4.7 solutions of triangles
TRANSCRIPT
Chapter 4.7 Solutions of Triangles
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Trigonometric Functions
Let be an angle whose initial side is the
positive axis. If , is a point on the
terminal side that is units away from the origin, then the si trigonometric functiox are defined as
sin cos
ns
y
r
x x y
r
tan
csc sec cot
yxr x
r r xy x y
2
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
,x yr
3
x
y
Right Triangles
a
b
c
A
B
C
2 2 2a b c 180A B C
4
x
y
a
b
c
A
B
C
,b asin
cos
tan
a oppA
c hyp
b adjA
c hyp
a oppA
b adj
5
2 2 2 180
sin cos tan
a b c A B C
opp adj oppA A A
hyp hyp adj
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Solving Triangles
To means to find the lengths of
all sides and the measures
solve a triang
of all ang
le
les.
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Example 4.7.1
Solve the triangle where 61.7 , 90 , and 106.2.
180 61.7 9
sin 61.7 cos 61.7106.2 106.2
106.2sin 61.7 93.5 106.2cos 61.7
0
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50.3
180
sin co
.
s
3
A B C
opp adjA A
hy
A
a b
a
C c
p hyp
b
B
61.7A 90C
106.2c a
B
b
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Angle of Elevation
The angle between the horizontal and a line
of sight above the horizontal is cal
angle of eleva
led an
tion.
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Angle of Depression
The angle between the horizontal and a line
of sight below the horizontal is call
angle of depress
ed an
ion.
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Example 4.7.2
To measure cloud altitude at night, a vertical beam of light is directed on a spot on the cloud. From a point 135 ft away from the light source, the angle of elevation to the spot is found to be 67.35o. Find the distance of the cloud from the ground.
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Representation:
Let be the distance of the
cloud from the ground.
Formulation:
tan 67.35135
Solving:
135tan 67.35 324 ft
h
h
h12
Conclusion:
The cloud is approximately
324 ft above the ground.
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Example 4.7.3
An aerial photographer who photographs farm properties for a real estate company has determined from experience that the best photo is taken at a height of approximately 475 ft and a distance of 850 ft from the farmhouse. What is the angle of depression from the plane to the house.
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Representation:
Let be the angle of depression from
the plane to the house.
Formulation:
475sin
850
B
B15
Solving:
475Arcsin 34
850
Conclusion:
The angle of depression from the airplane is
approximately 34 .
B
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Oblique Triangles
A triangle with no right angle is call obled ique.
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a b
cAB
C
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Law of Sines
In any triangle, the ratio of a side and the
sine of the opposite angle is a constant.
sin sin sin
sin sin sin
a b c
A B C
A B C
a b c19
a b
cAB
C
Law of Cosines
2 2 2
2 2 2
In a triangle, the square of a side is the sum ofthe squares of the other two sides, minus twicethe product of those sides and the cosine of theincluded angle.
2 cos
2 cos
c a b ab C
a b c bc A
2 2 2 2 cosb a c ac B
20
a b
cAB
C
Example 4.7.4 Solve the following triangles.
1. 12, 45 , 75
180
180
sin s
45 75 60
12
sin45 sin75
12sin7516.4
sin45
in
a
A B C
a
A C
B
c
A
c
C
c
21
12, 45 , 75
60
16.4
sin sin
12
sin45 sin75
12sin7514.7
sin45
a A C
B
c
a b
A B
b
b
22
2 2 2
2 22
2 2 2
2 2
2 2
2 2
2
2
2
2. 32, 48, 125.2
32 48 2 32 48 cos125.2
71.4
32 71.4 48 2 71.4 48 cos
2 71.4 48 cos 71.4 48
71.4 48cos
2 71.4 48
71.4 48Arccos 21.5
2 71.4
2 co
8
s
c
4
2 os
a c B
b
b
A
A
b a c ac B
a b c A
A
bc
A
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32, 48, 125.2
71.4
21.5
180 21.5 125.2 3
80
3.3
1A B
a
b
A
C
C
c B
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2 2 2
2
2 2
2 2
2 2 2
2
3. 3.5, 4.7, 2.8
3.5 4.7 2.8 2 4.7 2.8 cos
2 4.7 2.8 cos 4.7 2.8 3.5
4.7 2.8 3.5cos
2 4.7 2.8
47.80
2 cos
a b c
A
a b c bc
A
A
A
A
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2 2 2
2 2
2
2
2 2
2 2
2
3.5, 4.7, 2.8
47.80
4.7 3.5 2.8 2 3.5 2.8 cos
2 3.5 2.8 cos 3.5 2.8 4.7
3.5 2.8 4.7cos
2 3.5 2.8
95
2 co
.86
s
a b c
A
B
B
B
b a c ac B
B
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3.5, 4.7, 2.8
47.80
95.86
180 47.80 95.86 36.34
180A B
a b c
A
B
C
C
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End of Chapter 4.7
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