the 1900 Paris Expo.
•Took 26 months to build.
•The structure is very light and only weighs 7 300 tonnes.
•18 000 pieces, 2½ million rivets.
•1665 steps.
•Some tricky equations had to be solved for its design. 21
( ) tan ( ) ( ) ( )2
H H
x xf x cons tx H x xw x f x dx
The Trigonometric Ratios
A
BC
hypotenuse
opposite
A
B C
hypotenuse
opposite
adjacent
adjacent
Opposite
Sine AHypotenuse
OSinA
H
Adjacent
Cosine AHypotenuse
CosA
AH
Opposite
Tangent AAdjacent
TanO
AA
Make up a Mnemonic!
S O CH A H T O A
The Trigonometric Ratios (Finding an unknown side).
Example 1. In triangle ABC find side CB.
70o
A
BC
12 cmDiagrams
not to scale.
S O H C A H T O A
Opp
07012CB
Sin
0 11.12 70 (13 ) Sin CB pcm d
Example 2. In triangle PQR find side PQ.
22o
P
QR7.2 cm
S O H C A H T O A
0 7.222Cos
PQ 0
7.222
PQCos
1 )7.8 (cmPQ dp
Example 3. In triangle LMN find side MN.
75o
LM
N
4.3 m
S O H C A H T O A
0 4.375Tan
MN 0
4.375
MNTan
1 )1.2 (mMN dp
xo
43.5 m
75 m
Anytime we come across a right-angled triangle containing 2 given sides we can calculate the ratio of the sides then look up (or calculate) the angle that corresponds to this ratio.
Sin 30o = 0.50
Cos 30o = 0.87Tan 30o = 0.58
True Values (2 dp)
S O H C A H T O A
The Trigonometric Ratios (Finding an unknown angle).
0 43.575
0.58Tanx
30o
Example 1. In triangle ABC find angle A.A
BC
12 cm
11.3 cm
Diagrams not to scale.
The Trigonometric Ratios (Finding an unknown angle).
S O H C A H T O A
11.3Sin
12A
0 ( deg )70 Angle A nearest ree
Sin-1(11.3 12) =
Key Sequence
S O H C A H T O A
4.3
1.2Tan N
N (neares7 t degree4 )oAngle
Example 2. In triangle LMN find angle N. L
M
N
4.3 m
1.2 mTan-1(4.3 1.2) =
Key Sequence
S O H C A H T O A
7.2 Q
7.8Cos
23 ( degree)oAngle Q nearest
Example 3. In triangle PQR find angle Q. P
QR7.2 cm
7.8 cm
Cos-1(7.2 7.8) =
Key Sequence
Applications of Trigonometry
A boat sails due East from a Harbour (H), to a marker buoy (B), 15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. Make a sketch of the trip and calculate the bearing of the harbour from the lighthouse to the nearest degree.
15
6.4
Tan L
HB
L
15 miles
6.4 miles 0Angle 66.9L
360 66.9 293oBearing
SOH CAH TOA
9.5
12
Sin L
12 ft9.5 ft
A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. Calculate the angle that the foot of ladder makes with the ground.
Applications of Trigonometry
Lo
SOH CAH TOA
52oAngle L
Not to Scale
P
570 miles
W
430 miles
Q
An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 430 miles North to a point P before turning left and flying for 570 miles to a second point Q, West of W. It then returns to base.
(a) Make a sketch of the flight.
(b) Find the bearing of Q from P.
Applications of Trigonometry
SOH CAH TOA
430
Cos P570
0180 41 221Bearing
P 41oAngle
Angles of Elevation and Depression.
An angle of elevation is the angle measured upwards from a horizontal to a fixed point. The angle of depression is the angle measured downwards from a horizontal to a fixed point.
Horizontal25o
Angle of elevation
Horizontal
25oAngle of depression
Explain why the angles of elevation and depression are always equal.
A man stands at a point P, 45 m from the base of a building that is 20 m high. Find the angle of elevation of the top of the building from the man.
Applications of Trigonometry
20 m
45 m P
SOH CAH TOA
20
P45
Tan
02Angle P (4 deg )nearest ree
A 25 m tall lighthouse sits on a cliff top, 30 m above sea level. A fishing boat is seen 100m from the base of the cliff, (vertically below the lighthouse). Find the angle of depression from the top of the lighthouse to the boat.
100 m
55 m
DC
100
C55
Tan
00 29 D 90 61.2 ( deg )Angle nearest ree
C 61.2oAngle
D
Or more directly since the angles of elevation and depression are equal.
55
D 29100
oTan Angle DSOH CAH TOA
A 22 m tall lighthouse sits on a cliff top, 35 m above sea level. The angle of depression of a fishing boat is measured from the top of the lighthouse as 30o. How far is the fishing boat from the base of the cliff?
x m
57 m
30o
6057x
Tan
57
30Tanx
60o
99m
57 60
= (nearest m)
x Tan
57
9930
x mTan
Or more directly since the angles of elevation and depression are equal.
30o
SOH CAH TOA