4.7 solutions of triangles

Chapter 4.7 Solutions of Triangles

1

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

6

Solving Triangles

To means to find the lengths of

all sides and the measures

solve a triang

of all ang

le

les.

7

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

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

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

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

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.

13

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

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

16

Oblique Triangles

A triangle with no right angle is call obled ique.

17

a b

cAB

C

18

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

23

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

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

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

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

End of Chapter 4.7

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