Homework, Page 484

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 1

Homework, Page 484Solve the triangle.

1.

180 180 60 45 75

sin 3.7sin 604.532

sin sin sin sin sin 45sin 3.7sin 75

5.054sin sin 45

75 ; 4.532; 5.054

C A B

a b c b Aa

A B C Bb C

cB

C a c

C

A Bb

3.7 a

60 45



5.

180 180 40 30 110

sin 10sin 4012.856

sin sin sin sin sin30sin 10sin110

18.794sin sin30

110 ; 12.856; 18.794

C A B

a b c b Aa

A B C Bb C

cB

C a c

40 0A b



9.

1 1

sin sin sin sinsin

sin 11sin32sin sin 20.053

17

180 32 20.053 127.947

sin 17sin127.94725.298

sin sin32

20.053, 127.947, 25.298

A B C b AB

a b c ab A

Ba

C

a Cc

A

B C c

32 17, 11A a b


Homework, Page 484State whether the given measurements determine zero, one, or two triangles.

13. 36 2, 7A a b

36 2, 7

sin 36 sin 36 7sin 36

zero triangles

A a b

hh b

b


Homework, Page 484State whether the given measurements determine zero, one, or two triangles.

17. 30 18, 9C a c

30 18, 9

sin 30 sin 36 18sin 30

one triangle

C a c

hh a

a


Homework, Page 484Two triangles can be formed using the given measurements. Find both triangles.

21. 68 19, 18C a c

1

1

68 , 19, 18

sin sin sin sinsin sin

19sin 68sin 78.152 or 180 78.152 101.848

18

180 68 78.152 33.848 or 10.152

sin 18sin33.848 18sin10.10.813 or

sin sin 68

C a c

A C a C a CA A

a c c c

A

B

a Bb

A

152

3.422sin 68


Homework, Page 484Decide whether the triangle can be solved using the Law of Sines. If so, solve it, if not, explain why not.

25.

Neither triangle can be solved using the Law of Sines, for the one on the left we need to know the length of the side opposite the known angle and for the one on the right, we have the same problem.

B

A C

a

23

19

56

B

A C

a

b

19

56


Homework, Page 484Respond in one of the following ways:

(a) State: “Cannot be solved with Law of Sines.”

(b) State: “No triangle is formed.”

(c) solve the triangle.

29.

No triangle is formed. The largest side of a triangle is opposite the largest angle and angle A must be the largest angle and side a is no the largest side.

136 , 15, 28A a b


Homework, Page 484Respond in one of the following ways:

(a) State: “Cannot be solved with Law of Sines.”

(b) State: “No triangle is formed.”

(c) solve the triangle.

33. 75 , 49, 48C b c

1

75 , 49, 48

49sin 75sin 80.418

48sin20.885

sin 75

C b c

B

a


Homework, Page 48437. Two markers A and B on the same side of a canyon rim are 56 ft apart. A third marker C, located on the opposite rim, is positioned so that

(a) Find the distance between C and A.

(b) Find the distance between the canyon rims.

56sin53180 72 53 55 54.597

sin55C b ft

72 and BAC 53ABC

sin 72 54.597sin 72 51.92554.597

dd ft


Homework, Page 48441. A 4-ft airfoil attached to the cab of a truck makes an 18º angle with the roof and angle β is 10º. Find the length of the vertical brace positioned as shown.

sin 28 4sin 28 1.8784

ll ft


Homework, Page 48445. Two lighthouses A and B are known to be exactly 20 mi apart. A ship’s captain at S measures the angle S at 33º. A radio operator measures the angle B at 52º. Find the distance from the ship to each lighthouse.

180 33 52 95

20sin5228.937

sin3320sin95

36.582sin33

A

AS mi

BS mi


Homework, Page 48449. The length x in the triangle is

(A) 8.6

(B) 15.0

(C) 18.1

(D) 19.2

(E) 22.6

180 95 53 32

12.0sin5318.085

sin32x

95

12.0

53

x


Homework, Page 48453. (a) Show that there are infinitely many triangles with AAA given if the sum of the three positive angles is 180º.

Consider the triangle formed with its base on a radius that is one-half the diameter of a semi-circle. If the opposite ends of the radius are connected to a point on the semi-circle, a triangle is formed. Since there are an infinite number of possible values of the radius, there must be an infinite number of possible triangles.


Homework, Page 48453. (b) Give three examples of triangles where A = 30º, B = 60º, and C = 90º.

(c) Give three examples where A = B = C = 60º.

1, 3, 2; 10, 10 3, 20

100, 100 3, 200

a b c a b c

a b c

1; 2; 3a b c a b c a b c


Homework, Page 48457. Towers A and B are known to be 4.1 mi apart on level ground. A pilot measures the angles of depression to the towers at 36.5º and 25º, respectively. Find distances AC and BC and the height of the aircraft. C

A B4.1 mi

2536.5

25 , 11.5 180 36.5 143.5

4.1sin 258.691

sin11.54.1sin143.5

12.233sin11.5

12.233sin 25 5.170

B C A

AC mi

BC mi

h mi

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

5.6

The Law of Cosines


Quick Review

2 2 2

2 2

Find an angle between 0 and 180 that is a solution to the equation.

1. cos 4 / 5

2. cos -0.25

Solve the equation (in terms of and ) for (a) cos and

(b) , 0 180 .

3. 7 2 cos

4. 4

A

A

x y A

A A

x y xy A

y x

4 cos

5. Find a quadratic polynomial with real coefficients that has no real zeros.

x A


Quick Review Solutions

2

Find an angle between 0 and 180 that is a solution to the equation.

1. cos 4 / 5

2. cos -0.25

Solve the equation (in terms of and ) for (a) cos and

(b) , 0 180 .

3.

36.87

104.48

7

A

A

x y A

A A

2 2 2 2

-12 2

2

2 2 2 2

-12

49 49(a) (b) cos

2 2

4 4(a) (b) cos

2 cos

4. 4 4 cos

5. Find a quadratic polynomial with real coefficients that has no

4

ea

4

r

x y xy A

y x x

x y x y

xy xy

y x y x

x xA

2One answer

l ze

:

r s.

2

o

x


What you’ll learn about

Deriving the Law of Cosines Solving Triangles (SAS, SSS) Triangle Area and Heron’s Formula Applications

… and whyThe Law of Cosines is an important extension of the Pythagorean theorem, with many applications.


Deriving the Law of CosinesC (x, y)

B (c, 0)cA

ba

C (x, y)

B (c, 0)cA

ba

C (x, y)

B (c, 0)cA

b a


Law of Cosines

2 2 2

2 2 2

2 2 2

Let be any triangle with sides and angles

labeled in the usual way. Then

2 cos

2 cos

2 cos

ABC

a b c bc A

b a c ac B

c a b ab C


Example Solving a Triangle (SAS)

Solve given that 10, 4 and 25 .ABC a b C


Example Solving a Triangle (SSS)

Solve given that 10, 4 and 2.ABC a b c


Area of a Triangle

1Area sin

21

Area sin21

Area sin2

bc A

ac B

ab C


Heron’s Formula

Let , , and be the sides of , and let denote

the , ( ) / 2. Then, the area of

is given by Area - .

a b c ABC s

a b c

ABC s s a s b s c

semiperimeter


Example Using Heron’s Formula

Find the area of a triangle with sides 10, 12, 14.


Example Finding the Area of a Regular Circumscribed Polygon

Find the area of a regular nonagon (9-sided) circumscribed about a circle of radius 10 in.


Example Surveyor’s ProblemTony must find the distance from point A to point B on opposite sides of a lake. He finds point C which is 860 ft from point A and 175 ft from point B. If he measures the angle at point C between points A and B as 78º, what is the distance between points A and B.


HomeworkHomework Assignment #1Review Section 5.6Page 494, Exercises: 1 – 53 (EOO)


What you’ll learn about

Two-Dimensional Vectors Vector Operations Unit Vectors Direction Angles Applications of Vectors

… and whyThese topics are important in many real-world applications, such as calculating the effect of the wind on an airplane’s path.


Directed Line Segment


Two-Dimensional Vector

A may be written as an ordered

pair of real numbers, denoted in as , .

The numbers and are the of the vector .

The o

a b

a b

two - dimensional vector

component form

components

standard representation

v

v

f the vector , is the arrow

from the origin to the point ( , ). The of is the

length of the arrow, and the of is the direction in

which the arrow is pointing. The vector

a b

a b magnitude

direction

v

v

= 0,0 , called the

, has zero length and no direction.

0

zero vector


Two-Dimensional Vector

1 1 2 2

2 1 2 1

If an arrow has initial point , and terminal point , , it

represents the vector , .

Any two arrows of the same length and pointing in the same

direction, represent the same vector. Tr

x y x y

x x y y

anslation of a vector

does not change either its magnitude nor its direction.


Initial Point, Terminal Point, Equivalent


Magnitude

1 1 2 2

2 2

2 1 2 1

2 2

If is represented by the arrow from , to , , then

.

If , , then .

x y x y

x x y y

a b v a b

v

v

v


Example Finding Magnitude of a Vector

Find the magnitude of represented by , where (3, 4)

and (5,2).

PQ P

Q

�� v


Vector Addition and Scalar Multiplication

1 2 1 2

1 1 2 2

Let , and , be vectors and let be a real number

(scalar). The (or ) is the vector

, .

Graphically, two vectors may be added by placi

u u v v k

u v u v

sum resultant of the vectors and

u v

u v

u v

ng the tail of one on the

head of the other. The vector obtained by connecting the tail of the

second to the head of the first is the resultant vector. This is sometimes

called the parallelogram met

1 2 1 2

hod.

In the case of the vector , the vector has the same magnitude and

opposite direction as .

The and the vector is

, , .

u u

u

k k u u ku ku

product of the scalar

k u

u


Example Performing Vector Operations

Let 2, 1 and 5,3 . Find 3 . u v u v

x

y


Unit Vectors

A vector with | | 1 is a . If is not the zero vector

10,0 , then the vector is a

| | | |

.

unit vector

unit vector in the direction

of

u u v

vu v

v v

v


Example Finding a Unit Vector

Find a unit vector in the direction of 2, 3 . v


Standard Unit Vectors

The two vectors 1,0 and 0,1 are the standard unit

vectors. Any vector can be written as an expression in terms

of the standard unit vector:

,

,0 0,

1,0 0,1

a b

a b

a b

a b

i j

v

v

i j


Resolving the Vector

If has direction angle , the components of can be computed

using the formula = cos , sin .

From the formula above, it follows that the unit vector in the

direction of is cos ,si

v v

v v v

vv u

vn .


Example Finding the Components of a Vector

Find the components of the vector with direction angle 120

and magnitude 8.

v


Example Finding the Direction Angle of a Vector

Find the magnitude and direction angle of 2,3 .

u


Velocity and Speed

The velocity of a moving object is a vector

because velocity has both magnitude and

direction. The magnitude of velocity is speed.


Example Writing Velocity as a Vector

An aircraft is flying on course 073 at 450 kts. Find the

component form of the aircraft's velocity.


Example Calculating the Effects of Wind Velocity

An aircraft is flying on course 103 at 450 kts. The wind at the

arrcraft's altitude is blowing from 060 at 75 kts. What are

the aircraft's course and speed made good?


Example Finding the Direction and Magnitude of the Resultant Force

Three forces with magnitudes 100, 50, and 80 lb act on an object

at angles of 50 , 160 , and 20 , respectively. Find the resultant

force.

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