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Homework, Page 484. Solve the triangle. 1. Homework, Page 484. Solve the triangle. 5. Homework, Page 484. Solve the triangle. 9. Homework, Page 484. State whether the given measurements determine zero, one, or two triangles. 13. Homework, Page 484. - PowerPoint PPT Presentation
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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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 2
Homework, Page 484Solve the triangle.
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 3
Homework, Page 484Solve the triangle.
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 4
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 5
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 6
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 7
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 8
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 9
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 10
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 11
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 12
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 13
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 14
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 15
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 16
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 18
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 19
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 20
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 21
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 22
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 23
Example Solving a Triangle (SAS)
Solve given that 10, 4 and 25 .ABC a b C
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 24
Example Solving a Triangle (SSS)
Solve given that 10, 4 and 2.ABC a b c
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 25
Area of a Triangle
1Area sin
21
Area sin21
Area sin2
bc A
ac B
ab C
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 26
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 27
Example Using Heron’s Formula
Find the area of a triangle with sides 10, 12, 14.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 28
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 29
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 30
HomeworkHomework Assignment #1Review Section 5.6Page 494, Exercises: 1 – 53 (EOO)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 31
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 32
Directed Line Segment
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 33
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 34
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 35
Initial Point, Terminal Point, Equivalent
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 36
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 37
Example Finding Magnitude of a Vector
Find the magnitude of represented by , where (3, 4)
and (5,2).
PQ P
Q
������������� �v
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 38
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 39
Example Performing Vector Operations
Let 2, 1 and 5,3 . Find 3 . u v u v
x
y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 40
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 41
Example Finding a Unit Vector
Find a unit vector in the direction of 2, 3 . v
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 42
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 43
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 .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 44
Example Finding the Components of a Vector
Find the components of the vector with direction angle 120
and magnitude 8.
v
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 45
Example Finding the Direction Angle of a Vector
Find the magnitude and direction angle of 2,3 .
u
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 46
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 47
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 48
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?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5- 49
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.