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Trig Lecture 4
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5/21/2018 Trig Lecture 4
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Lecture FourApplications of Trigonometry
Section 4.1Law of Sines
Oblique Triangle
A triangle that is not a right triangle, either acute or obtuse.
The measures of the three sides and the three angles of a triangle can be found if at least one side and
any other two measures are known.
The Law of Sines
There are many relationships that exist between the sides and angles in a triangle.
One such relation is called the law of sines.
Given triangle ABC
sin sin sinA B C
a b c
or, equivalently
sin sin sin
a b c
A B C
Proof
b
hA sin sin (1)h b A
a
hB sin sin (2)h a B
From (1) & (2)
h h
BaAb sinsin
ab
Ba
ab
Ab sinsin
b
B
a
A sinsin
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AngleSide - Angle (ASA or AAS)
If two angles and the included side of one triangle are equal, respectively, to two angles and the
included side of a second triangle, then the triangles are congruent.
Example
In triangleABC, 30A , 70B , and cma 0.8 . Find the length of side c.
Solution
)(180 BAC
)7030(180
100180
80
AC
ac
sinsin
CA
c a
sinsin
80sin30sin
8
cm16
Example
Find the missing parts of triangleABCif 32A , 8.81C , , and cma 9.42 .
Solution
)(180 CBB )8.8132(180
2.66
sin sin
a b
A B
sin
sin
a
Ab
B
32sin
2.66sin9.42
cm.147
sin sin
c a
C A
sin
sin
a
Ac
C
42.9sin 81.8
sin32
80.1 cm
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Example
You wish to measure the distance across a River. You determine
that C= 112.90,A= 31.10, and 347.6 tb f . Find the
distance aacross the river.
Solution
180B A C 180 31.10 112.90
36
sin sin
a b
A B
347.6
sin 31.1 sin 3 6
a
31.136
347.6sin
sina
305. 5 ta f
Example
Two ranger stations are on an east-west line 110 mi apart. A forest fire is located on a bearing N 42 E
from the western station atAand a bearing of N 15 E from the eastern station atB. How far is the fire
from the western station?
Solution
90 42 48BAC
90 15 105ABC
180 105 48 27C
sin sin
b c
B C
110
sin 105 sin 27
b
110sin105
sin 27
b
423b mi
The fire is about 234 miles from the western station.
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Example
Find distancexif a= 562ft., 7.5B and 3.85A
Solution
AB
ax
sinsin
Ax
Ba
sin
sin
3.85sin
7.5sin562
ft0.56
xA
B
a
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Example
A hot-air balloon is flying over a dry lake when the wind stops blowing. The balloon comes to a stop
450 feet above the ground at pointD. A jeep following the balloon runs out of gas at pointA. The
nearest service station is due north of the jeep at pointB. The bearing of the balloon from the jeep atA
is N 13E, while the bearing of the balloon from the service station at Bis S 19E. If the angle of
elevation of the balloon fromAis 12, how far will the people in the jeep have to walk to reach the
service station at pointB?
Solution
AC
DC12tan
12tan
DCAC
12tan
450
ft117,2
)1913(180 ACB
148
Using triangleABC
19sin148sin
2117AB
19sin
148sin2117AB
3, 400ft
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Ambiguous Case
SideAngleSide (SAS)
If two sides and the included angle of one triangle are equal, respectively, to two sides and the included
angle of a second triangle, then the triangles are congruent.
Example
Find angleBin triangle ABC if a= 2, b= 6, and 30A
Solution
a
A
b
B sinsin
aB
Absinsin
2
30sin6
5.1
1 sin 1
Since 1sinB
is impossible, no such triangle exists.
Example
Find the missing parts in triangle ABC if C= 35.4, a= 205 ft., and c= 314 ft.
Solution
c
CaA
sinsin
314
4.35sin205
3782.0
)3782.0(1sinA
22.2A
180 22.2 157.8A
35.4 157.8C A
193.2 180
4.122)4.352.22(180B
C
Bcb
sin
sin
4.35sin
4.122sin314
ft458
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Example
Find the missing parts in triangle ABC if a= 54 cm, b= 62 cm, andA= 40.
Solution
a
A
b
B sinsin
aB
Absinsin
54
40sin62
738.0
48)738.0(sin 1B 13248180B
)4840(180 C )13240(180 C
92 8
A
Cac
sin
sin
ACa
csin
sin
40sin
92sin54
40sin8sin54
cm84
cm12
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Area of a Triangle (SAS)
In any triangleABC, the areaAis given by the following formulas:
1
2sinA bc A 1
2 sinA ac B 1
2 sinA ab C
Example
Find the area of triangleABCif 24 40 , 27.3 , 52 40A b cm and C
Solution
180 24 40 52 40B
40 4060 60
180 24 52
102.667
sin sin
a b
A B
27.3
sin 24 40 sin 102 40
a
27.3sin 24 40
sin 102 40a
11.7 cm
1 sin2
A ac B
1 (11.7)(27.3)sin 52 402
2127 cm
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Example
Find the area of triangleABC.
Solution
1sin
2A ac B
1 34.0 42.0 sin 55 102
2586ft
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Number of Tr iangles Satisfying the Ambiguous Case (SSA)
Let sides aand band angleAbe given in triangleABC. (The law of sines can be used to calculate thevalue of sinB.)
1. If applying the law of sines results in an equation having sinB> 1, then no triangle satisfies the
given conditions.
2. If sinB= 1, then one triangle satisfies the given conditions andB= 90.
3. If 0 < sinB< 1, then either one or two triangles satisfy the given conditions.
a) If sinB= k, then let 11
sinB k and use1
B forBin the first triangle.
b)Let2 1
180B B . If2
180A B , then a second triangle exists. In this case, use2
B for
Bin the second triangle.
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Exercises Section 4.1Law of Sines
1. In triangleABC, 110B , 40C and inb 18 . Find the length of side c.
2. In triangleABC, 4.110A , 8.21C and inc 246 . Find all the missing parts.
3.
Find the missing parts of triangleABC,if 34B , 82C , and cma 6.5 .
4. Solve triangleABCifB= 5540, b= 8.94 m, and a= 25.1 m.
5. Solve triangleABC ifA= 55.3, a= 22.8ft., and b= 24.9 ft.
6. Solve triangleABC givenA= 43.5, a= 10.7 in., and c= 7.2 in.
7. If 26 , 22, 19,A s and r findx
8. A man flying in a hot-air balloon in a straight line at a constant rate of 5 feet per second, while
keeping it at a constant altitude. As he approaches the parking lot of a market, he notices that the
angle of depression from his balloon to a friends car in theparking lot is 35. A minute and ahalf later, after flying directly over this friends car, he looks back to see his friend getting into
the car and observes the angle of depression to be 36. At that time, what is the distance between
him and his friend?
9. A satellite is circling above the earth. When the satellite is directly above pointB, angleAis
75.4. If the distance between pointsBandDon the circumference of the earth is 910 miles and
the radius of the earth is 3,960 miles, how far above the earth is the satellite?
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10. A pilot left Fairbanks in a light plane and flew 100 miles toward Fort in still air on a course with
bearing of 18. She then flew due east (bearing 90) for some time drop supplies to a snowbound
family. After the drop, her course to return to Fairbanks had bearing of 225 . What was her
maximum distance from Fairbanks?
11. The dimensions of a land are given in the figure. Find the area of the property in square feet.
12. The angle of elevation of the top of a water tower from point A on the ground is 19.9. From
point B, 50.0 feet closer to the tower, the angle of elevation is 21.8. What is the height of the
tower?
13. A 40-ft wide house has a roof with a 6-12 pitch (the roof rises 6 ft for a run of 12 ft). The owner
plans a 14-ft wide addition that will have a 3-12 pitch to its roof. Find the lengths of AB and BC
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14. A hill has an angle of inclination of 36. A study completed by a states highway commission
showed that the placement of a highway requires that 400 ft of the hill, measured horizontally, be
removed. The engineers plan to leave a slope alongside the highway with an angle of inclination
of 62. Located 750 ft up the hill measured from the base is a tree containing the nest of an
endangered hawk. Will this tree be removed in the excavation?
15. A cruise missile is traveling straight across the desert at 548 mph at an altitude of 1 mile. A
gunner spots the missile coming in his direction and fires a projectile at the missile when the
angle of elevation of the missile is 35. If the speed of the projectile is 688 mph, then for what
angle of elevation of the gun will the projectile hit the missile?
16. When the ball is snapped, Smith starts running at a 50angle to the line of scrimmage. At the
moment when Smith is at a 60angle from Jones, Smith is running at 17 ft/sec and Jones passes
the ball at 60 ft/sec to Smith. However, to complete the pass, Jones must lead Smith by the angle
. Find (find in his head. Note that can be found without knowing any distances.)
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17. A rabbit starts running from pointAin a straight line in the direction 30from the north at 3.5
ft/sec. At the same time a fox starts running in a straight line from a position 30ftto the west of
the rabbit 6.5ft/sec. The fox chooses his path so that he will catch the rabbit at point C. In how
many seconds will the fox catch the rabbit?
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Section 4.2- Law of Cosines
Law of Cosines(SAS)
2 2 22 cosa b c bc A
2 2 2 2 cosb a c ac B
2 2 22 cosc a b ab C
Derivation
222 )( hxca
222
2 hxcxc (1)
222hxb (2)
From (2):
(1) 222 2 bcxca
cxbca 2222 (3)
bxA cos
xAb cos
(3) Acbbca cos2222
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Example
Find the missing parts in triangleABCifA= 60, b= 20 in, and c= 30 in.
Solution
Abccba cos2222
60cos)30)(20(23020
22
700
26a
a
AbB
sinsin
26
60sin20
6662.0
)6662.0(sin 1
B
42
BAC 180
4260180
78
Example
A surveyor wishes to find the distance between two inaccessible
pointsAandBon opposite sides of a lake. While standing at point
C, she finds thatAC= 259 m,BC= 423 m, and angleACB=
13240. Find the distanceAB.
Solution
2 2 2 2 cosAB AC BC AC BC C
2 2259 423 2 259 423 cos 132 40
394510
628AB
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Law of Cosines(SSS) - Three Sides
2 2 2cos
2
b c aAbc
2 2 2cos
2
a c bBac
2 2 2cos
2
a b cCab
Example
Solve triangleABCif a= 34 km, b= 20 km, and c = 18 km
Solution
bc
acb
A 2cos
222
)18)(20(2341820
222
6.0
)6.0(cos 1 A
127
OR
ab
cbaC
2cos
222
)20)(34(2182034
222
91.0
1 cos (0.91)C
25
a
AcC
sinsin
34
127sin18
4228.0
)4228.0(sin 1C
25
180 A CB
25127180
28
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Example
A plane is flying with an airspeed of 185 miles per hour with heading 120. The wind currents are
running at a constant 32 miles per hour at 165clockwise from due north. Find the true course and
ground speed of the plane.
Solution
120180
60
165360
60165360
135
cos2222
WVWVWV
135cos)32)(185(232185 22
621,43
mphWV 210
210sin
32
sin
210
135sin32sin
1077.0
)1077.0(sin 1
6
The true course is:
120 120 6 126 .
The speed of the plane with respect to the ground is 210 mph.
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Example
Find the measure of angleBin the figure of a roof truss.
Solution
2 2 2
cos2
a c bB
ac
2 2 211 9 62(11)(9)
2 2 21 11 9 6cos2(11)(9)
B
33
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Herons Area Formula (SSS)
If a triangle has sides of lengths a, b, and c, with semi-perimeter
12
s a b c
Then the area of the triangle is:
A s s a s b s c
Example
The distance as the crow flies from Los Angeles to New York is 2451 miles, from New Yorkto
Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the
triangular region having these three cities as vertices? (Ignore the curvature of Earth.)
Solution
The semiperimetersis:
12
s a b c
12 2451 331 2427
2604.5
A s s a s b s c
2604.5 2604.5 262451 3304.5 2604.51 2427
2401,700 mi
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Exercises Section 4.2- Law of Cosines
1. If a= 13 yd., b= 14 yd., and c= 15 yd., find the largest angle.
2. Solve triangleABCif b= 63.4 km, and c= 75.2 km,A = 124 40
3. Solve triangleABCif 42.3 , 12.9 , 15.4A b m and c m
4. Solve triangleABCif a= 832 ft.,b= 623 ft., and c = 345 ft.
5. Solve triangleABCif 9.47 , 15.9 , 21.1a ft b ft and c ft
6. The diagonals of a parallelogram are 24.2 cm and 35.4 cm and intersect at an angle of 65.5.
Find the length of the shorter side of the parallelogram
7. An engineer wants to position three pipes at the vertices of a triangle. If the pipesA, B, and C
have radii 2 in, 3 in, and 4 in, respectively, then what are the measures of the angles of thetriangleABC?
8. A solar panel with a width of 1.2 m is positioned on a flat roof. What is the angle of elevation
of the solar panel?
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9. Andrea and Steve left the airport at the same time. Andrea flew at 180 mph on a course with
bearing 80, and Steve flew at 240 mph on a course with bearing 210. How far apart were
they after 3 hr.?
10. A submarine sights a moving target at a distance of 820 m. A torpedo is fired 9ahead of the
target, and travels 924 m in a straight line to hit the target. How far has the target moved fromthe time the torpedo is fired to the time of the hit?
11. A tunnel is planned through a mountain to connect pointsAandBon two existing roads. If
the angle between the roads at point Cis 28, what is the distance from pointAtoB? Find
CBAand CABto the nearest tenth of a degree.
12. A 6-ft antenna is installed at the top of a roof. A guy wire is to be attached to the top of the
antenna and to a point 10 ft down the roof. If the angle of elevation of the roof is 28, thenwhat length guy wire is needed?
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13. On June 30, 1861, Comet Tebutt, one of the greatest comets, was visible even before sunset.
One of the factors that causes a comet to be extra bright is a small scattering angle . When
Comet Tebutt was at its brightest, it was 0.133 a.u. from the earth, 0.894 a.u. from the sun, and
the earth was 1.017 a.u. from the sun. Find the phase angle and the scattering angle forComet Tebutt on June 30, 1861. (One astronomical unit (a.u) is the average distance between
the earth and the sub.)
14. A human arm consists of an upper arm of 30 cm and a lower arm of 30 cm. To move the hand
to the point (36, 8), the human brain chooses angle1 2
and to the nearest tenth of a
degree.
15. A forest ranger is 150 ft above the ground in a fire tower when she spots an angry grizzly bear
east of the tower with an angle of depression of 10. Southeast of the tower she spots a hiker
with an angle of depression of 15. Find the distance between the hiker and the angry bear.
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Section4.3Vectors and Dot Product
Notation The quanti ty is
V a vector
V a vector
V a vector
AB a vector
x Scalar
|V| Magnitude of vector V, a scalar
Equalityfor Vectors
The vectors are equivalent if they have the same magnitude and the same direction.1 2
V V
A = B C = D A E A E
Standard Position
A vector with its initial point at the origin is called a posit ion vector.
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Magnitudeof a Vector
The length or magni tude of a vectorcan be written:
2 2V a b
22yVxVV
Direction Angleof a Vector
The direction angle satisfies tan ba
, where a0.
ZeroVector
A vector has a magnitude of zero 0V and has no defined direction.
Example
Draw the vector V= (3, -4) in standard position and find its magnitude.
Solution
22baV
22 )4(3
5
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Example
Find the magnitude and direction angle for u= 3,2.
Solution
223 2u
13
1tan
y
x
1 2tan3
33.7
360 33.7
326.3
Horizontal & Vertical Vector Components
The horizontal and vertical components, respectively, of a vector Vhaving magnitude |V| and
direction angle are given by:
cos sinandx y
V V V V
xV is the hor izontal vector componentof V
52cosVVx
52cos15
2.9
yV is the vertical vector componentof V
52sinVVy
52sin15 12
52
y
x
Vx
V = 15Vy
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Example
The human cannonball is shot from cannon with an initial velocity of 53 miles per hour at an angle
of 60from the horizontal. Find the magnitude of the horizontal and vertical vector components of
the velocity vector.
Solution
60cos53xV
hrmi /27
60sin53yV
hrmi /46
Addition and Subtraction of VectorsThe sum of the vectors Uand V(U + V) is called the resul tant vector.
UV = U + (-V)
The sum of a vector Vand its oppositeVhas magnitude 0 and is called the zero vector.
U
V
U + V
U
V
U + V
U
V
U - V
60
y
x
Vx
53Vy
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Addition and subtraction with AlgebraicVectors
5,32,6 VU
52,36
7,3
5,32,6 VU
52),3(6
3,9
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ScalarMultiplication
Example
3,233 V
9,6
Example
If 3-,5U and 4,6V , find:
a.
VU
b. VU 54
Solution
a. 4,63,5VU
43,65
1,1
b.
4,653,5454 VU
20,3012,20
2012),30(20
32,3020
50, 32
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Component Vector Form
The vector that extends from the origin to the point (1, 0) is called the unit horizontal vector and is
denoted by i.
The vector that extends from the origin to the point (0, 1) is called the unit vertical vector and is
denoted byj.
Example
Write the vector 4,3V in terms of the unit vectors iandj.
Solution
jiV 43
Algebraic Vectors
If iis the unit vector from (0, 0) to (1, 0), andjis the unit vector from (0, 0) to (0, 1), then any
vector Vcan be written as
,V a b a b i j
Where aand bare real numbers. The magnitude of Vis
22
baV
cos sinanda V b V
, cos , sinV a b V V
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Example
Vector Vhas its tail at the origin, and makes an angle of 35with the positivex-axis. Its magnitude
is 12. Write Vin terms of the unit vectors iandj.
Solution
12cos35 9.8a
12sin35 6.9b
9.8 6.9V i j
Example
If 5 3U i j and 6 4V i j
a. VU
5 3 6 4U V i j - i j
i j
b. VU 54
4 5 4 5 3 6 4-U V - 5i j i j
3020 12 20 i j + i j
50 32 i j
Example
Vector whas magnitude 25.0 and direction angle 41.7. Find the horizontal and verticalcomponents.
Solution
cosa w
25cos41.7
18.7
sinb w
25sin 41.7
16.6
18.7, 16.6w
Horizontal component: 18.7
Vertical component: 16.6
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Force
When an object is stationary (at rest) we say it is in a state of static equilibrium.
When an object is in this state, the sum of the forces acting on the object must be equal to the zero
vector 0.
Example
A traffic light weighing 22 pounds is suspended by two wires. Find the magnitude of the tension in
wireAB, and the magnitude of the tension in wireAC.
Solution
75sin
22
45sin
1T
75sin45sin22
1T
lb16
75sin22
60sin
2T
75sin
60sin222T
20lb
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Example
Danny is 5 years old and weighs 42 pounds. He is sitting on a swing when his sister Stacey pulls
him and the swing back horizontally through an angle of 30and then stops. Find the tension in theropes of the swing and the magnitude of the force exerted by Stacey.
Solution
H H i
42W W j j
cos 60 sin 60T T Ti j
0T H W Static equilibrium 0all forces
cos 60 sin 60 42 0T T Hi j i j
cos 60 sin 60 42 0T H Ti i j j
cos 60 sin 60 42 0T H Ti j
sin60 42 0T cos60 0T H
sin60 42T
42sin60
T
48T lb
cos60H T
48cos60
24 lb
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The DOTProduct
The dot product(or scalar product) of two vectors U ai bj and V ci dj is written U V
and is defined as follows:
)()( djcibjaiVU
ac bd
The dot product is a real number (scalar), not a vector.
It is helpful to find the angle between two vectors.
Finding the work done by a force
Example
Find each of the following dot products
a.
3,4 2,5U V when U and V
3, 4 2, 5U V
3(2) 4(5)
26
b. 1, 2 3, 5
1, 2 3, 5 3 10
13
c. 6 3 2 7S W when S i j and W i j
12 21S W
9
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Finding the Angle Between Two Vectors
The dot product of two vectors is equal to the product of their magnitudes multiplies by the cosine
of the angle between them. That is, when is the angle between two nonzero vectors Uand V, then
cosU V U V
cos U V
U V
Example
Find the angle between the vectors Uand V.
a. 2,3 3,2U and V
b. 6 4U i j and V i j
Solution
a) 2,3 3,2U and V
cos U V
U V
2( 3) 3(2)
2 2 2 22 3 ( 3) 2
6 6
13 13
013
0
1cos (0) 90
b) 6 4U i j and V i j
cos U V
U V
6(1) ( 1)(4)
2 2 2 2
6 ( 1) 1 4
6 4
37 17
2
25.08
0.0797
1cos (0.0797) 85.43
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Perpendicular Vectors
If Uand Vare two nonzero vectors, then
0U V U V
Two vectors are perpendicular if and only if (iff) their dot product is 0.
Example
Which of the following vectors are perpendicular to each other?
8 6U i j 3 4V i j 4 3W i j
Solution
8 6 3 4U V i j i j
24 24
0 Uand Vare perpendicular
8 6 4 3U W i j i j
32 18
50 Uand Ware not perpendicular
3 4 4 3V W i j i j
12 12
0 Vand Ware perpendicular
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Work
Work is performed when a force (constant) is used to move an object a certain distance.
d: displacement vector.
V: Represents the component of Fthat is the same direction of d,
is sometimes called the projectionof onto d.
cosV F
Work V d
cosF d
cosF d
F d
DefinitionIf a constant force Fis applied, and the resulting movement of the object is represented by the
displacement vector d, then the work performed by the force is
Work F d
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Example
A force F= 35i12j(in pounds) is used to push an object up a ramp. The resulting movement of
the object is represented by the displacement vector d= 15i+ 4j(in feet). Find the work done by the
force.
Solution
Work F d
(35)(15) ( 12)(4)
480 ft lb
Example
A shipping clerk pushes a heavy package across the floor. He applies a force of 64 pounds in a
downward direction, making an angle of 35with the horizontal. If the package is moved 25 feet,
how much work is done by the clerk?
Solution
30cosFFx
30cos64
dxFW .
25.35cos64
lbft 1300
F
25 ft.
30
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Airspeed and Groundspeed
The airspeedof a plane is its speed relative to the air
The groundspeedof a plane is its speed relative to the ground.
The groundspeedof a plane is represented by the vector sum of the airspeed and windspeed vectors.
Example
A plane with an airspeed of 192 mph is headed on a bearing of 121. A north wind is blowing (from
north to south) at 15.9 mph. Find the groundspeed and the actual bearing of the plane.
Solution
121BCO AOC
The groundspeed is represented by |x|.
2 2 2192 15.9 2(192)(15.9) cos121x
40,261
200.7x mph
The planes groundspeed is about 201 mph.
sin sin121
15.9 200.7
15.9sin121sin
200.7
1 15.9sin121200.7sin 3.89
The planes groundspeed is about 201 mph on abearing of 121 + 4 = 125.
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Exercises Section4.3Vectors and Dot Product
1. Let u= 2, 1and v= 4, 3. Find the following.
a) u+ v
b)
2uc) 4u3v
2. Given: 2.24,8.13 V , find the magnitudes of the horizontal and vertical vector
components of V,x y
V and V , respectively
3. Find the angle between the two vectors u= 3, 4and v= 2, 1.
4. A bullet is fired into the air with an initial velocity of 1,800 feet per second at an angle of 60
from the horizontal. Find the magnitude of the horizontal and vertical vector component as of
the velocity vector.
5. A bullet is fired into the air with an initial velocity of 1,200 feet per second at an angle of 45
from the horizontal.
a) Find the magnitude of the horizontal and vertical vector component as of the velocity
vector.
b) Find the horizontal distance traveled by the bullet in 3 seconds. (Neglect the resistance of
air on the bullet).
6. A ship travels 130 km on a bearing of S 42E. How far east and how far south has it traveled?
7. An arrow is shot into the air with so that its horizontal velocity is 15.0 ft./sec and its vertical
velocity is 25.0 ft./sec. Find the velocity of the arrow?
8. An arrow is shot into the air so that its horizontal velocity is 25 feet per second and its vertical
is 15 feet per second. Find the velocity of the arrow.
9. A plane travels 170 miles on a bearing of N 18 E and then changes its course to N 49 E and
travels another 120 miles. Find the total distance traveled north and the total distance traveled
east.
10. A boat travels 72 miles on a course of bearing N 27 E and then changes its course to travel 37
miles at N 55E. How far north and how far east has the boat traveled on this 109-mile trip?
11. A boat is crossing a river that run due north. The boat is pointed due east and is moving
through the water at 12 miles per hour. If the current of the river is a constant 5.1 miles per
hour, find the actual course of the boat through the water to two significant digits.
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12. Two forces of 15 and 22 Newtons act on a point in the plane. (A newtonis a unit of force that
equals .225 lb.) If the angle between the forces is 100, find the magnitude of the resultant
vector.
13. Find the magnitude of the equilibrant of forces of 48 Newtons and 60 Newtons acting on a
pointA, if the angle between the forces is 50. Then find the angle between the equilibrant and
the 48-newton force.
14. Find the force required to keep a 50-lb wagon from sliding down a ramp inclined at 20 to the
horizontal. (Assume there is no friction.)
15. A force of 16.0 lb. is required to hold a 40.0 lb. lawn mower on an incline. What angle does
the incline make with the horizontal?
16. Two prospectors are pulling on ropes attached around the neck of a donkey that does not want
to move. One prospector pulls with a force of 55 lb, and the other pulls with a force of 75 lb. If
the angle between the ropes is 25, then how much force must the donkey use in order to stayput? (The donkey knows the proper direction in which to apply his force.)
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17. A ship leaves port on a bearing of 28.0 and travels 8.20 mi. The ship then turns due east and
travels 4.30 mi. How far is the ship from port? What is its bearing from port?
18. A solid steel ball is placed on a 10incline. If a force of 3.2 lb in the direction of the incline is
required to keep the ball in place, then what is the weight of the ball?
19.
Find the amount of force required for a winch to pull a 3000-lb car up a ramp that is inclined
at 20.
20. If the amount of force required to push a block of ice up an ice-covered driveway that is
inclined at 25is 100lb, then what is the weight of the block?
21. If superman exerts 1000 lb of force to prevent a 5000-lb boulder from rolling down a hill and
crushing a bus full of children, then what is the angle of inclination of the hill?
22.
If Sisyphus exerts a 500-lb force in rolling his 4000-lb spherical boulder uphill, then what isthe angle of inclination of the hill?
23. A plane is headed due east with an air speed of 240 mph. The wind is from the north at 30
mph. Find the bearing for the course and the ground speed of the plane.
24. A plane is headed due west with an air speed of 300 mph. The wind is from the north at 80
mph. Find the bearing for the course and the ground speed of the plane.
25. An ultralight is flying northeast at 50 mph. The wind is from the north at 20 mph. Find the
bearing for the course and the ground speed of the ultralight.
26. A superlight is flying northwest at 75 mph. The wind is from the south at 40 mph. Find the
bearing for the course and the ground speed of the superlight.
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27. An airplane is heading on a bearing of 102with an air speed of 480 mph. If the wind is out of
the northeast (bearing 225) at 58 mph, then what are the bearing of the course and the ground
speed of the airplane?
28. In Roman mythology, Sisyphus revealed a secret of Zeus and thus incurred the gods wrath.
As punishment, Zeus banished him to Hades, where he was doomed for eternity to roll a rockuphill, only to have it roll back on him. If Sisyphus stands in front of a 4000-lb spherical rock
on a 20incline, then what force applied in the direction of the incline would keep the rock
from rolling down the incline?
29. A trigonometry student wants to cross a river that is 0.2 mi wide and has a current of 1 mph.
The boat goes 3 mph in still water.
a) Write the distance the boats travels as a function of the angle .
b) Write the actual speed of the boat as a function of and .
c) Write the time for the trip as a function of . Find the angle for which the student willcross the river in the shortest amount of time.
30. Amal uses three elephants to pull a very large log out of the jungle. The papa elephant pulls
with 800 lb. of force, the mama elephant pulls with 500 lb. of force, and the baby elephant
pulls with 200 lb. force. The angles between the forces are shown in the figure. What is the
magnitude of the resultant of all three forces? If mama is pulling due east, then in what
direction will the log move?
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Section 4.4Trigonometric Form of Complex Numbers
1 i
The graph of the complex numberx=yiis a vector (arrow) that extends from the origin out to thepoint (x, y)
Horizontal axis: realaxis
Vertical axis: imaginaryaxis
Example
Graph each complex number: 2 4i , 2 4i , and 2 4i
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Example
Graph each complex number: 1, , 1,i and i
Example
Find the sum of 62iand43i. Graph both complex numbers and their resultant.
Solution
(62i) + (43i) = 642i3i
= 25i
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Definition
The absolute valueor modulusof the complex number z x yi is the distance from the origin
to the point (x, y). If this distance is denoted by r, then
2 2r z x yi x y
The argumentof the complex number z x yi denoted )arg( z is the smallest possible angle
from the positive real axis to the graph ofz.
r
xcos cosrx
r
ysin sinry
yixz
irr )sin(cos
)sin(cos ir is called the trigonometricfrom ofz.
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Definition
If iyxz is a complex number in standard form then the tri gonometri c formforzis given by
(cos sin ) z r i r cis
Where r is the modulus or absolute value ofzandis the argument ofz.
We can convert back and forth between standard form and trigonometric form by using the
relationships that follow
For cisririyxz )sin(cos
22yxr
x
yand
r
y
r
x tan,sin,cos
Example
Write iz 1 in trigonometric form
Solution
The modulus r:
21)1( 22
r
1cos
2
xr
1sin
2
y
r
135
iyxz
)135sin135(cos2 i
1352cis
In radians:
4
32
cisz
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Example
Write 602cisz in rectangular form.
Solution
602cisz
)60sin60(cos2 i
2
3
2
12 i
31 i
Example
Express 2 cos300 sin300i in rectangular form.
Solution
312 2
2 cos 300 sin 300 2i i
1 3i
Example
Find the modulus of each of the complex numbers 5i, 7, and 3 + 4i
Solution
For z = 5i= 0 + 5i 550 22
zr
Forz= 7= 7 + 0i 70722
zr
For 3 + 4i 543 22
r
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Product Theorem
If 1 1 1cos sinr i and 2 2 2cos sinr i are any two complex numbers, then
1 1 1 2 2 2 1 2 1 2 1 2cos sin cos sin cos sinr i r i r r i
1 1 2 2 1 2 1 2r cis r cis r r cis
2 2i ia b a b a b
a b a bi a bi
Example
Find the product of 3 cos45 sin 45i and 2 cos135 sin135i . Write the result in rectangularform.
Solution
3 cos 45 sin 45 2 cos135 sin135i i
3 2 cos 45 135 45 135sini
6 cos180 sin180i
6 1 .0i
6
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Quotient Theorem
If 1 1 1cos sinr i and 2 2 2cos sinr i are any two complex numbers, then
1 2 1 2
cos sin1 1 1 1 cos sin
cos sin 22 2 2
r i ri
rr i
1 1 11 2
2 2 2
r cis r
cisr cis r
Example
Find the quotient
10 60
5 150
cis
cis
. Write the result in rectangular form.
Solution
10 60 10 60 150
55 150
ciscis
cis
2 210cis
2 cos 210 sin 210i
3 1
2 22 i
3 i
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Exercises Section 4.4Trigonometric Form of Complex Numbers
1. Write 3 i in trigonometric form. (Use radian measure)
2. Write 3 4i in trigonometric form.
3. Write 21 20i in trigonometric form.
4. Write 11 2i in trigonometric form.
5. Write 4 cos30 sin30i in standard form.
6. Write 74
2 cis in standard form.
7. Find the quotient
20 75
4 40
cis
cis
. Write the result in rectangular form.
8.
Divide1 2
1 3 3z i by z i . Write the result in rectangular form.
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Section 4.5Polar Coordinates
To reach the point whose address is (2, 1), we start from origin and travel 2 units right and then 1
unit up. Another way to get to that point, we can travel 5 units on the terminal side of an angle in
standard position and this type is calledPolar Coordinates.
Definitionof Polar Coordinates
To define polar coordinates, let an originO(called the pole) and an initial rayfrom O. Then each
pointPcan be located by assigning to it a polar coordinate pair ,r in which r gives the
directed from OtoPand gives the directed angle from the initial ray to yay OP.
Polar Coordinates
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Example
A point lies at (4, 4) on a rectangular coordinate system. Give its address in polar coordinates
,r
Solution
2 24 4r
32
4 2
1 44
tan
1tan 1
45
The address is 4 2 , 45
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Example
Graph the points 3,45 , 432,
, 34,
, and 5, 210 on a polar coordinate system
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Example
Give three other order pairs that name the same point as 3, 60
3, 60 , 3, 240 , 3, 120 , 3, 300
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Polar Coordinates and Rectangular Coordinates
To Convert Rectangular Coordinates to Polar Coordinates
Let 2 2r x y and tan y
x
Where the sign of rand the choice of place the point (r, ) in the same quadrant as (x, y)
To Convert Polar Coordinates to Rectangular Coordinates
Let cosx r sinand y r
Example
Convert to rectangular coordinates. (4, 30 )
Solution
cosx r
4cos30
32
4
2 3
siny r
4sin30
14 2 2
The point (2 3, 2) in rectangular coordinates is equivalent to (4, 30 ) in polar coordinates.
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Example
Convert to rectangular coordinates 342,
.
Solution
3
4
2 cosx
1
22
1
3
42 siny
1
22
1
The point (1, 1) in rectangular coordinates is equivalent to 342,
in polar coordinates.
Example
Convert to rectangular coordinates (3, 270 ) .
Solution
3cos270x
3(0)
0
3sin270y
3( 1)
3
The point (0, 3) in rectangular coordinates is equivalent to (3, 270 ) in polar coordinates.
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Example
Convert to polar coordinates (3, 3) .
Solution
2 23 3r
9 9
3 2
33
tan 1
1tan 1
45
The point 3 2 , 45 is just one.
Example
Convert to polar coordinates ( 2, 0) .
Solution
4 0r
2
1 02
tan
0
The point 2, 180r
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Example
Convert to polar coordinates ( 1, 3) .
Solution
1 3r
2
311
tan
120
The point 2, 120r
Example
Write the equation in rectangular coordinates2
4sin2r
Solution
24sin2r sin 2 2 sin cos
4 2sin cos cos sin yx
r r
8 y xr r
28
xy
r
4 8r xy 2 2 2
r x y
2
2 28x y xy
Example
Write the equation in polar coordinates4
x y
Solution
cos sin 4r r cos sinx r y r
cos sin 4r
4cos sin
r
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Exercises Section 4.5Polar Coordinates
1. Convert to rectangular coordinates 2, 60
2. Convert to rectangular coordinates
2, 225
3. Convert to rectangular coordinates 4 3, 6
4. Convert to polar coordinates 3, 3 0 0 360r
5. Convert to polar coordinates 2, 2 3 0 0 360r
6. Convert to polar coordinates 2, 0 0 0 2r
7. Convert to polar coordinates
1, 3 0 0 2r
8. Write the equation in rectangular coordinates2
4r
9. Write the equation in rectangular coordinates 6cosr
10. Write the equation in rectangular coordinates2
4cos2r
11. Write the equation in rectangular coordinates cos sin 2r
12. Write the equation in polar coordinates 5x y
13. Write the equation in polar coordinates 2 2 9x y
14. Write the equation in polar coordinates 2 2 4x y x
15. Write the equation in polar coordinates y x
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Section 4.6 - De Moivres Theorem
De MoivresTheorem
If cos sinr i is a complex number, then
cos sin cos sinn
nni r i nr
n n
rcis r c sni
Example
Find 8
1 3i and express the result in rectangular form.
Solution
11 3
3
xi
y
22
1 3 2r
3
1tan 3
is in QI, that implies: 60
1 3 2 60i cis
Apply De Moivres theorem:
8 8
1 3 2 60i cis
82 c .608is
256 c 480is 480 360 120
256 c 120is
31256
2 2i
128 128 3i
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nth
Root Theorem
For a positive integer n, the complex number a+ biis an nth
root of the complex numberx+ iyif
na bi x yi
If nis any positive integer, ris a positive real number, andis in degrees, then the nonzero
complex number cos sinr i has exactly ndistinct nth roots, given by
cos sinn r i or n r cis
Where 0,1, 2, ,360 , 1kk nn
360 k
n n
2 , 0,1, , , 12k nkn
2 kn n
Example
Find the two square root of 4i. Write the roots in rectangular form.
Solution
04
4
xi
y
2 20 4 4r
4
0tan
2
4 42
i cis
The absolute value: 4 2
Argument: 2 22
22 2 2 4
kk
k
Since there are twosquare root, then k= 0 and 1.
0 (0)4 4
If k
5 1 (1)4 4If k
The square roots are: 52 24 4
cis and cis
2 2 cos sin4 4 4
cis i
2 2
22 2
i
2 2i
5 5 52 2 cos sin4 4 4
cis i
2 2
22 2
i
2 2i
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Example
Find all fourth roots of 8 8 3i . Write the roots in rectangular form.
Solution
88 8 3
8 3
xi
y
22( 8) 8 3 16r
8 3
8tan 3 120
8 8 3 16 120i cis
The fourth roots have absolute value: 416 2
120 360 30 90
4 4
k k
Since there are fourroots, then k= 0, 1, 2, and 3.
0 30 0090 ( ) 3If k
1 30 9 ) 210 ( 1 0If k
2 30 90 (2) 210If k
3 30 9 ) 030 ( 3 0If k
The fourth roots are: 2 30 , 2 120 , 2 210 , 2 300cis cis cis and cis
3 12 30 2 cos30 sin30 2 32 2
c i iis i
312 120 2 cos120 sin120 2 1 32 2
cis i i i
3 12 210 2 cos 210 sin212
30 22
cis i i i
312 300 2 cos300 sin 30 2 1 302 2
c i iis i
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Example
Find all complex number solutions of 5 1 0x . Graph them as vectors in the complex plane.
Solution
5 51 0 1x x
There is one real solution, 1, while there are five complex solutions.
1 1 0i
2 21 0 1r
0
1tan 0 0
1 1 0cis
The fifth roots have absolute value: 1 1 1
0 360 0 72 725 5
k k k
Since there are fifthroots, then k= 0, 1, 2, 3, and 4.
0 72 ) 00(If k
1 7 ) 7202 (If k
2 72 (2) 144If k
3 72 (3) 216If k
4 72 (4) 288If k
Solution: 0 , 72 , 144 , 216 , 288cis cis cis cis and cis
The graphs of the roots lie on a unit circle. The roots are equally spaced about the circle, 72
apart.
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Exercises Section 4.6 - De Moivres Theorem
1. Find 81 i and express the result in rectangular form.
2.
Find 10
1 i and express the result in rectangular form.
3. Find fifth roots of 1 3z i and express the result in rectangular form.
4. Find the fourth roots of 16 60z cis
5. Find the cube roots of 27.
6. Find all complex number solutions of 3 1 0x .
7. Find 52 30cis