Unidad 6 - Trigonometría

8/12/2019 Unidad 6 - Trigonometra

1/29

4 ESO - MATEMTICAS - UNIDAD 6 (Trigonometra)

Pg!6 " 1 ( ) ( )( )

( ) === 295780,571416,3

18018011

radradrad

"45'1757

Pg!6 " 2 a) ( )

( ) = radrad

318060

( )rad05,1 #)

( )( ) = rad

rad

3

2

180120

( )rad09,2

$) ( )

( ) = radrad

6

7

180210

( )rad67,3 %)

( )( ) = rad

rad

4

3

180135

( )rad36,2

e) ( )

( ) = radrad

6

11

180330

( )rad76,5

Pg!6 " 3 a) ( ) ( ) ( )== 5

180180

5 radrad

36 #) ( )

( ) ( ) = 7

540180

7

3

radrad

"34'877

$) ( ) ( )=

radrad

1803

7420 %) ( ) ( )

=rad

rad

1804 720

e) ( ) ( )=

radrad

180

6

11330

Pg!6 " 4 ( )

( )radrad

A9

2

18040

== ( )

( ) =

radradB

18054,1 "8'1488

( )=+= "8'148840180C "52'4551 ( )+= 54,192C ( )rad90,0

Pg!6 " 5 ( )

==180

65 rad

CA

( )rad36

13 ==

36

13DB ( )rad

36

23

Pg!6 " 6 &i#ro'io'Net

Pg!* " 7 1316925144512 22 ==+=+=a

13

12=senB 13

5cos =B

5

12=tgB

13

5=senC 13

12cos =B

12

5=tgB

Pg!* " 8 a) 534528 22 =+=a

== CsenB cos 5328

==senCBcos 5345

==ctgCtgB 4528

==ctgBtgC 2845

#) 5530254873 22 ===c

== CsenB cos73

48

== BsenC cos73

55

==ctgCtgB55

48

==ctgBtgC48

55

$) 91215 22 ==b

== BsenC cos15

12 == CsenB cos

15

9 ==ctgBtgC

9

12 ==ctgCtgB

12

9

Pg!* " 9 ngulo 6 30 4 90 3 120

Seno 21

+ 0,5 1

1 23

0,87

Co'eno2

3 0,87 0 02

1

+ 0,5

,.!


2/29


Tangente 3

1 0,58 3 1,73

..!


3/29


Pg!! " 10 22 sentg + =

=

=

2

2

2

2

22

2

2

cos

cos11

cos

1

cossensensen

sen

22tgsen

Pg!! " 11 21

1tg

+ + =+

=+

2

22

2

2coscos

1sen

sen

sen 2

1

sen

Pg!! " 12 =+=

+=

2241

4

1

tg

tgsen17

424,0

=+

=+

=22

41

1

1

1cos

tg 17

197,0

Pg!! " 13 =+=

+=

2275,01

75,0

1

tg

tgsen 6,0 =+

=+

=22

75,01

75,0

1

1cos

tg8,0

Pg!! " 14 === 22 554,01cos1 sen 83,0 ===554,0

554,01

cos

cos1 22

tg 50,1

Pg!! " 15 ==10

6sen 6,0 === 22 6,011cos sen 8,0 =

=

21 sen

sentg 5,1

==10

8sen 8,0 === 22 8,011cos sen 6,0

=

21 sen

sentg 67,0

Pg!! " 16 a) 22cos sen + ( ) = 221 sensen 221 sen

#)

2

22 2

tg

sentg ++ =+=+

22

2

2

2

cos/2121sen

sen

tg

sen2

cos21+

Pg!! " 17 a) tgtg 1+ + =+=+=+

sensen

sensen

sensen

coscoscos

coscos/1

cos

22

cos1sen

#)

2

1

tg + =

+=

+=

22

22

2

2

22

2

2

2

cos

1coscos

coscos

cos

sen

sen

sen

sen

sen

+ =

+=

+

2

2

2

2

2 11cos

1cos

tgtg

tg 22 coscos

+

tg

$) cos11

tg + ==

cos

1cos

cos

1

cos

1

sensen sen1

%) 22 cos

11+

sen+ =

+

22

22

cos

cos

sen

sen 22 cos

1

sen

e) 2cos + ( ) ( )=+= 2222 coscos1cos sen

422 coscos +sen

Pg!! " 18

cos21

211

2

+

++

sen

tgtg + ( )

=+

++=

++

cos21

cos2cos1

cos21

cos2cos1

22

22

2

sen

sensensen

sen

sensen

2

1

sen

Pg!! " 19 1cos

cos1cos

2

22 =

+

01cos2cos 24 =+ 1cos2 = 1cos =

&a re/a$i0n e' $ierta $1an%o 0= 2 180=

3.!


4/29


Pg,, " 20 En . $1a%rante5 cos 2 tg'on negatio'

37

35

1369

1225

37

121cos

2

==

= 37

35cos = ==

cos

sentg

35

12

Pg,, " 21 a) += 45180225 3cuadrante = 45225 sensen2

2

#) = 60360300 4cuadrante == 60300 sensen2

3

$) = 30180210 2cuadrante ( ) == 30210 tgtg3

1

%) = 30360330 4cuadrante ( ) === 30cos30cos)330cos(2

3

e) ( ) += 303602750 1cuadrante == 30cos750cos2

3

) = 120360480 3cuadrante ( ) == 60480 tgtg 3

Pg,, " 22 a) 53,6360/2350 ( ) 19036062350 +=

#) 99,11360/4315 ( ) 355360114315 ++=

$) 11,1360/400 ( ) 403601400 +=

Pg,, " 23 209070 = =70sen 20cos =70cos 20sen 70tg 201 tg

20180160 = =160sen 20sen =160cos 20cos 160tg20tg

20180200 += =200sen 20sen =200cos 20cos 200tg20tg

20360340 = =340sen 20sen

=340cos 20cos =340tg20tg

Pg,, " 24 4cuadrante ( ) == 253/281sen 53/45 =tg 28/45

Pg,, " 25 3cuadrante ( )=

+=

236771

3677sen

85

77

( ) 236771

1cos

+= +

85

36

Pg,, " 26 = 30180150 2cuadrante ( ) 330180150 cuadrante=== 30150 sensen 21 == 30cos150cos 23 == 30150 tgtg

31

( ) == 30150 sensen 21 ( ) == 30cos150cos 23 ( ) == 30150 tgtg31

Pg,, " 27 345180225 cuadrante+= ( ) 245180225 cuadrante+=== 45225 sensen 22 == 45cos225cos 22 == 45225 tgtg

1

( ) == 45225 sensen 22 ( ) == 45cos225cos 22 ( ) == 45225 tgtg1

Pg,, " 28 &o' tring1/o' re$tng1/o' %i#17a%o' tienen /a%o' 8er8en%i$1/are' entre '

S1' ng1/o' ag1%o' 'on ig1a/e' 2 /a' 9i8oten1'a' 'in i%:nti$a' Por tanto5 ta/e' tring1/o' tienen ig1a/e' /o' /a%o' $orre'8on%iente' ( ) cos90 =+sen ( ) sen=+90cos ( ) tgtg 190 =+

4.!


5/29


Pg,, " 29 &i#ro'io'Net

;.!


6/29


Pg,. " 30 3133 ==tgx 30=x kx 18030 +=

Pg,. " 31 a) 43sen 6820,0 #) 33cos 8387,0 $) =radtg 5,2 7470,0 %) =radsen18415,0

e) 16,0arcsen rad160691,0 "48'1310 ) 99,0arccos

rad141539,0 "35'68

g) 25,0arctg rad244979,0 "11'214 9) 1arcsen =rad570796,190

Pg,. " 32 a) == 1arcsenx 90 kx 36090 +=

#) == 1arccosx 180 kx 360180 +=

$) == 22arccosx 45 kx 36045 +=

%) == 0arctgx 1800 kx 1800 +=

e) == 3arctgx 12060 kx 18060 +=

) == 211802 arcsenx 15030 = 1802xkk 36015036030 ++

kx 3601502 += kx 18075 += kx 360302 += kx 18015 +=

Pg,3 " 33 CHA 48tgxh = CHB ( ) 2430 tgxh += ( ) 243048 tgxtgx +=

244824

30tgtg

tgx

= 24482448

30tgtg

tgtgh

=

4452,01106,1

4452,01106.130

( )m29,22

Pg,3 " 34 a) == 4390B 47 = 435 senc ( )cm41,3 = 43cos5b ( )cm66,3

#) =+= 22 86b ( )cm10 ( )= 68arctgA "48'753 ( )= 86arctgB"12'5236

Pg,3 " 35 AnchuraRoa lAlturarboh= 40tgah = ( ) 284 tgah =+

( ) 28440 tgatgah +== 284028

4tgtg

tga

= 28404028

4tgtg

tgtgh

=

5317,08391,0

5317,04a ( )m92,6

=

5317,08391,0

8391,05317,04h ( )m80,5

Pg,; " 36 ( ) =+= 6035180A 85

( )[ ]=

+=

6035180

354sen

senb

( )603535

4+

sen

sen

9962,0

5736,04 ( )cm30,2

( )[ ]=

+=

6035180

604sen

senc

( )603560

4+

sen

sen

9962,0

8660,04 ( )cm48,3

Pg,; " 37 ( ) 50cos64264 22 +=c =+ 15,2185,303616 ( )cm60,4

6.!


7/29


( )

( )50cos64264

50cos46arccos

22 +

=A 7454,0arccos

60,4

43,3arccos '4841A

( )

( )50cos64264

50cos64arccos

22 +

=B 0311,0arccos

60,4

14,0arccos '1288B

Pg,; " 38 ( )44cos404524045 22 +=x

=+ 38,103562,258916002025 ( )m18,23

*.!


8/29


Pg,; " 39 a) Tring1/o i'0'$e/e'ABC5 $on #a'e 26=c 2 /a%o' ig1a/e' 36=ba

3611,036

13

36

226coscos == BA "54'4968=BA

( ) =+= ABAC 2180180 "12'2042#) Tring1/o i'0'$e/e'ABC5 $on #a'e 5=a 2 /a%o' ig1a/e' cb

=4003,0

5,2

4,66cos

25cb 25,6 ( ) === 2,474,662180A "0'1247

Pg,; " 40 E' e/ ter$er /a%o %e 1n tring1/o5 $on 1n ng1/o %e *;5 $12o' /a%o' 'on

( ) ( )

( ) ( ) ( )kmkm

hhkm 67,1

6

100

min60

min10100 = ( )

( )

( ) ( ) ( )kmkm

hhkm 5,1

6

90

min60

min1090 ==

( ) + 7690,475cos5,167,125,167,1 22x ( )km18,2

Pg,; " 41 Se toma5 $omo #a'e %e/ tring1/o5 e/ /a%o 100 S1 a/t1ra e' $ateto5 o81e'to a 60 5 $on 9i8oten1'a 120

( ) == 33000601201002

1senreatringulo ( )m15,5196

Pg,; " 42 &a a#ert1ra %e 8ata' (+ , $m) e' e/ ra%io5 /1ego '1 %imen'i0n en ( )cm5210 =

E/ 'emi-ng1/o %e a#ert1ra 2/ e' o81e'to a/ $ateto 5 $on 9i8oten1'a 10

( )=2sen2

1

10

5

2==

sen 30

2=

60=

Pg,; " 43 38

3038

sensenA= 7798,038

30

38 = sensenA "45'14511=A

"28'451282 =A

( )=+= 11 38180 AC "15'4590 ( )+= 22 38180 AC"32'1413

=

38

30 11

sen

senCc ( )cm72,48 =

38

30 22

sen

senCc ( )cm16,11

Pg,6 " 44 136432 =++p ( )cmp 5,6=

( ) ( ) ( ) = 4375,2865,645,635,65,6S ( )233,5 cm

Pg,6 " 45 E/ 8entgono tiene ; tring1/o' ig1a/e'5 %e a/t1ra h ng1/o $entra/ %e 1n /a%o + 725360 = Mita% + 36 h+ $ateto5 $on otro $ateto + ; o81e'to a 36

==7265,0

125

36

125

36

510

2

15

tgtg

S ( )265,172 cm

Pg,6 " 46 Ca%a 1no %e /o' ! /a%o' ig1a/e' %e/ enegono mi%e 963 ( )cm7 Ca%a 1no %e '1' /a%o' '1#tien%e 1n ng1/o $entra/ %e 409360 =


9/29


&a a8otema e' 1n $ateto5 $on otro $ateto o81e'to a ng1/o %e 20240 =

==20

25,110

20

277

2

19

tgtgS ( )291,302 cm

Pg,6 " 47 a) rea %e 1n tring1/o = 31021 ( )215 cm

rea 'e$tor 2336075 ( )289,5 cm

rea 'om#rea%a ( )= 89,5152 ( )222,18 cm

!.!


10/29


#) &a%o' tring1/oABC ( )cma 7= ( )cmb 8= ( )cmc 9=

6667,0144

96

982

798cos

722

=+

=A "23'1148=A

5238,0126

66

792

879cos

222

=+

=B "43'2458=B

2857,011632872 987cos222

= +=C "54'2373=C

( ) ( ) ( ) ( ) ( ) ( ) === 70091281271212cpbpappStringulo 83,26

( ) ( ) ( )[ ]++= 222sec 340,73441,58519,48360

tores43,24

rea tring1/o $1ri/neo 43,2483,26 + ( )24,2 cm

Pg,* " 48 =a%io #a'e (r) $ateto o81e'to a 285 $on 9i8oten1'a 4$m

= 4695,04284 senr ( )cm88,1

[ ] + 288,1488,1 totalS ( )273,34 cm

Pg,* " 49 =a%io %e &1na 62 10384 = r 2/19103 =r ( )km739.1

3105763,4

380000

1739

2

=

sen 5244.022622,0 = "427'310

Pg,* " 50 >eneratri? (g) @i8oten1'a5 $on $ateto %e 5 $m5 o81e'to a 20

=

3420,0

5

20

5

sen

g ( )cm62,14 +

= 62,14

2

510lateralA ( )

264,109 cm

Pg,* " 51 Cara' e1i/tera' =

= 3252

355

2

14A ( )230.43 cm

Pg,* " 52 &i#ro'io'Net

Pg,< " 53 == 22 2025x 15 70=B ( )== 15270b 40 20=h 115=L

=+= 115202

4070V ( )3126500 cm

Pg,< " 54 a) ( ) ( )+=+=+= 8824 33

CilindroCuboV ( )313,89 cm

#) = 2040 tgr 56,14 4056,143

12

2V ( )3760.17 cm

Pg,< " 55 A8otema (a) $ara a/t1ra tring1/o e1i/tero /a%o * 2

37=a

A/t1ra (h) tetrae%ro $ateto $on 9i8oten1'a (a) 2 otro $ateto (a/3) ah2

6=

4

32aAbase = ==

=

12

27

12

2

3

6

4

3

3

1 3

3

2

aaa

V ( )342,40 cm

,.!


11/29


Pg,< " 56 ( )

=

= 33

302

610

3022

31000

tgtgL ( )cm48,13 = 30

2

2tg

Lh

( )cm94,7

,,.!


12/29


Pg,< " 57 &i#ro'io'Net

Pg,, " 58 a) =180

30

rad( )rad

6

( )rad5236,0 #) =

180270

rad( )rad

2

3 ( )rad7124,4

$) =180

240rad

( )rad3

4( )rad1888,4 %) =

180135

rad ( )rad4

3

( )rad3562,2 e) =

18090

rad( )rad

2

( )rad5708,1 ) =180

300

rad( )rad

3

5( )rad2360,5

Pg,, " 59 a) = radrad

1805 900 #) =

rad

rad

180

8'3022 $) =

rad

rad

180

6

5150

%) =rad

rad

180

3

4240 e) =

rad

rad

180

4

7315 ) =

rad

rad

180

11

7

"44'32114

Pg,, " 60 a) ( )+= 3601450 90

#) ( )+= 36051800 0

$) ( )+= 3602720 0

%) ( )+= 3601540 180 e) ( )+= 36031300 220 ) ( )+= 3602900 180

Pg,, " 61 a) ( )+= 3601480 120 =180

120rad

( )rad3

2( )rad0949,2

#) ( )+= 36031235 155 =180

155rad

( )rad36

31( )rad7053,2

$) ( )+= 3602930 210 =180

210rad

( )rad6

7 ( )rad6652,3

%) ( )+= 36041440 0 =180

0rad

( )rad0

Pg,, " 62 a) ( )+=

214

19

8

3#) ( )+=

23

3

22

3

4$) ( )+=

22

2

11

2

3

%) ( )+= 22041 e) ( )+=

216

17

6

5

Pg,, " 63 a) =110

36036 #) =5

10

360180 $) = 3

36

108E %) =

4

36

144A

Pg,, " 64 10=a 8=b 6=c Crnguloeno

== 106senC 6,0 == 10

8cosC 8,0 == 86tgC 75,0

Pg,, " 65 a) =63,3

3cosB '1634=B #) =

8,4

2cosC '2265=C

$) =4

2tgB '3426=B %) =

1,1

95,2tgC '3369=C

Pg,, " 66 ==25

2/24cos 48,0 =

21 sensen 88,0 48,0

88,0tg 83,1

Pg,, " 67 12,16,20 === cba 222 121620 +=

90=A

20/12cos =B "48'753=B 20/16cos =C "12'5236=C

,..!


13/29


Pg,, " 68 a) 921,0 #) 434,0 $) 225,0 %) 309,0 e) 810,0 ) 414,2

Pg,, " 69 a) '117=A #) '5075=B $) '2051=C

Pg,, " 70 a) radA 016,1= #) radB 870,0= $) radC 507,0=

Pg,, " 71 === 64,06,01cos 2 8,0 ===

8,0

6,0

6,01

6,0

2tg 75,0

Pg,, " 72 === 36,08,01 2sen 6,0 ===8,0

6,0

8,0

8,01 2tg 75,0

Pg,, " 73 ( )

=+

=2

51

5sen

6

5 913,0( )

=+

=2

51

1cos

6

1408,0

Pg,,, " 74 a) ( ) ==+=+= 1cos

cos

cos

cos1

1

22

2

22

sensencos

#)( )

==+

=2

2

2

11 222 xsenxsenxsen

xsen2

$) =

=

+=

2222

cos

1

cos

cos

senxxsenxx

xxsen

xsenx 221

cos

1

Pg,,, " 75 a) ==

2

2

2

coscos

sen

2sen

#) ==

==

22

coscos

cos/

cos.

sen

sen

sen

sen

21 sen

$) ==+

=

+

22

2

2

222

2

2

coscos

1cos

cos

coscos

cos1

sensen1

Pg,,, " 76 a) 4315 cuadrante #) 2850 cuadrante $) 4340 cuadrante %) 2150 cuadrante e) 2118 cuadrante ) 2460 cuadrante

g) 3190 cuadrante 9) 3230 cuadrante i) 1721 cuadrante

Pg,,, " 77 a) 10 2 #) 30 4 $) 10 3 %) 20 3

Pg,,, " 78 16250,05625,00625,075,025,0cos 2222

=+=++ sen No

Pg,,, " 79 a) =

=

2

7

61sen

7

13 515,0 ==7/13

7/6tg

13

6664,1

#) == 2375,01cos 859375,0 927,0 =tg 859375,0375,0

405,0

$)( )

=+

=2

21

2sen

3

2 816,0 ( )

=+

=2

21

1tg

3

1 577,0

,3.!


14/29


Pg,,, " 80 a) =

=

2

7

41sen

7

33 821,0 =

=7/4

7/33tg

4

33 436,1

#) ( ) == 29,01cos 19,0 436,0 =

=19,0

9,0tg

19,0

9,0 065,2

$)( )

=

+

=2

21

2sen

3

2 816,0 ( )

=

+

=2

21

1cos

3

1 577,0

Pg,,, " 81 a) 58,165 "54'34165 #) 41,277 "43'24277 $) 59,246

"32'35246

Pg,,, " 82 a) 5931,01416,35931,05931,0 5485,2

#) 6841,028322,66841,026841,0 5991,5

$) ++ 9828,01416,39828,09828,0 1244,4

Pg,,, " 83 a) ( ) === 60120180 sensen2

3 #) == 60sen2

3

$) ( ) === 30150180 sensen2

1

%) ( ) === 3030180 sensen2

1

e) ( ) ==+= 45cos45180cos2

2 ) ( ) === 30cos150180cos

2

3

g) ( ) ==+= 4545180 sensen2

2 9) ( ) ==+= 30cos360330cos

2

3

i) ( ) ==+= 45360315 tgtg 1 7)

( ) ==== 30210720930 tgtgtg3

1

B) ( ) === 60360300 sensen2

3 /) ( ) ==+= 45360315 tgtg 1

Pg,,, " 84 135 ( ) === 45135180135 sensensen2

2

( ) === 45cos135180cos135cos2

2

=

==2/2

2/2

135cos

135135 sen

tg 1

225 ( ) ==+= 4545180225 sensensen2

2

( ) ==+= 45cos45180cos225cos2

2

=

==

2/2

2/2

225cos

225225 sen

tg 1

Pg,,, " 85 ( ) ( ) ( ) ( ) ===+=+ 2951180360180180 sensensensen 9/53

( ) ( ) ( ) ==+=+ cos180cos360180cos180cos 9/5

,4.!


15/29


( ) ( ) ( ) ==+=+ tgtgtgtg 180360180180 5/53

,;.!


16/29


Pg,,, " 86 a) ( ) ( ) ===+ 213121180 sensen 13/5

#) ( )( )

==

=

==13/5

13/12

13121

13/12

cos1

cos190

22

tgtg

5

12

$) ( ) == cos180cos 13/12

%) ( ) ( ) === 213121 sensen 13/5

Pg,,, " 87 a) = 30180x 150 0 210

#) 2/1cos =x = 60x 60 0 60

$) 60=x 0 24060180 =+=x = 30270x 240 0 300

%) 45=x 0 22545180 =+=x = 90135x 45 0 225

Pg,,, " 88 a) rad1071,1 0 =++ 1416,31071,11071,1 rad0345,2

#) 1cos =x =x rad1416,3

$) rad9442,0 0 9442,01416,39442,0 rad1974,2

%) 4/1=senx x rad

2527,0

0 2527,01416,32527,0 8889,2

Pg,,, " 89 a) == 6090C 30 === 211260cos12c ( )cm6 == 3012 senb 36 ( )cm39,10

#) =CB 45 =+= 22 1111a 211 ( )cm56,15

$) == 4090B 50 = 40/9 senc ( )cm00,14 = 40/9 tgb( )cm73,10

%) = 22 1023a ( )cm71,20 == 2310cosAsenB '4625=B '1464=C

Pg,,. " 90 a) == 5590B 35

= 55/18 senc ( )cm74,14 = 55/18 tgc

( )cm50,10

#) == 6,010/6senB "12'5236=B = 6,0cosA "48'753=C ==== 6436100610 22a ( )cm8

$) == 5/615/18tgA "40'1150=A == 6/518/15tgB

"20'4839=B

=+=+= 5492253241518 22c ( )cm43,23

Pg,,. " 91 ( ) === 3614421212 22Altura 108 ( )cm39,10

Pg,,. " 92 ( )== 280cos/16Altura 40cos/16 ( )m26,12

Pg,,. " 93 4'=BA 20=aBC == 204cBA 54 ( )cm94,8

( ) ==== 320804005420 22b 58 ( )cm89,17

==5

1

20

54cosB "58'2663=B ==

5

1cosBsenC

"54'3326=C

Pg,,. " 94 E/ rom#o orma5 $on '1' %iagona/e'5 4 tring1/o' re$tng1/o' ig1a/e' @i8oten1'a + lado%e/ rom#o L

,6.!


17/29


Cateto + mita% %e /a diagonal $ono$i%a + 2/8 + ( )cm4 ng1/o $om8ren%i%o + 26

26cos/4=L 8988,0/4 ( )cm45,4

,*.!


18/29


Pg,,. " 95 90=A 90=B ==

3

7

912

7tgD "5'4866=D

"55'1123907

390 ++= arctgC "55'11113=C o #ien

)(360 DBAC ++=

Pg,,. " 96 a) 95=A ( )mb 20= ( )mc 18=

( ) +++= 7521,6232440095cos182021820 22a ( )m05,28

( ) ( ) = 05,28/9520/ senasenAbsenB 7103,0 "40'1545=B

( ) ( ) = 05,28/9518/ senasenAcsenC "20'4439

Pg,,. " 97 a) ( ) ++= 7408,1342592,1256419656cos8142814 22a ( )cm61,11

( ) = 9997,061,115614 sensenB "36'989=B ( ) 5713,061,11/568 sensenC "24'5034=C #) ( ) =+= 7545180A 60

3

50

2/3

25

60

25

7545=

sensen

c

sen

b

= 453

50senb ( )cm41,20 = 75

3

50senc ( )cm88,27

$) 5711,02024

14444842116

22462

382246cos

222

+

=

+=A "11'1055A

1124,01672

21161444484

38222

463822cos

22

=+

=

+=B "22'2796=B

46382

224638cos

222

+

=C + 8799,03496

48421161444

+ "27'2228=C

%) 42=A 65=C ( )cmb 14= =+= )6542(180B 73

654273

14

sen

b

sen

a

sen ==

=65

4214

sen

sena ( )cm80,9

=

73

6514

sen

senb ( )cm27,13

Pg,,. " 98 19=a 13=c 50=B

( ) ++= 4629,2125371,31736116950cos191321913 22b ( )cm58,14

05396,008,379

4629,20

08,379

3611694629,212

1358,142

191358,14cos

222

+

+A '5486A

7300,104,554

4629,404

04,554

1694629,212361

58,14192

1358,1419cos

222

+

+C '643C

Pg,,. " 99 a) 3=a 2=b 140=C

( ) +++= 1925,221925,949140cos23223 22c ( )cm71,4

140

71,423

sensenBsenA= 4094,0

71,4

1403 sensenA '1024A

,


19/29


2729,071,4

1402 sensenB '5015B

,!.!


20/29


#) Con 19=a el problema es imposible, pues resulta 5892,1senA Se toma( )cma 9=

9=a 8=b 62=B 62

89

sensenA= 9933,0

8

629

= sen

senA

'22831 A ( )+= 11 180 ABC '38341 C

=62

8 12

sen

senCc

( )cmc 13,51

'38962 =A ( )+= 22 180 ABC '22212 C

=62

82

2sen

senCc

( )cmc 30,32

Pg,,. " 100 De/ tring1/oABC mita% 8ara/e/ogramo5 'e $ono$en 45=B 25=C ( )cmb 18=

E/ ng1/o A o81e'to a /a %iagona/ a e' ( )+= 2545180A 110=A

45

25

110 sensen

a= =

45

110.25

sen

sena

( )cma 93,19

Pg,,. " 101Por i'0'$e/e' 87,126==+ CD

Como ng1/o' a/terno' interno' 66,38= 11,88=

( )=+= 11,8866,38180 13,53

ABD =11,8813,53

81,12

sen

AB

sen

=

13,53

11,8881,12

sensen

AB ( )cmAB 16

BCD =47,1487,126

81,12

sen

CD

sen

=

87,126

47,1481,12

sensen

CD ( )cmCD 4

=66,3887,126

81,12

senBC

sen =

87,12666,3881,12

sensenDABC

( )cmDABC 19

Pg,,. " 102 &o' re$tng1/o' 'on ig1a/e' entre ' e ig1a/e' a /a mita% %e/ $1a%ra%o &o' tring1/o' 'on ig1a/e' entre ' e ig1a/e' a /a mita% %e/ $1a%ra%o Por tanto5 /a re/a$i0n %e rea' entre re$tng1/o' 2 $1a%ra%o' e' ig1a/ a 1

Pg,,. " 103 Sea ABC $on 90=A 4,14= pro!ecci"n de b 6,25= pro!ecci"n de c =+=+ 6,254,14a ( )cm40

= ab 2

= 404,14b ( )cm24 = ac

2

= 406,25c ( )cm32

=== 32242121 cbrea ( )2384cm

Pg,,. " 104 Un /a%o e' "24'413384,28 sen E/ otro /a%o e' "24'4133cos84,28

( )+= "24'4133cos"24'413384,282 senpermetro ( )cm80

= "24'4133cos"24'413384,28 2 senrea ( )288,383 cm

Pg,,. " 105 ng1/o $entra/ o$t0gono reg1/ar + =8/360 45

A8otema $ateto o81e'to a =2/45 5,22

$on otro $ateto %e =2/20( )cm10 = 4142,0/105,22/10 tgapotema ( )cm14,24

..!


21/29


14,2420218rea ( )237,1931 cm

Pg,,. " 106 a'e (R)tring1/o i'0'$e/e' $on ng1/o 70 2 /a%o' ig1a/e' ( )cm7

( )= 2/7072 senR ( )cm03,8

= 03,81416,322 Riarcun#erencLongitudCi ( )cm45,50

.,.!


22/29


Pg,,. " 107 &a a/t1ra %e/ 8ara/e/ogramo e' = 726 senh ( )cm71,5

71,59rea ( )236,51 cm

Pg,,. " 108 Ter$er ng1/o %e/ tring1/o + ( ) =+ 35115180 30

&a%o' %e'$ono$i%o' 115opuestox 30opuesto!=

30115353011535

20

3011535

20

sensensen

permetro

sensensen

!x

sen

!

sen

x

sen ++

++++==

++

=5736,0

9799,120

35

301153520

sen

sensensenpermetro ( )cm90

Pg,,3 " 109 a) Ter$era %imen'i0n 8ara/e/e88e%o = 609 tgh ( )cm59,15

[ ]+ 59,159492 2rea$otal ( )224,723 cm

59,1599Volumen ( )379,1262 cm

#) >eneratri? $ono = 8660,0/1260/12 seng ( )cm86,13

=a%io #a'e = 7321,1/1260/12 tgr ( )cm93,6

[ ]+ 86,1393,622193,6 2 rea$otal ( )263,452 cm

1293,631 2

Volumen ( )35,603 cm

Pg,,3 " 110 Dimetro (2r) #a'e = 47,3208,262 senr ( )cm14 ( )cmr 7=

A/t1ra (h) $i/in%ro = '4732cos08,26h ( )cm22

= 2272

Volumen ( )3

64,3386 cm

Pg,,3 " 111 A/t1ra ortoe%ro = 43,1818 tgh ( )cm6

&ongit1% ortoe%ro = 96,2318 tgL ( )cm8

[ ] ( )[ ]++ 681828182rea$otal ( )2600 cm

6812Volumen ( )

3576cm

Pg,,3 " 112 =a%io (r) #a'e = 3610 senr ( )cm88,5

A/t1ra (h) $ono = 36cos10h ( )cm09,8

[ ]+ 09,888,522188,5 2 rea$otal ( )206,258 cm

09,888,531 2

Volumen ( )391,292 cm

Pg,,3 " 113 &a #a'e triang1/ar tiene $omo $ateto' 38,6713 sen 2 38,67cos13

( ) ( ) = 2438,67cos1338,671321 senVolumen ( )330 cm

Pg,,3 " 114 = 7321,16,2606,2 tgh ( )m50,4...!


23/29


Pg,,3 " 115 El problema est incompleto, por no deinir el trin!ulo ABC, del "ue se dan s#lo dos datos$ Su planteo puede interpretarse como una subestimaci#n de la %eodesia, por e&emplo$

Pg,,3 " 116 Se '18one 1e e/ $9i$o e' 1n enano integra/ (%e a/t1ra + 0)

2

1

50

25=tg = 5,0arctg "54'3326

.3.!


24/29


Pg,,3 " 117 a) = 7321,3507550 tgd ( )m60,186

#) 60tg%

d =

==7321,1

60,186

60

7550

60 tg

tg

tg

d% ( )m73,107

Pg,,3 " 118 E' e/ ng1/o ag1%o $om8ren%i%o entre 9i8oten1'a 3,5m 2 $ateto 2m 5714,05,3/2cos = '955

Pg,,3 " 119 &a a/t1ra %e e'$a/era e' $ateto5 $on 9i8oten1'a 0,7 2 ng1/o $om8ren%i%o 25

= 9063,07,025cos7,0h ( )m63,0 + 5,163,0alcance 13,2 > 2 A/#apuede cambiar /a #om#i//a

Pg,,3 " 120 A/t1ra i'1a/i?a%a $on teo%o/ito + h ( )mtgh 8,151051,01506150 =

+ 2,18,15%lAltura$ota ( )m17

= 5,2175,2%$&AAlturaES$A ( )m5,14

Pg,,4 " 121 En e/ tring1/oABC 135=A 30=B ( )mc 1061= ( ) ( )=+=+= 30135180180 CBC 15

15

1061

135 sensena

=

=15

1351061

sen

sena

( )m71,2898

= 5773,071,289830tgah ( )m57,1673

Pg,,4 " 122 E/ tring1/o orma%oABC tiene 60=B 30=C ( )ma 40= ( ) ==+= 901803060180A 90 %'()$E*&SAa 40 (Centro

) E/ tring1/o 'AB e' e1i/tero (i'0'$e/e' $on 6 en /a #a'e) En $on'e$1en$ia5 'e tiene =ABc ( )m20

= 8660,0206020 senh ( )m32,17

Pg,,4 " 123 = 8122,022,14969,35cos22,149bleroLongitud$a ( )cm19,121

= 5834,022,14969,3522,149 senleroAnchura$ab ( )cm05,87

F1an puede ba!are/ ta#/ero

Pg,,4 " 124 ( ) ( )

( )( )

( )( )( )

=seg

h

km

m

h

kmsegentoDespla+ami

3600

1

1

1000

1

6006 ( )m1000

malnguloVisu 1000 Comprendido entre catetos 1350 2 1000

7407,01350/1000 =tg "44'3136

Pg,,4 " 125 EnABC ( )mb 246= 49=A 63=B ( )=+= 6349180C 68

6863246

49 sen

c

sensen

a

==

=63

49246

sen

sena ( )m37,208

=

63

68246

sen

senc ( )m99,255

.4.!


25/29


EnACD ( )md 246= ( )ma 240'= 85=D ( )85cos240'2'240246 222 += cc 02196'83,41'2 = cc

&a Gni$a 'o/1$i0n /i%a e' ( )mc 78,78'

.;.!


26/29


Pg,,4 " 126 &o' %o' 'egmento' %e AB 'e %e'ignan x e !

&o' ng1/o' o81e'to' a ta/e' 'egmento' 'on = 5590 35 2 = 489042

=h

xtg 35 35tghx = =

h

!tg 42 42tgh! = ( )km!x 120+

( )4253120 tgtgh += ++= 9004,07002,0

120

4235

120

tgtgh ( )km98,74

Pg,,4 " 127 AL$&RAh D'S$A*C'Ad=

=d

htg 28 28tgdh =

=d

htg

1214 1412 tgdh =

( ) ( )142812 tgtgdhh =

=2824,0

12

1428

12

tgtgd ( )m5,42

= 5317,05,4228tgdh ( )m6,22

Pg,,4 " 128 AL$&RAh A*C%&RAx=

( ) =h

tg 33,1

5290 =7813,0

33,1

38

33,1

tgh ( )m70,1

( ) +=

95290

h

xtg

+

+=

07813

70,10

7813,0

970,1

38

9

tg

hx ( )m70,13

Pg,,4 " 129 Un 8/ano meri%iano %a /a 'e$$i0n %e /a ig1ra En ABC tiene 90=A ( )cma 20= ( )22 50cmha =

BC 'e %ii%e 8or ah

en( )AlturaCono

2( )resto

= /50ah ( )cm4 222

4 +=b 222 4 +=c 222 acb +

22222 4420 +++= ( ) 22 2032400 += 032402 2 =+ 016202 =+ ( )cm16,19

= 16,195031oVolumenCon ( )333,319 cm

Pg,,; " 130 0122 =+ tgxxtg ( ) 01 2 =tgx 1=tx 4/1 =x 4/542 =+=x ==+=+ 46454

21 xx 23 pci"n c#

Pg,,; " 131 105521 == senA 8,0=sen "48'7531 =

"12'521262 =

@a2 %o' tring1/o' pci"n b#

Pg,,; " 132 ( )xxAC

==

120cos

13

cos

46 ( )xx = 120cos46cos13

[ ]senxsenxx = 120cos120cos46cos13

= senxxx

2

3cos

2

146cos13 senxxx 323cos23cos13 =+

9037,0323

36

=tgx 7419,0

9037,01

1cos

2

+=x

.6.!


27/29


= 7410,0/46AC 62 pci"n b#

.*.!


28/29


Pg,,; " 133 2cos

1=

tg cos21 =sen ( ) 22 cos41 =sen

( ) 22 1421 sensensen =+ 0325 2 = sensen ( ) 5/41 =sen

&a 'o/1$i0n 0cos1 == sen no e' a%mi'i#/e5 en /a eH8re'i0n ini$ia/

&a otra 'o/1$i0n %a 5/3=sen ( ) == 2

531cos 5/4 =tg 4/3

===+4

2

4

3

4

5

cos

1tgx

x5,0 pci"n d#

Pg,,; " 134( )

222

1

cos1

1

cos

1

1 sen

xxxx

= 21

senx

= pci"n a#

Pg,,; " 1 a) =

rad

rad 5

3180

108 #) =

rad

rad 4

15180

675 $) ( ) =

rad

rad

6

1801080

Pg,,; " 2 a) = 36

180

rad

5

rad63,0

#) = 100

180

rad

9

5 rad75,1

$) = 310

180

rad

18

31 rad41,5

Pg,,; " 3 a) = 624 tgAC 52,7 8779,162cos4 =B% = B%BC /42 52,8

#) h= 5,612,54 =h 15,3 $) == 6290! 28

Pg,,; " 4 a) 3,0=sen 18090


29/29


Pg,,; " 8 a) 90=A ( )cmc 18= 35=C == 3590B 55 = 35/18 sena ( )cm38,31 = 35/18 tgb

( )cm71,25

#) 90=A ( )cma 19= 65=B == 6590C 25 = 65cos19c ( )cm03,8 = 6519 senb

( )cm22,17

Pg,,; " 9 a) ( )cmb 16= 34=B 30=C =+= )3034(180A 116

3034

16

116 senc

sensena ==

=

34

11616

sen

sena 7,25

=

34

3016

sen

senb

3,14

#) ( )cmb 20= ( )cmc 12= 85=A ( ) += 85cos122021220 22a ( )cm4.,22 senCsenBsen

1220

85

4,22==

=4,22

88520 sensenB 8895,0 '4862B

=4,22

8512 sensenC 5337,0 '1232C

Pg,,; " 10 90=A 90=B

En tring1/oABC 15=a 12=b == 2590A 65

senBsen

12

65

15 = 15

6512 sensenB

= "22'2646B

( )+= "22'264665180C "38'3168

( )+= "22'2646cos151221512 22AB ( )cm11En tring1/oACD 90=D 25=A 12=AC

=25

12

senCD ( )cm39,28

Pg,,; " 11 = 38,67cos26r ( )cm10 = 38,6726 senh ( )cm24

[ ] ( ) =+=+= 360260100261022110 2rea$otal ( )297,1130 cm

== 800241031 2Volumen ( )327,2513 cm

Unidad 6 - Trigonometría

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