Transcript Trigonometry 1 (Right-angled Triangles)

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.
?
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
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
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
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.
?
45o
What’s he
going to do
next?
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.
?
45o
What’s he
going to do
next?
324 m
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
Trigonometry
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. 1 H 2
H
2 x
f (x )  cons tantx (H  x ) 

x
xw (x )f (x )dx
324 m
The Trigonometric Ratios
A
adjacent
C
B
hypotenuse
Sine A 
B
opposite
opposite
hypotenuse
C
Opposite
Hypotenuse
SinA 
O
H
Adjacent
Hypotenuse
CosA 
A
H
Opposite
Adjacent
TanA 
O
A
Cosine A 
Tangent A 
adjacent
Make up a Mnemonic!
A
S O H C A H T O A
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
CB
Diagrams
Sin 700 
70o
12 cm
12
not to
scale.
12Sin 700  CB  11.3 cm (1dp )
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
 MN 
Tan 750 
Tan 750
MN
75o
MN  1.2 m (1dp )
N
The Trigonometric Ratios (Finding an unknown angle).
True Values (2 dp)
Sin 30o = 0.50
Cos 30o = 0.87
Tan 30o = 0.58
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.
S O H C A H T O A
Tanx 0 
xoo
30
75 m
43.5
 0.58
75
43.5 m
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
12 cm
C
11.3 cm
11.3
Sin A 
12
Key Sequence
Sin-1(11.3  12) =
0

Angle
A

70
(nearest deg ree )
B
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
o

Angle
N

7
4
(nearest degree)
N
to scale.
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 
-1(7.2  7.8) =
Cos
7.8
Q
R
7.2 cm
 Angle Q  23o (nearest degree)
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.
H
15 miles
B
15
Tan L 
6.4
 Angle L  66.90
6.4 miles
Bearing  360  66.9  293o
L
SOH CAH TOA
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
o
 Angle L  52
12 ft
9.5 ft
Lo
SOH CAH TOA
Applications of Trigonometry
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.
Not to Scale
P
(a) Make a sketch of the flight.
(b) Find the bearing of Q from P.
570 miles
430
Cos P 
570
430 miles
 Angle P  41o
Bearing  180  41  221
0
Q
W
SOH CAH TOA
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
Explain why the angles of
elevation and depression are
always equal.
25o
Angle of elevation
Horizontal
25o
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.
Tan P 
20
45
Angle P  240 (nearest deg ree )
20 m
45 m
P
SOH CAH TOA
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 deg ree )
C
D
55 m
100 m
D
Or more directly since the angles of elevation
and depression are equal.
SOH CAH TOA
Tan D 
55
 Angle D  29o
100
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
SOH CAH TOA
Or more directly since the angles of elevation
and depression are equal. Tan 30  57
x
x 
57
 99m
Tan 30