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??1?1-1????ABC??CB??B???AC????C??? ????C????????AB??????????????

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????????????????????C B ?.????

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??????????????????????????????????????????????Rt?ABC???BC=a,AC=b,AB=c, ?????????????????

abc?sinA??sinB??sinC?1?ccc

abc????csinAsinBsinCC abc????????ABC????sinAsinBsinC?

??1?????????????????????????????????? ???????????????????

??1?1-3??1???ABC??????????AB????CD??????????????

?CD=asinB?bsinA,?

????

??asinA?bsinB?csinC??bsinB?? Ac B sinAsinBsinC

?2???ABC????????????????????????????? ??2????????

?????????????????????????

abc

???????????????A?????j?AC? ???????? AB?AC?CB

??????????????? j?AB?j?(AC?CB)

?????????????????j?AB?j?AC?j?CB

??????????0jABcos?90?A??0?jCBcos?900?C?

?csinA?asinC???????????ac ??????abcbc?????C?j?BC??? ?? ???sinAsinBsinC

???????????????

?????????????????????????????a

sinA?b

sinB?c

sinC

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?1?????????????????????????????????????

?????k?a?ksinA?b?ksinB?c?ksinC?

?2?a

sinAsinBsinCsinA

???????????????? ?b?c???a?bsinB?csinC?bsinB?asinA?csinC

???????????????????????a?bsinA? sinB

???????????????????????????????sinA?sinB?

?????????????????????????????????

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?1???ABC????A?32.00?B?81.80?a?42.9cm??????

????????????? ab

C?1800?(A?B)?1800?(32.00?81.80)?66.20?

asinB42.9sin81.80

??????? b???80.1(cm)? sin32.00

asinC42.9sin66.20

??????? c???74.1(cm). sin32.0??????????????????????

?????ABC?????????????

?1?A?45?C?30?c?10cm? ?2?A?60?B?45?c?20cm ?2? ??ABC????a?20cm?b?28cm?A?400???????????10??????1cm??

?????????

????

bsinA28sin400

sinB???0.8999.??00?B?1800???B?640??0 B?116.

? ?B?640?? C?108?0A(?B0?)10?800??(4?064asinC20sin760

c???30(cm). sin400

? ?B?1160??C?108?0A?(B0?)01?8?0?(4?01asinC20sin240

c???13(cm). sin40

??????????????????????????????

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?4????2??

??????ABC??a

sinAsinB?.?????????????

?1??????????b?csinC?k(k>o)???k??ABC?????? a?b?c?k?k?0?? sinAsinBsinCsinA?sinB?sinC

?a?ksinA?b?ksinB?c?ksinC(k?0) a?b?c?

?2???????????

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1.2???????? ????

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3? ????

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(2)?1?????A?B???????????????????????A???????????????C???AC????55m??BAC=51???ACB=75???A?B?????(???

0.1m)

??1??ABC????????????????????????

??2????????????????????????

?????????????????????????????????????????????AB????AC?????????????????????????????AC????????????AB??

?????????? AB=ACsin?ACBsin?ABC

sin?ABC55sin75? = 55sin75? ? 65.7(m) sin54?sin(180??51??75?) AB =ACsin?ACB=55sin?ACB= sin?ABC

?:A?B???????65.7?

????????A?B??????C??????a km,??A????C????30????B????C???60???A?B?????????

???????????????? ???2a km

?2????A?B?????????????????????A?B????????? ??????1???????????????????????????????????????????C?D????????????????????????????????????????AC?BC?????????????AB????

???????????????C?D???CD=a????C?D???????BCA=??

? ACD=???CDB=???BDA =????ADC??BDC?????????AC =

BC = asin(???)= asin(???) sin[180??(?????)]sin(?????)asin?asin?=

sin[180??(?????)]sin(?????)

???AC?BC?????ABC???????????AB??????AB = AC2?BC2?2AC?BCcos?

????????????????????????????????

?????????????40??C?D??????BCA=60???ACD=30???CDB=45???BDA =60?

?????????????2???????AB=206

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2

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a?2Rsin

A?b?2RsinB?c?2RsinC sinA?

a2R

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sinB?

12

b2R

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b?c?a

2bc

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2

2

cosA??cosB?

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2ac

222

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a?b?c

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222

3??? ??

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iii. ?an?1?pan?q????an?1?m?p(an?m)??m,??????an?m?

?Sn?f(an)

iv. ?Sn?f(an

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?Sn?1?f(an?1)?Sn?2an?1

???Sn?2an?1??a1???????????????an?1?2an?1?2an

?Sn?1?2an?1?1

2.?????

? ???an?1?an=d????,??????????????? ? ??d?0??an???n??????

d?0??a

n????????d?0??an????????

n(n?1)2

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d?

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iii. ?an??????,?Sn,S2n?Sn,S3n?S2n,K??????????qn? iv.G?a,b?????,G??ab 4.?????????:

?.???:?an?2n?3,an?3n?1

nn?1

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1

2

3

4

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3

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n

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1

2

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1.??????:

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a?b,b?c?a?c

a?b?

??a?c?b?d c?d?

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a?b,c?R?a?c?b?c,??:

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a

?b?a?b?a?b?0?

?ac?bc;?ac?bc;????ac?bd?0

c?0?c?0?c?d?0?

? a?b?0?an?bn?0;a?b?0?2.???????????:

a?

b?0

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2

????f?x??ax2?bx?c???????,????????????????? ?.??????????????.

?????x2?2ax?8?0??????0?1?????????4,5?????

f(0)?0?f(1)?0?f(4)?0?f(5)?0

3.???????

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?a?0,b?0?ab?????a+b????? ?a?0,b?0?a+b?????

ab????? ????????:

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?.????????????

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?.A+B+C=?,?

A?B2

?

?

2

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C2

?sin

A?B2

?cos

C2

?.??ABC?, a?b?c , a?b?c ; A?B?sinA?sinB,

A?B?cosA?cosB, a ?b? A?B

?.??ABC??????A?B?

?

2

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?

2

,A+C ?

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2

;

a2?b2?c2?b2?c2?a2?a2?c2?b2 2????????????.

(2R??ABC??????)

a?2Rsin

A?b?2RsinB?c?2RsinC sinA?

a2R

?

sinB?

12

b2R

? sinC?

12

c2R

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acsinB

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2

2

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2

2

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absinC?

2

bcsinA?

2

2

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b?c?a

2bc

2

2

2

cosA??cosB?

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222

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222

3??? ??

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iii. ?an?1?pan?q????an?1?m?p(an?m)??m,??????an?m?

?Sn?f(an)

iv. ?Sn?f(an)???a

1?????an?1?an??????

S?f(a)n?1?n?1?Sn?2an?1

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?Sn?1?2an?1?1

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n?1?an=d????,??????????????? ? ??d?0??an???n??????

d?0??an????????d?0??a

n????????

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d?

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?.???:?an?2n?3,an?3n?1

?.?????:?an?3n?2n?1?2n?5???????3n???2n?1???2n?5????????????????

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?

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Sn?5???7???9???????(3n?1)??

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1

2

3

4

2

3

n?1

n

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n

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n

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1

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2,a3,???,an???n+2???????? ??Sn?a1?a2?????an?????Sn?

32n?

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1.??????:

? a?b,b?c?a?c

?

a?b,c?R?a?c?b?c,??:

a?b?

??a?c?b?d c?d?

a

?b?a?b?a?b?0?

?

??ac?bc;??ac?bc;??ac?bd?0

c?0?c?0?c?d?0?

? a?b?0?an?bn?0;a?b?0?2.??????? ???????

a?

b?0

?a?0,b?0?ab?????a+b?????

?a?0,b?0?a+b?????ab?????


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