Download - 4.7 solutions of triangles
![Page 1: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/1.jpg)
Chapter 4.7 Solutions of Triangles
1
![Page 2: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/2.jpg)
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
![Page 3: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/3.jpg)
-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
![Page 4: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/4.jpg)
x
y
Right Triangles
a
b
c
A
B
C
2 2 2a b c 180A B C
4
![Page 5: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/5.jpg)
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
![Page 6: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/6.jpg)
2 2 2 180
sin cos tan
a b c A B C
opp adj oppA A A
hyp hyp adj
6
![Page 7: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/7.jpg)
Solving Triangles
To means to find the lengths of
all sides and the measures
solve a triang
of all ang
le
les.
7
![Page 8: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/8.jpg)
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
28
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
8
![Page 9: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/9.jpg)
Angle of Elevation
The angle between the horizontal and a line
of sight above the horizontal is cal
angle of eleva
led an
tion.
9
![Page 10: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/10.jpg)
Angle of Depression
The angle between the horizontal and a line
of sight below the horizontal is call
angle of depress
ed an
ion.
10
![Page 11: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/11.jpg)
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.
11
![Page 12: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/12.jpg)
Representation:
Let be the distance of the
cloud from the ground.
Formulation:
tan 67.35135
Solving:
135tan 67.35 324 ft
h
h
h12
![Page 13: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/13.jpg)
Conclusion:
The cloud is approximately
324 ft above the ground.
13
![Page 14: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/14.jpg)
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.
14
![Page 15: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/15.jpg)
Representation:
Let be the angle of depression from
the plane to the house.
Formulation:
475sin
850
B
B15
![Page 16: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/16.jpg)
Solving:
475Arcsin 34
850
Conclusion:
The angle of depression from the airplane is
approximately 34 .
B
16
![Page 17: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/17.jpg)
Oblique Triangles
A triangle with no right angle is call obled ique.
17
![Page 18: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/18.jpg)
a b
cAB
C
18
![Page 19: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/19.jpg)
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
![Page 20: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/20.jpg)
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
![Page 21: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/21.jpg)
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
![Page 22: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/22.jpg)
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
![Page 23: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/23.jpg)
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
23
![Page 24: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/24.jpg)
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
24
![Page 25: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/25.jpg)
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
25
![Page 26: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/26.jpg)
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
26
![Page 27: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/27.jpg)
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
27
![Page 28: 4.7 solutions of triangles](https://reader031.vdocuments.us/reader031/viewer/2022032022/55ac02c61a28ab7c718b48a6/html5/thumbnails/28.jpg)
End of Chapter 4.7
28