232-42072155.aus m trig polar coords solutions aus
TRANSCRIPT
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7/27/2019 232-42072155.AUS M Trig Polar Coords Solutions AUS
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Trigonometry
&
PolarCoordina
tes
www.mathlecs.com.au
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1SERIES TOPIC NUMBER
M 3
Knowing MoreSolutions
5. FindthecoordinatesofthepointsA,B,CandDcorrectto3decimalplaces.
a
c
b
d
A:
C:
B:
D:
(to d.p.)
, ,
. , .
cos sinA x y A
A
45 45
0 707 0 707 3
` c c=
=
^ ^
^
h h
h
(to d.p.)
, ,
. , .
cos sinC x y C
C
250 250
0 342 0 940 3
` c c=
= - -
^ ^
^
h h
h
(to d.p.)
, ,
. , .
cos sinB x y B
B
170 170
0 985 0 174 3
` c c=
= -
^ ^
^
h h
h
(to d.p.)
, ,
. , .
cos sinD x y D
D
315 315
0 707 0 707 3
` c c=
= -
^ ^
^
h h
h
cos
sin
x
y
45
45
45
c
c
c
i=
=
=
cos
sin
x
y
250
250
250
c
c
c
i=
=
=
cos
sin
x
y
170
170
170
c
c
c
i=
=
=
cos
sin
x
y
315
315
315
c
c
c
i=
=
=
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1SERIES TOPIC NUMBER
M4
Knowing MoreSolutions
6. Completethetableforthefollowingpointsroundedto2d.p.whereappropriate:
Point r iPolarcoordinates
(r,i)x= rcosi y = rsini (x, y) Coordinates
A 2 30c A(2, 30c) 2 30
1.73
cos 3c =
=
sin2 30 1c = . ,A 1 73 1^ h
B 4 85c B(4, 85c) .cos4 85 0 35c = .sin4 85 3 98c = . , .B 0 35 3 98^ h
C 2 110c C(2, 110c) .cos2 110 0 68c = - .sin2 110 1 88c = . , .C 0 68 1 88-^ h
D 4 230c D(4, 230c) .cos4 230 2 57c = - .sin4 230 3 06c = - . . .D 2 57 3 06- -^ h
E 3 300c E(3, 300c)
.
.
cos cos3 300 3 60
3 0 50
1 50
c c=
=
=
^ h
3 300 3 60
.
sin sin
2
3 3
2 60
c c= -
= -
= -
. , .E 1 50 2 60-^ h
F 5 340c F(5, 340c) .cos3 340 2 82c = .sin3 340 1 03c = - . , .F 2 82 1 03-^ h
1
2
3
4
5
-1
-2
-3
-4
-5
1 2 3 4 5- 5 - 4 - 3 - 2 -1
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1SERIES TOPIC NUMBER
M6
Knowing MoreSolutions
,ri^ h cosx r i= siny r i= ,x y^ h
,2 45c
^ h cosx 2 45 1c
= = siny 2 45 1c
= = ,1 1^ h
,3 60c^ h .cosx 3 602
31 5c= = =
2
3.siny 3 60 3 2 60c= = =c m . , .1 50 2 60^ h
,5 130c^ h .cosx 5 130 3 21c= = - .siny 5 130 3 83c= = . , .3 21 3 83-^ h
,4 200c^ h 4 200 3.76cosx c= = - .siny 4 200 1 37c= = - . , .3 76 1 37- -^ h
,6 315c^ h .cos cosx 6 315 6 452
64 24c c= = = = .siny 6 315 4 24c= = - . , .4 24 4 24-^ h
,7 270c^ h cosx 7 270 7 0 0c= = =^ h siny 7 270 7c= = - ,0 7-^ h
9. Completethetablebelowtondtherectangularcoordinates(to2d.p.):
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1SERIES TOPIC NUMBER
M 7
Using Our KnowledgeSolutions
i360c
iArclengthcoveredby i iinRadians
30c
360
30
12
1
c
c
= r r121
2 6# r
r
= r r6 6'r r
=
60c360
60
6
1
c
c= 2 r r
6
1
3# r
r= r r
3 3'
r r=
90c360
90
4
1
c
c= 2 r r
4
1
2# r
r= r r
2 2'
r r=
012 c360
120
3
1
c
c= 2 r r
3
1
3
2# r
r= r r
3
2
3
2'
r r=
2 04 c360
240
3
2
c
c= 2 r r
3
2
3
4# r
r= r r
3
4
3
4'
r r=
027 c360270
43
c
c= 2 r r
43
23# r r= r r
23
23'r r=
030 c360
300
6
5
c
c= 2 r r
6
5
3
5# r
r= r r
3
5
3
5'
r r=
036 c360
3601
c
c= 2 r r
1
12# r r= 2 2r r'r r=
10.Usethecircleandtheaboveformulatocompletethetablebelow(ignorethelastcolumnfornow):
11.Completethelastcolumnofthetableonthepreviouspage.Usethetabletodrawalltheanglesinradians
ontheaxesbelow:
radians306
cr
=
radians063
cr
=
radians902
cr
=
radians0123
2c
r=
radians2 043
4c
r= radians030
3
5c
r=
radians360 2c r=
radians0272
3c
r=
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1SERIES TOPIC NUMBER
M8
Using Our KnowledgeSolutions
12. Usethisformulatoconvertthefollowinganglesfromdegreestoradians(leaveanswersasfracons):
a
a
a
c
c
c
e
e
b
b
d
d
f
f
b
b
rad
60 60180
3
#c cc
r
r
=
=
3
2
3
2 180120#
cc
r r
r= =
2702
3
2
3 180#
cc
r r
r= =
rad
40 40180
9
2
#c cc
r
r
=
=
rad
225 225180
4
5
#c cc
r
r
=
=
rad
90 90180
2
#c cc
r
r
=
=
4 4
18045#
cc
r r
r= =
3
4
3
4 180240#
cc
r r
r= =
1506
5
6
5 180#
cc
r r
r= =
3154
7
4
7 180#
cc
r r
r= =
rad
100 100180
9
5
#c cc
r
r
=
= rad
315 315180
4
7
#c cc
r
r
=
=
13. Todevelopaformulatoconvertradianstodegrees,makeicthesubjectoftheformulaabove:
(degrees) =(radians) #180cr
14. Usethisformulatoconvertthefollowinganglesfromradianstodegrees(1decimalplace):
15.
Find the arc length L.
cmL r 44
#ir
r= = =
cmL r 6 3
4
8#ir
r= = =
cm.L r 4 86
54#i
rr= = =
cm2 2L r r r#i r r= = =
Find the perimeter of the following sector.
6
FindL and Pin the following diagram. Find the arc length of the enre circle below.
(Hint: Convert the angle to radians rst)
This is the formula for circumference of a circle.
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1SERIES TOPIC NUMBER
M 9
Using Our KnowledgeSolutions
Idenfy the major and minor segments on the diagram
16. Usethediagramontheletoanswerthefollowingquesons:
a
b
c
d
Find the area of the sector: cm6r2
1
2
16
3
2 2 2
#ir
r= =^ h
3r
6 cm
Minor segment
Major segment
Find the area of the minor segment
Area sector cm cm ( d.p.)6 6 18.85 2r2
1
2
1
3
2 2 2 2i
rr= = = =^ h
Area triangle cm or cm ( d.p.)60 18 9 15.59 2sin sinr2
1
2
16
2
33
2 2 2 2ci= = = =^ ch m
Find the area of the triangle
Area minor segment = Area sector - area triangle cm cm ( d.p.)6 9 3.26 23 22r= - =
17. Findthearclengthofasectorsubtendinganangleof6
5randaradiusof12cm.
6
5rcm10
L r
126
5
i
r
r
=
=
=
` j
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1SERIES TOPIC NUMBER
M10
Using Our KnowledgeSolutions
18. Asectorwitharadiusof4cmhasanareaof cm4 2r .Whatistheangleinsidethesector?
Use the formula for the area of a sector, and solve for i.
rad
4
4
4 8
8 4
2
14
2
16
8
4
2
2` r i
r i
r i
i r
ir
ir
=
=
=
=
=
=
^ h
19. Findtheoverlappingareaofthe2circlesbelowifx3
r= and y
3
2r= (to2d.p.):
x y
12 cm
3 cm
The overlapping area is the sum of2 minor segments. Use the
formula for the area of minor segment for each of these.
Area
cm
12 3sin sin2
1
3 3 2
1
3
2
3
2
723 2
3
2
9
3
2
2
3
24 36 3 34
93
27 34
144 9
274
1533
2 2
2
`r r r r
r r
r r
r
r
= - + -
= - + -
= - + -
= -+
= -
` `
c c
`
`
j j
m m
j
j
20. Whatistheperimeteroftheoverlappingareainthepreviousqueson?
The perimeter of the overlapping area is the sum of 2 arc lengths. Use the arc length formula for each side.
Perimeter
cm6
123
33
2
3
12 6
3
18
`r r
r r
r
r
= +
=+
=
=
` `j j
First substute A 4r= and r=4.
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1SERIES TOPIC NUMBER
M 11
Using Our KnowledgeSolutions
21. FindtheareaoftheregionbelowifMisthecentreofthecircleandABisthediameter:
23c
23c
16cm
D
B
A
M
C
By symmetry the two triangles are exactly the same,
since AMB+ is a straight angle so
AMD AMC23 23 180c c c+ ++ = + =
AMD AMC
180 23
157
`
c c
c
+ +=
= -
=
Total area
cm ( d.p.).
sin sin2
116 157
2
116 157
2
116
180
46
202 79 2
2 2 2
2
` c cr
= + +
=
^ ^ ^ `h h h j
Convert the angle of46c to radians: rad46180
46c
r=
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1SERIES TOPIC NUMBER
M12
Thinking MoreSolutions
i sini cosi tani
0 0 1 0
6r
2
1
2
3
3
1
3r
2
3
2
13
4r
2
1
2
11
2r
1 0 Undened
4
5r
2
1-
2
1- 1
2
3r-1 0 Undened
22. Usetheexactraosabovetocompletethetable:
23. Completetheradianvaluesforthequadrantsontheaxes:
24. Completethefollowingtables:
rad2
90cr ^ h
rad2
3270c
r ^ h
rad0 0c^ hrad 180cr ^ h
a bRuleindegrees Ruleinradians
sin cos90c i i- =^ h sin cos2
ri i- =` j
cos sin90c i i- =^ h cos sin2
ri i- =` j
tan cot90c i i- =^ h tan cot2
ri i- =` j
Ruleindegrees Ruleinradians
sin sin180c i i- =^ h sin sinr i i- =^ h
cos cos180c i i- = -^ h cos cosr i i- = -^ h
tan tan180c i i- = -^ h tan tanr i i- = -^ h
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1SERIES TOPIC NUMBER
M 13
Thinking MoreSolutions
c dRuleindegrees Ruleinradians
sin sin180c i i+ = -^ h sin sinr i i+ = -^ h
cos cos180c i i+ = -^ h cos cosr i i+ = -^ h
tan tan180c i i+ =^ h tan tanr i i+ =^ h
Ruleindegrees Ruleinradians
sin sin360c i i- = -^ h sin sin2r i i- = -^ h
cos cos360c i i- =^ h cos cos2r i i- =^ h
tan tan360c i i- = -^ h tan tan2r i i- = -^ h
24.
25. Usewhatyouknowaboutquadrantstocompletethefollowingtable:
Expressionofi i(radians) Quadrantofi SimplicaonRule
sin6
r
6r
1st
sin6
r
cos2
r
2r
1st
cos2
r
sin4
3r
4
3r2nd
sin sin sin4
3
4
3
4
rr
r r= - =` j
tan
12
11r
12
11r2nd
tan tan tan
12
11
12 12
rr
r r= - = -
` j
cos4
5r
4
5r3rd
cos cos cos4
5
4 4
rr
r r= + = -` j
tan3
2r
3
2r2nd
tan tan tan3
2
3 3
rr
r r= - = -` j
cos10
9r
10
9r2nd
cos cos cos10
9
10 10
rr
r r= - =-` j
sin 10
19r
10
19r4
th
sin sin sin10
192 10 10
rr
r r= - = -` j
tan4
7r
4
7r4
thtan tan tan
4
72
4 4
rr
r r= - = -` j
sin10
r
10
r1
stsin
10
r
cos7
8r
7
8r3rd
cos cos cos7
8
7 7
rr
r r= + = -` j
tan8
15r
8
15r4
thtan tan tan
8
152
8 8
rr
r r= - = -` j
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1SERIES TOPIC NUMBER
M14
Thinking MoreSolutions
iasafracon iasadecimal Quadrant
2
r1.5707... 1
st
6r
0.5235... 1st
3
4r4.1887... 3
rd
4
3r2.3561... 2
nd
4
7r5.4977.. 4
th
Point r iPolarCoordinates
,r i^ hcosx r i= siny r i=
,x y^ hCoordinates
A 56r
,56r` j cos5
6 2
5 3r= 5 sin
6 2
5r= ,
25 3
25c m
B 42r ,
24 r` j cos4
20
r= 4sin
24
r= ,0 4^ h
C 36
7r ,673 r` j 3 cos
6
7
2
3 3r= - 3 sin
6
7
2
3r=- ,
23 3
23
- -c m
D 54
7r,47
5r` j cos5
4
7
2
5r= 5sin
4
7
2
5r=- ,
2
5
2
5-c m
E 2 r ,2 r^ h cos2 2r = - 2 sin 0r = ,2 0-^ h
26. Canyoucompletethetablebelow?
4
5
-
-
-3
-4
-5
-
--
-
3
2
1
1 2 3 4 55 4 2 1
1
-3
27. Completethetableforthefollowingpoints:
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1SERIES TOPIC NUMBER
M 15
Thinking MoreSolutions
,x y^ h Quandrant r x y2 2= + tanAx
y1=
- ` j i ,r i^ h
,3 4-^ h 4th 5r 3 42 2= + - =^ h . ...tan3
40 927
1 -= -
-
` j.A2 5 36r+ = , .5 5 36^ h
3 , 6^ h 1st 3r 6 32 2
= + =^ ^h h3
0.9 ...tan tan6
2 551 1
= =- -c m .A 0 96= , .3 0 96^ h
,6 8-^ h 2nd r 6 8 102 2= - + - =^ ^h h 0.927...tan6
81
-=-
- ` j .A 2 21r+ = , .10 2 21^ h
,12 9- -^ h 3rd r 12 9 152 2= - + - =^ ^h h 0.927...tan9
121
-
-=
- ` j .A 4 07r+ = , .15 4 07^ h
5, 0-^ h - 5r 5 02 2= - + =^ ^h h tan5
00
1
-=
- ` j r ,5 r^ h
,0 5^ h - 5r 0 52 2= + =^ ^h h undefinedtan0
51=
- ` j2r ,5
2r` j
,1 3- -^ h 3rd r 1 3 222
= - + - =^ ^h h tan1
3
3
1 r
-
-=
- c m A3
4r
r+ = ,2
34r` j
2 , 7-^ h 4th r 2 7 32 2
= + - =^ ^h h . ...tan2
71 0799
1 -= -
- c m .A 2 5 20r+ = , .3 5 20^ h
,r i^ h cosx r i= siny r i= ,x y^ h
,43r` j cos4
34
2
12
r= =` j sin4
34
2
32 3
r= =c m ,2 2 3^ h
,24r` j 1cos2
4
r= sin2
41
r= ,1 1^ h
,345r` j cos3
4
5
2
3r=- sin3
4
5
2
3r=- ,
2
3
2
3- -c m
,28
15r` j . ...cos2 815 1 8477r = . ...sin2 815 0 7653r = - (to d.p.). , . 21 85 0 77-^ h
,12r` j cos
20
r= sin
21
r= ,0 1^ h
,3 r^ h cos3 3r = - sin3 0r = ,3 0-^ h
,523r` j cos5
2
30
r= 5 5sin
2
3r= - 0, 5-^ h
28. Usetherulesforeachquadranttocompletethefollowingtable(to2d.p.):
29. Completethetablebelowtondtherectangularcoordinates(Setcalculatorstoradianmode):
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1SERIES TOPIC NUMBER
M16
Notes
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