TRIGONOMETRY (applications)

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Sharp Tutor Mon, Jan 17, 2022 09:27 PM UTC

We will be discussing some applications of trigonometry related to measurements of heights and distances in the natural world.

1. 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.
Discuss how the following
sequence of diagrams allows
us to determine the height
of the Eiffel Tower without
actually having to climb it.
?

2. 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.
Discuss how the following
sequence of diagrams allows
us to determine the height
of the Eiffel Tower without
actually having to climb it.
30o

3. 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.
Discuss how the following
sequence of diagrams allows
us to determine the height
of the Eiffel Tower without
actually having to climb it.
35o

4. 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.
Discuss how the following
sequence of diagrams allows
us to determine the height
of the Eiffel Tower without
actually having to climb it.
40o

5. 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.
Discuss how the following
sequence of diagrams allows
us to determine the height
of the Eiffel Tower without
actually having to climb it.
?
What’s he going
to do next?
45o

6. 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.
Discuss how the following
sequence of diagrams allows
us to determine the height
of the Eiffel Tower without
actually having to climb it.
?
What’s he going
to do next?
45o
324 m

7. 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.
324 m
45o
324 m

8. Trigonometry
Eiffel Tower Facts:
•Designed by Gustave Eiffel.
•Completed in 1889 to celebrate the centenary of 324 m
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
•18 000 pieces, 2½ million rivets.
•1665 steps.
•Some tricky equations had to be solved for its
design. 1 H f 2 (x )  cons tantx (H  x )  H xw (x )f (x )dx
2
x x

9. The Trigonometric Ratios
A
hypotenuse Opposite O
adjacent Sine A  SinA 
Hypotenuse H
C opposite B Adjacent A
Cosine A  CosA 
Hypotenuse H
B opposite C Opposite O
Tangent A  TanA 
Adjacent A
adjacent
hypotenuse
Make up a Mnemonic!
A
S O H C A H T O A

10. The Trigonometric Ratios (Finding an unknown side).
Example 1. In triangle ABC find side CB. S O H C A H T O A
A
0 CB Diagrams
70 o
12 cm Sin 70 
12 not to
12Sin 700 CB 11.3 cm (1dp ) scale.
C B
Opp
Example 2. In triangle PQR find side PQ. S O H C A H T O A
P 7.2 7.2
Cos 220   PQ 
PQ Cos 220
22o Q PQ 7.8 cm (1dp )
R
7.2 cm
Example 3. In triangle LMN find side MN. S O H C A H T O A
L 4.3 m
4.3 4.3
M Tan 750   MN  0
MN Tan 75
75o MN 1.2 m (1dp )
N

11. The Trigonometric Ratios (Finding an unknown angle).
True Values (2 dp) Anytime we come across a right-angled
triangle containing 2 given sides we can
Sin 30o = 0.50
calculate the ratio of the sides then
Cos 30o = 0.87 look up (or calculate) the angle that
corresponds to this ratio.
Tan 30o = 0.58
S O H C A H T O A
Tanx 0 
43.5
0.58
43.5 m
75
xoo
30
75 m

12. The Trigonometric Ratios (Finding an unknown angle).
Example 1. In triangle ABC find angle A. S O H C A H T O A
A
Key Sequence
12 cm 11.3
Sin A 
12 Sin-1(11.3  12) =
0
C B  Angle A  70 (nearest degree )
11.3 cm
Example 2. In triangle LMN find angle N. S O H C A H T O A
L 4.3 m
Key Sequence
M 4.3
Tan N  Tan-1(4.3  1.2) =
1.2
1.2 m
Diagrams not
N
o
to scale.  Angle N  74 (nearest degree)
Example 3. In triangle PQR find angle Q. S O H C A H T O A
P 7.8 cm Key Sequence
7.2
Cos Q 
Q 7.8 Cos-1(7.2  7.8) =
R
7.2 cm  Angle Q 23o (nearest degree)

13. 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
H miles
B
15
Tan L 
6.4 6.4
miles
 Angle L 66.90
Bearing 360  66.9 293o
L
SOH CAH TOA

14. Applications of Trigonometry
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.
9.5
Sin L 
12
12 ft
o
 Angle L 52 9.5
ft
Lo
SOH CAH TOA

15. Applications of Trigonometry
An AWACS aircraft takes off from RAF
Not to Scale P
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.
570 miles
430 miles
430
Cos P 
570
 Angle P  41o
0
Q W
Bearing 180  41 221
SOH CAH TOA

16. 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.
Horizontal
Angle of depression 25o
Explain why the angles of
elevation and depression are
always equal.
Angle of elevation
25o
Horizontal

17. Applications of Trigonometry
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.
20
Tan P 
45
Angle P 240 (nearest degree )
20 m
45 m P
SOH CAH TOA

18. 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
Tan C 
55
Angle C 61.2o
Angle D  90  61.20 290 (nearest degree )
D
C
55 m
D
100 m
Or more directly since the angles of elevation
and depression are equal.
55
Tan D   Angle D 29o
SOH CAH TOA 100

19. 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
Tan 60 
57
x 57Tan 60
=99m (nearest m)
30o
60o
57 m
30o
xm
Or more directly since the angles of elevation
and depression are equal. Tan 30  57
x
SOH CAH TOA  x 
57
 99m
Tan 30

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