Trig Lecture 4

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.



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

Documents

Trig Lecture 4