trigonometria final exam review

Trigonometry 8th ed Lial Hornsby Schneider Trig Final Exam Review F07 OrsquoBrien

1

Trigonometry Final Exam Review Chapters 7 8 9 Note A portion of Exam 3 will cover Chapters 1 ndash 6 so be sure you rework problems from the first and second exams and from the Exam 1 and Exam 2 Reviews Directions Work these problems with no resources other than the departmental formula sheet and a graphing calculator Your book notebook homework solutions manual etc should be closed Read and carefully follow all directions Show all of your work on all problems Anytime you are asked to perform a calculation manually or to give an exact answer you may not use a calculator Chapter 7 Applications of Trigonometry and Vectors

1 Determine the remaining sides and angles of the triangle ABC Show all work and or support your answer

2 Determine the remaining sides and angles of the triangle ABC Show all work and or support your answer

C = 7183deg B = 4257deg a = 2614

3 A ship is sailing due north At a certain point the bearing of a lighthouse 125 km distant is N 388deg E Later on the captain notices that the bearing of the lighthouse has become S 442deg E How far did the ship travel between the two observations of the lighthouse Show all work and or support your answer

4 Mark notices that the bearing of a tree on the opposite bank of a river flowing north is 11545deg Lisa is on the same bank as Mark but 4283 m away She notices that the bearing of the tree is 4547deg The two banks are parallel What is the distance across the river Show all work and or support your answer

5 Find the area of the triangle ABC Show all work and or support your answer

A = 3050deg b = 1300 cm C = 11260deg

6 A real estate agent wants to find the area of a triangular lot A surveyor takes measurements and finds that two sides are 521 m and 213 m and the angle between them is 422deg What is the area of the triangular lot Show all work and or support your answer

7 Determine the number of triangles ABC possible with the given parts Show all work and or support your answer

a) b = 60 a = 82 B = 100deg b) a = 35 b = 30 A = 40deg c) B = 54deg c = 28 b = 23

8 Find the unknown angles in triangle ABC if the triangle exists Show all work and or support your answer

a) C = 41deg20 b = 259 m c = 384 m b) B = 482deg a = 890 cm b = 697 cm

c) B = 743deg a = 859 m b = 783 m

9 Solve the triangle ABC if possible Show all work and or support your answer

C = 523deg a = 325 yd c = 598 yd


2


A = 38deg40 a = 972 km b = 118 km

11 Solve each triangle and then find the area of the triangle Approximate values to the nearest tenth Show all work and or support your answer

a) b)

12 Solve each triangle and then find the area of the triangle Show all work and or support your answer

a) C = 283deg b = 571 in a = 421 in b) a = 189 yd b = 214 yd c = 325 yd

13 The sides of a parallelogram are 40 cm and 60 cm One angle is 58deg while another is 122deg Find the lengths of the diagonals of the parallelogram Show all work and or support your answer

14 A weight is supported by cables attached to both ends of a balance beam as shown in the figure What angles are formed between the beam and the cables Show all work and or support your answer

15 Using the vectors in the figure below draw a sketch to represent

a) c + d b) d ndash c c) 3c

16 Use the vectors in the adjacent figure to find a) a + b b) a ndash b c) ndash a Show all work and or support your answer

17 For vectors u and w with angle θ between them sketch the resultant u + w 8=u 12=w θ = 20deg

18 Find the magnitude and direction angle for vector v = 8- 15 Show all work and or support your answer

19 Vector v has magnitude 415=v and direction angle 0327α prime= o Find the horizontal (a) and vertical (b) components of v Show all work and or support your answer


3

20 Write vector v in component form b a Show all work and or support your answer

21 Two forces act at a point in the plane One is 19 newtons the other is 32 newtons The angle between the forces is 118deg Find the magnitude of the resultant force Show all work and or support your answer

22 Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the adjacent figure Round your answer to the nearest tenth Show all work and or support your answer

23 Given u = 5 2minus and v = 3 4 find ndash2u + 4v Write your answer in the form ai + bj Show all work and or support your answer

24 Find the dot product of 3- 2 and 5 6 Show all work and or support your answer

25 Find the angle between the vectors -5i + 12j and 3i + 2j Show all work and or support your answer

26 Given u = 1 2minus v = 4 3 and w = 12 5minus evaluate ( )wvu minussdot Show all work and or support your answer

27 Determine whether the vectors 2 1 and 3 6minus are orthogonal Show all work and or support your answer

28 Two tugboats are pulling a disabled speedboat into port with forces 1240 lb and 1480 lb The angle between these forces is 282deg Find the magnitude and direction of the equilibrant Show all work and or support your answer

29 Two forces of 128 lb and 253 lb act at a point The equilibrant force is 320 lb Find the angle between the forces Show all work and or support your answer

30 A force of 176 lb makes an angle of 78deg50 with a second force The resultant of the two forces makes an angle of 41deg10 with the first force Find the magnitudes of the second force and the resultant Show all work and or support your answer

31 A 130-lb force keeps a 600-lb object from sliding down an inclined ramp Find the angle of the ramp Show all work and or support your answer

32 Find the magnitude of force necessary to keep a 3500-pound car from sliding down a ramp of 5deg Show all work and or support your answer

33 Two people are carrying a box One person exerts a force of 150 lb at an angle of 624deg with the horizontal The other person exerts a force of 114 lb at an angle of 549deg Find the weight of the box Show all work and or support your answer


4

34 A ship leaves port on a bearing of 340deg and travels 104 miles The ship then turns due east and travels 46 miles How far is the ship from port and what is its bearing from port Show all work and or support your answer

35 A pilot is flying at 168 mph She wants her flight path to be on a bearing of 57deg40 A wind is blowing from the south at 271 mph Find the bearing the pilot should fly and find the planersquos groundspeed Show all work and or support your answer

36 A pilot wants to fly on a bearing of 749deg By flying due east he finds a 42-mph wind blowing from the south puts him on course Find the groundspeed and airspeed Show all work and or support your answer

Chapter 8 Complex Numbers Polar Equations and Parametric Equations

37 Write 288minus as the product of a real number and i Show all work and or support your answer

38 Solve 4x23x 2 minus=+ and express all nonreal complex solutions in terms of i Show all work and or support your answer

39 Perform the indicated operations and then simplify Show all work and or support your answer

a) 40

10minus

b) 8

612 minussdotminus

40 Perform the indicated operation and then simplify Write your answers in standard form Show all work and or support your answer

a) ( ) ( ) ( )iii 54324 +minus++minusminusminus b) ( )( )ii 2432 minus+minus c) ii

5132

minus+

41 Simplify each power of i Show all work and or support your answer

a) 25i b) 13minusi

42 Find the sum of i65minus and i32+minus by graphing each number in the complex plane and finding their resultant Show all work and or support your answer

43 Write the complex number 3[cos 150deg + i sin 150deg] in rectangular form Show all work and or support your answer

44 Write the complex number ndash5 ndash 5i in trigonometric form r(cos θ + i sin θ) with θ in the interval [0deg 360deg) Show all work and or support your answer

45 Find each product and write it in rectangular form On part b round to four decimal places Show all work and or support your answer

a) ( )[ ] ( )[ ]oooo 120sin 120cos5300sin 300cos8 ii +sdot+ b) (12 cis 185deg) (3 cis 125deg)


5

46 Find the quotient ( )( )oo

oo

150sin150cos2

120sin120cos4

i

i

+

+ amp write it in rectangular form Show all work and or support your answer

47 Convert the numerator and denominator of 31

2ii

minusminus to trigonometric form Then find the quotient and write it in

rectangular form Show all work and or support your answer

48 Find each power Write each answer in rectangular form Show all work and or support your answer

a) [ ]3100 3 ocis b) ( )522 iminusminus

49 Find all cube roots of each complex number Leave answers in trigonometric form Show all work and or support your answer

a) o300 27 cis b) iminus3

50 Find all complex solutions of each equation Leave answers in trigonometric form Show all work and or support your answer

a) 0x 4 =minus i b) 016x 4 =+

51 For each pair of polar points plot the point give two other pairs of polar coordinates (including one with a negative r) and give the rectangular coordinates for the point Show all work and or support your answer

a) ( )o135 2minus b) ⎟⎠⎞

⎜⎝⎛ minus

35 3 π

52 Given the rectangular coordinates ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

21

23 give two pairs of polar coordinates for the point including one with a

negative r Select θ so it is in the interval [0deg 360deg) Show all work and or support your answer

53 For each rectangular equation give its equivalent polar equation and sketch its graph Show all work and or support your answer

a) 3x ndash 2y = 6 b) 9yx 22 =+


6

54 Match each equation with its corresponding graph

a r = a cos θ A

b r = a sin θ B

c r = a plusmn b sin θ with 1 ltba C

d r = a plusmn b cos θ with 1 =ba D

e r = a cos 4θ E

f r = a sin 5θ F

g r = θsin b θ acos

cminus

G

h r2 = 4 sin 2θ H


7

55 Identify the type of polar equation and draw a complete graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer

a) r = 2 + 2 cos θ b) r = 4 cos 2θ

56 For each polar equation give its equivalent rectangular equation and sketch its graph Include a t-chart with at least 5 key points with your graph Show all work and or support your answer

a) r = 2 sin θ b) sinθ13r

minus=

Chapter 9 Exponential and Logarithmic Functions

57 Graph the following exponential function Give the equation of the horizontal asymptote a 5-point t-chart with the ldquokey pointrdquo at the center and an accurate graph with one notch on each axis labeled Show all of your work

( ) 12y 4x +minus= minus

58 Solve the following exponential equations algebraically Show all your work and give an exact answer

a 49

32 x

=⎟⎠

⎞⎜⎝

⎛minus

b 6 x 4 x 82 minus+ = c 125x 43=

59 Find the balance A for P dollars invested at rate r for t years and compounded n times

a P = $2000 r = 65 t = 10 years compounded quarterly

b P = $3000 r = 7 t = 8 years compounded continuously

60 Find the required annual interest rate to the nearest tenth of a percent for $48000 to grow to $7818694 when compounded semiannually for 5 yr Round your decimal answer to four decimal places and convert to a percent Show all of your work

61 Solve the following logarithmic equations algebraically Show all your work and give an exact answer

a 2161log x minus= b 5xlog

3= c x4log

16=

62 Graph the following logarithmic function Give the equation of the vertical asymptote the exponential form of the equation a 5-point t-chart with the ldquokeyrdquo point at the center and an accurate graph with one notch on each axis labeled Show all of your work

y = ( ) 42xlog 3 minus+

63 Use the properties of logarithms to expand or condense the following log expressions Show all of your work

a 4

3

bz

xylog b zlog32ylog

31xlog 2 aaa minusminus

64 Find the pH for lye soap which has a hydronium ion concentration of 141032 minustimes moles per liter

65 Find the [ ]+OH3 for drinking water which has a pH of 65


8

66 Use the change of base theorem to find an approximation for 15log21 Round your answer to four decimal places

67 Given f(x) = 3 x evaluate the following Show all of your work

a f (log 3 7) b f [log 3 (ln 3)] c f [log 3 (2 ln 3)]

68 a Find the decibel rating of a sound having an intensity of 100000 I0

b If the intensity of the sound is doubled by how much is the decibel rating increased

69 a Find the Richter scale rating for an earthquake having an amplitude of 1000000 I0

b Express the magnitude of a 67 earthquake (on the Richter scale) in terms of I0

70 Solve

a 64 13x =minus b x5x 32 =+ c 1116e 34x =minus+ d 50004

06110004t=⎟

⎠

⎞⎜⎝

⎛ +

71 a log (2x ndash 1) + log 10x = log 10 b ln x + ln (x ndash 1) = 2

c ( ) 12log3xlog4xlog 666 =minusminus d log (11x + 9) = 3 + log (x + 3)

72 Elena McDuff wants to buy an $8000 painting She has saved $6000 Find the number of years (to the nearest tenth) it will take for her savings to grow to $8000 at 57 compounded monthly

73 At what interest rate will $2500 grow to $467104 if invested for 10 years and interest is compounded quarterly

Answers

Chapter 7 Applications of Trigonometry and Vectors

1 A = 372deg a = 178 m c = 244 m

2 A = 6560deg b = 1942 cm c = 2727 cm

3 278 km

4 2934 m

5 6594 cm2

6 373 m2

7 a) 0 b) 1 c) 2

8 a) B asymp 26deg30 A asymp 112deg10 b) A1 asymp 722deg C1 asymp 596deg A2 asymp 1078deg C2 asymp 240deg c) no such triangle

9 A asymp 255deg B asymp 1022deg b asymp 739 yd

10 B1 asymp 49deg20 C1 asymp 92deg00 c1 asymp 155 km B2 asymp 130deg40 C2 asymp 10deg40 c2 asymp 288 km


9

11 a) A asymp 223deg B asymp 1082deg C asymp 495deg area = 2units 152231 asymp

b) A asymp 567deg C asymp 683deg b asymp 882 area = 2units 3680613547079 asymp

12 a) c asymp 283 in A asymp 449deg B asymp 1068deg area = 2units 57 7325368851 asymp

b) C asymp 107deg20 B asymp 39deg00 A asymp 33deg40 area = 2units 193040 372645000 asymp

13 52 cm 88 cm

14 264deg and 363deg

15 a) b) c)

16 a) 10 0 b) 2 8 c) 6- 4minus

17

18 17v = θ = 3319deg

19 a asymp 137 b asymp 711

20 v = 34 4

21 29 newtons

22 244 lb

23 20i + 2j

24 ndash3

25 7893deg

26 ndash24

27 orthogonal

28 2640 lb at an angle of 1672deg with the 1480-lb force

29 701deg

30 magnitude of second force 190 lb magnitude of the resultant 283 lb

31 125deg

32 305 lbs

33 226 lb

34 135 mi 504deg

35 bearing 65deg30 groundspeed 181 mph

36 groundspeed 161 mph airspeed 156 mph


10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3

40 a) i+minus10 b) i162+minus c) i21

21+minus

41 a) i b) ndash i 42 i33minus

43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i

49 a) 3 cis 100deg 3 cis 220deg 3 cis 340deg b) ( )oo 110sin 110 cos23 i+ ( )oo 230sin 230 cos23 i+

( )oo 350sin 350 cos23 i+

50 a) cos 225deg + i sin 225deg cos 1125deg + i sin 1125deg cos 2025deg + i sin 2025deg cos 2925deg + i sin 2925deg

b) 2 cis 45deg 2 cis 135deg 2 cis 225deg 2 cis 315deg

51 a) ( )o495 2minus (2 315deg) ( )2 2 minus

b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus

= b) r = 3 or r = ndash3

54 a F b A c D d G e H f B g C h E


11

55a

55b

56a

56b

57 HA y = 1 ldquokey pointrdquo (4 0) 58 a x = 2 b x = 11 c x = 625

59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12

62 VA x = ndash2 exponential form of the equation 23x 4 y minus= +

x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2

63 a zlog 4xlog21ylog 3 bbb minus+ b

3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069

67 a 7 b ln 3 asymp 10986 c 2 ln 3 = ln 32 = ln 9 asymp 21972

68 a 50 b about 3 decibels

69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


2


A = 38deg40 a = 972 km b = 118 km

11 Solve each triangle and then find the area of the triangle Approximate values to the nearest tenth Show all work and or support your answer

a) b)

12 Solve each triangle and then find the area of the triangle Show all work and or support your answer

a) C = 283deg b = 571 in a = 421 in b) a = 189 yd b = 214 yd c = 325 yd

13 The sides of a parallelogram are 40 cm and 60 cm One angle is 58deg while another is 122deg Find the lengths of the diagonals of the parallelogram Show all work and or support your answer

14 A weight is supported by cables attached to both ends of a balance beam as shown in the figure What angles are formed between the beam and the cables Show all work and or support your answer

15 Using the vectors in the figure below draw a sketch to represent

a) c + d b) d ndash c c) 3c

16 Use the vectors in the adjacent figure to find a) a + b b) a ndash b c) ndash a Show all work and or support your answer

17 For vectors u and w with angle θ between them sketch the resultant u + w 8=u 12=w θ = 20deg

18 Find the magnitude and direction angle for vector v = 8- 15 Show all work and or support your answer

19 Vector v has magnitude 415=v and direction angle 0327α prime= o Find the horizontal (a) and vertical (b) components of v Show all work and or support your answer


3
















4








a) 40

10minus

b) 8

612 minussdotminus



5132

minus+


a) 25i b) 13minusi







5


oo

150sin150cos2

120sin120cos4

i

i

+



2ii








a) 0x 4 =minus i b) 016x 4 =+


a) ( )o135 2minus b) ⎟⎠⎞

⎜⎝⎛ minus

35 3 π


⎞⎜⎜⎝

⎛minusminus

21




a) 3x ndash 2y = 6 b) 9yx 22 =+


6


a r = a cos θ A

b r = a sin θ B



e r = a cos 4θ E

f r = a sin 5θ F


cminus

G

h r2 = 4 sin 2θ H


7


a) r = 2 + 2 cos θ b) r = 4 cos 2θ



minus=





a 49

32 x

=⎟⎠

⎞⎜⎝

⎛minus

b 6 x 4 x 82 minus+ = c 125x 43=







3= c x4log

16=




a 4

3

bz

xylog b zlog32ylog





8








70 Solve


06110004t=⎟

⎠

⎞⎜⎝

⎛ +





Answers


1 A = 372deg a = 178 m c = 244 m

2 A = 6560deg b = 1942 cm c = 2727 cm

3 278 km

4 2934 m

5 6594 cm2

6 373 m2

7 a) 0 b) 1 c) 2





9





13 52 cm 88 cm


15 a) b) c)

16 a) 10 0 b) 2 8 c) 6- 4minus

17

18 17v = θ = 3319deg


20 v = 34 4

21 29 newtons

22 244 lb

23 20i + 2j

24 ndash3

25 7893deg

26 ndash24

27 orthogonal


29 701deg


31 125deg

32 305 lbs

33 226 lb

34 135 mi 504deg




10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3


21+minus


43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i


( )oo 350sin 350 cos23 i+




b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus




11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


3
















4








a) 40

10minus

b) 8

612 minussdotminus



5132

minus+


a) 25i b) 13minusi







5


oo

150sin150cos2

120sin120cos4

i

i

+



2ii








a) 0x 4 =minus i b) 016x 4 =+


a) ( )o135 2minus b) ⎟⎠⎞

⎜⎝⎛ minus

35 3 π


⎞⎜⎜⎝

⎛minusminus

21




a) 3x ndash 2y = 6 b) 9yx 22 =+


6


a r = a cos θ A

b r = a sin θ B



e r = a cos 4θ E

f r = a sin 5θ F


cminus

G

h r2 = 4 sin 2θ H


7


a) r = 2 + 2 cos θ b) r = 4 cos 2θ



minus=





a 49

32 x

=⎟⎠

⎞⎜⎝

⎛minus

b 6 x 4 x 82 minus+ = c 125x 43=







3= c x4log

16=




a 4

3

bz

xylog b zlog32ylog





8








70 Solve


06110004t=⎟

⎠

⎞⎜⎝

⎛ +





Answers


1 A = 372deg a = 178 m c = 244 m

2 A = 6560deg b = 1942 cm c = 2727 cm

3 278 km

4 2934 m

5 6594 cm2

6 373 m2

7 a) 0 b) 1 c) 2





9





13 52 cm 88 cm


15 a) b) c)

16 a) 10 0 b) 2 8 c) 6- 4minus

17

18 17v = θ = 3319deg


20 v = 34 4

21 29 newtons

22 244 lb

23 20i + 2j

24 ndash3

25 7893deg

26 ndash24

27 orthogonal


29 701deg


31 125deg

32 305 lbs

33 226 lb

34 135 mi 504deg




10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3


21+minus


43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i


( )oo 350sin 350 cos23 i+




b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus




11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


4








a) 40

10minus

b) 8

612 minussdotminus



5132

minus+


a) 25i b) 13minusi







5


oo

150sin150cos2

120sin120cos4

i

i

+



2ii








a) 0x 4 =minus i b) 016x 4 =+


a) ( )o135 2minus b) ⎟⎠⎞

⎜⎝⎛ minus

35 3 π


⎞⎜⎜⎝

⎛minusminus

21




a) 3x ndash 2y = 6 b) 9yx 22 =+


6


a r = a cos θ A

b r = a sin θ B



e r = a cos 4θ E

f r = a sin 5θ F


cminus

G

h r2 = 4 sin 2θ H


7


a) r = 2 + 2 cos θ b) r = 4 cos 2θ



minus=





a 49

32 x

=⎟⎠

⎞⎜⎝

⎛minus

b 6 x 4 x 82 minus+ = c 125x 43=







3= c x4log

16=




a 4

3

bz

xylog b zlog32ylog





8








70 Solve


06110004t=⎟

⎠

⎞⎜⎝

⎛ +





Answers


1 A = 372deg a = 178 m c = 244 m

2 A = 6560deg b = 1942 cm c = 2727 cm

3 278 km

4 2934 m

5 6594 cm2

6 373 m2

7 a) 0 b) 1 c) 2





9





13 52 cm 88 cm


15 a) b) c)

16 a) 10 0 b) 2 8 c) 6- 4minus

17

18 17v = θ = 3319deg


20 v = 34 4

21 29 newtons

22 244 lb

23 20i + 2j

24 ndash3

25 7893deg

26 ndash24

27 orthogonal


29 701deg


31 125deg

32 305 lbs

33 226 lb

34 135 mi 504deg




10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3


21+minus


43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i


( )oo 350sin 350 cos23 i+




b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus




11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


5


oo

150sin150cos2

120sin120cos4

i

i

+



2ii








a) 0x 4 =minus i b) 016x 4 =+


a) ( )o135 2minus b) ⎟⎠⎞

⎜⎝⎛ minus

35 3 π


⎞⎜⎜⎝

⎛minusminus

21




a) 3x ndash 2y = 6 b) 9yx 22 =+


6


a r = a cos θ A

b r = a sin θ B



e r = a cos 4θ E

f r = a sin 5θ F


cminus

G

h r2 = 4 sin 2θ H


7


a) r = 2 + 2 cos θ b) r = 4 cos 2θ



minus=





a 49

32 x

=⎟⎠

⎞⎜⎝

⎛minus

b 6 x 4 x 82 minus+ = c 125x 43=







3= c x4log

16=




a 4

3

bz

xylog b zlog32ylog





8








70 Solve


06110004t=⎟

⎠

⎞⎜⎝

⎛ +





Answers


1 A = 372deg a = 178 m c = 244 m

2 A = 6560deg b = 1942 cm c = 2727 cm

3 278 km

4 2934 m

5 6594 cm2

6 373 m2

7 a) 0 b) 1 c) 2





9





13 52 cm 88 cm


15 a) b) c)

16 a) 10 0 b) 2 8 c) 6- 4minus

17

18 17v = θ = 3319deg


20 v = 34 4

21 29 newtons

22 244 lb

23 20i + 2j

24 ndash3

25 7893deg

26 ndash24

27 orthogonal


29 701deg


31 125deg

32 305 lbs

33 226 lb

34 135 mi 504deg




10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3


21+minus


43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i


( )oo 350sin 350 cos23 i+




b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus




11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


6


a r = a cos θ A

b r = a sin θ B



e r = a cos 4θ E

f r = a sin 5θ F


cminus

G

h r2 = 4 sin 2θ H


7


a) r = 2 + 2 cos θ b) r = 4 cos 2θ



minus=





a 49

32 x

=⎟⎠

⎞⎜⎝

⎛minus

b 6 x 4 x 82 minus+ = c 125x 43=







3= c x4log

16=




a 4

3

bz

xylog b zlog32ylog





8








70 Solve


06110004t=⎟

⎠

⎞⎜⎝

⎛ +





Answers


1 A = 372deg a = 178 m c = 244 m

2 A = 6560deg b = 1942 cm c = 2727 cm

3 278 km

4 2934 m

5 6594 cm2

6 373 m2

7 a) 0 b) 1 c) 2





9





13 52 cm 88 cm


15 a) b) c)

16 a) 10 0 b) 2 8 c) 6- 4minus

17

18 17v = θ = 3319deg


20 v = 34 4

21 29 newtons

22 244 lb

23 20i + 2j

24 ndash3

25 7893deg

26 ndash24

27 orthogonal


29 701deg


31 125deg

32 305 lbs

33 226 lb

34 135 mi 504deg




10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3


21+minus


43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i


( )oo 350sin 350 cos23 i+




b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus




11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


7


a) r = 2 + 2 cos θ b) r = 4 cos 2θ



minus=





a 49

32 x

=⎟⎠

⎞⎜⎝

⎛minus

b 6 x 4 x 82 minus+ = c 125x 43=







3= c x4log

16=




a 4

3

bz

xylog b zlog32ylog





8








70 Solve


06110004t=⎟

⎠

⎞⎜⎝

⎛ +





Answers


1 A = 372deg a = 178 m c = 244 m

2 A = 6560deg b = 1942 cm c = 2727 cm

3 278 km

4 2934 m

5 6594 cm2

6 373 m2

7 a) 0 b) 1 c) 2





9





13 52 cm 88 cm


15 a) b) c)

16 a) 10 0 b) 2 8 c) 6- 4minus

17

18 17v = θ = 3319deg


20 v = 34 4

21 29 newtons

22 244 lb

23 20i + 2j

24 ndash3

25 7893deg

26 ndash24

27 orthogonal


29 701deg


31 125deg

32 305 lbs

33 226 lb

34 135 mi 504deg




10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3


21+minus


43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i


( )oo 350sin 350 cos23 i+




b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus




11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


8








70 Solve


06110004t=⎟

⎠

⎞⎜⎝

⎛ +





Answers


1 A = 372deg a = 178 m c = 244 m

2 A = 6560deg b = 1942 cm c = 2727 cm

3 278 km

4 2934 m

5 6594 cm2

6 373 m2

7 a) 0 b) 1 c) 2





9





13 52 cm 88 cm


15 a) b) c)

16 a) 10 0 b) 2 8 c) 6- 4minus

17

18 17v = θ = 3319deg


20 v = 34 4

21 29 newtons

22 244 lb

23 20i + 2j

24 ndash3

25 7893deg

26 ndash24

27 orthogonal


29 701deg


31 125deg

32 305 lbs

33 226 lb

34 135 mi 504deg




10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3


21+minus


43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i


( )oo 350sin 350 cos23 i+




b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus




11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


9





13 52 cm 88 cm


15 a) b) c)

16 a) 10 0 b) 2 8 c) 6- 4minus

17

18 17v = θ = 3319deg


20 v = 34 4

21 29 newtons

22 244 lb

23 20i + 2j

24 ndash3

25 7893deg

26 ndash24

27 orthogonal


29 701deg


31 125deg

32 305 lbs

33 226 lb

34 135 mi 504deg




10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3


21+minus


43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i


( )oo 350sin 350 cos23 i+




b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus




11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


10


37 i 212

38 i32

32x plusmnminus=

39 a) i21 b) ndash3


21+minus


43 i23

233+minus

44 ( )oo sin225 icos225 25 +

45 a) 32020 i+ b) 308580 + 185414 i

46 iminus3

47 i21

23minusminus

48 a) i2

3272

27minus b) 128 + 128i


( )oo 350sin 350 cos23 i+




b) ⎟⎠⎞

⎜⎝⎛

3π 3 ⎟

⎠⎞

⎜⎝⎛minus

34π3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

233

23

52 (1 210deg) (-1 30deg)

53 a) 2sinθ3cosθ

6rminus




11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


11

55a

55b

56a

56b


59 a $381112 b $525202

60 10

61 a x = 4 b x = 243 c 21x =

x y 2

43

3 21

4 0 5 -1 6 -3


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63


12


x y

917minus -6

35minus -5

-1 -4 1 -3 7 -2


3 2

2

ayz

xlog

64 135

65 71032 minustimes

66 ndash39069



69 a 6 b about 5000000 I0

70 a 7642 b 85476 c ndash5767 d 270246

71 a 1 b 32639 c 45 d Oslash

72 51 years

73 63

trigonometria final exam review

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