unidad 6 - trigonometría
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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
,.!
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Tangente 3
1 0,58 3 1,73
..!
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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.!
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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.!
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Pg,, " 29 &i#ro'io'Net
;.!
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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 +=
$) == 22arccosx 45 kx 36045 +=
%) == 0arctgx 1800 kx 1800 +=
e) == 3arctgx 12060 kx 18060 +=
) == 211802 arcsenx 15030 = 1802xkk 36015036030 ++
kx 3601502 += kx 18075 += kx 360302 += kx 18015 +=
Pg,3 " 33 CHA 48tgxh = CHB ( ) 2430 tgxh += ( ) 243048 tgxtgx +=
244824
30tgtg
tgx
= 24482448
30tgtg
tgtgh
=
4452,01106,1
4452,01106.130
( )m29,22
Pg,3 " 34 a) == 4390B 47 = 435 senc ( )cm41,3 = 43cos5b ( )cm66,3
#) =+= 22 86b ( )cm10 ( )= 68arctgA "48'753 ( )= 86arctgB"12'5236
Pg,3 " 35 AnchuraRoa lAlturarboh= 40tgah = ( ) 284 tgah =+
( ) 28440 tgatgah +== 284028
4tgtg
tga
= 28404028
4tgtg
tgtgh
=
5317,08391,0
5317,04a ( )m92,6
=
5317,08391,0
8391,05317,04h ( )m80,5
Pg,; " 36 ( ) =+= 6035180A 85
( )[ ]=
+=
6035180
354sen
senb
( )603535
4+
sen
sen
9962,0
5736,04 ( )cm30,2
( )[ ]=
+=
6035180
604sen
senc
( )603560
4+
sen
sen
9962,0
8660,04 ( )cm48,3
Pg,; " 37 ( ) 50cos64264 22 +=c =+ 15,2185,303616 ( )cm60,4
6.!
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( )
( )50cos64264
50cos46arccos
22 +
=A 7454,0arccos
60,4
43,3arccos '4841A
( )
( )50cos64264
50cos64arccos
22 +
=B 0311,0arccos
60,4
14,0arccos '1288B
Pg,; " 38 ( )44cos404524045 22 +=x
=+ 38,103562,258916002025 ( )m18,23
*.!
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Pg,; " 39 a) Tring1/o i'0'$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 =
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&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
!.!
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#) &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
,.!
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Pg,< " 56 ( )
=
= 33
302
610
3022
31000
tgtgL ( )cm48,13 = 30
2
2tg
Lh
( )cm94,7
,,.!
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4 ESO - MATEMTICAS - UNIDAD 6 (Trigonometra)
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
,..!
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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.!
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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.!
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( ) ( ) ( ) ==+=+ tgtgtgtg 180360180180 5/53
,;.!
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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.!
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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
,*.!
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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
,
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2729,071,4
1402 sensenB '5015B
,!.!
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#) 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 $ateto %e =2/20( )cm10 = 4142,0/105,22/10 tgapotema ( )cm14,24
..!
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14,2420218rea ( )237,1931 cm
Pg,,. " 106 a'e (R)tring1/o i'0'$e/e' $on ng1/o 70 2 /a%o' ig1a/e' ( )cm7
( )= 2/7072 senR ( )cm03,8
= 03,81416,322 Riarcun#erencLongitudCi ( )cm45,50
.,.!
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4 ESO - MATEMTICAS - UNIDAD 6 (Trigonometra)
Pg,,. " 107 &a a/t1ra %e/ 8ara/e/ogramo e' = 726 senh ( )cm71,5
71,59rea ( )236,51 cm
Pg,,. " 108 Ter$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 $ateto' 38,6713 sen 2 38,67cos13
( ) ( ) = 2438,67cos1338,671321 senVolumen ( )330 cm
Pg,,3 " 114 = 7321,16,2606,2 tgh ( )m50,4...!
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Pg,,3 " 115 El problema est incompleto, por no deinir el trin!ulo ABC, del "ue se dan s#lo dos datos$ 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.!
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4 ESO - MATEMTICAS - UNIDAD 6 (Trigonometra)
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.!
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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'
.;.!
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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.!
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= 7410,0/46AC 62 pci"n b#
.*.!
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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 $) == 6290! 28
Pg,,; " 4 a) 3,0=sen 18090
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Pg,,; " 8 a) 90=A ( )cmc 18= 35=C == 3590B 55 = 35/18 sena ( )cm38,31 = 35/18 tgb
( )cm71,25
#) 90=A ( )cma 19= 65=B == 6590C 25 = 65cos19c ( )cm03,8 = 6519 senb
( )cm22,17
Pg,,; " 9 a) ( )cmb 16= 34=B 30=C =+= )3034(180A 116
3034
16
116 senc
sensena ==
=
34
11616
sen
sena 7,25
=
34
3016
sen
senb
3,14
#) ( )cmb 20= ( )cmc 12= 85=A ( ) += 85cos122021220 22a ( )cm4.,22 senCsenBsen
1220
85
4,22==
=4,22
88520 sensenB 8895,0 '4862B
=4,22
8512 sensenC 5337,0 '1232C
Pg,,; " 10 90=A 90=B
En tring1/oABC 15=a 12=b == 2590A 65
senBsen
12
65
15 = 15
6512 sensenB
= "22'2646B
( )+= "22'264665180C "38'3168
( )+= "22'2646cos151221512 22AB ( )cm11En tring1/oACD 90=D 25=A 12=AC
=25
12
senCD ( )cm39,28
Pg,,; " 11 = 38,67cos26r ( )cm10 = 38,6726 senh ( )cm24
[ ] ( ) =+=+= 360260100261022110 2rea$otal ( )297,1130 cm
== 800241031 2Volumen ( )327,2513 cm
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