Τριγωνομετρία
181
,
-
Upload
clevermind-private-institute -
Category
Documents
-
view
7 -
download
1
description
Το βιβλίο αυτό προορίζεται για τους μαθητές των τάξεων Α και Β Λυκείου που επιλέγουν ενισχυμένα μαθηματικά.Βασική επιδίωξη του βιβλίου είναι να γνωρίσει ο μαθητής τις διάφορες έννοιες και μεθόδους της τριγωνομετρίας, να εμβαθύνει σε αυτές, χωρίς να υπεισέρχεται σε πολύπλοκες διερευνήσεις και να τις χρησιμοποιεί σε άλλους κλάδους των μαθηματικών και των θετικών επιστημών γενικότερα.
Transcript of Τριγωνομετρία
, ' ' ( 2 , 3 ) - 1999 ISBN:9963-0-4314-3 :2003 :2004,2005, 2006,2007,2008 :
- - . , : ) , ) . , ' , . . . , , , . , , , . : , - , , , , ........... . ................ . .... ... ................... .. 1 1. ......................... ......... ........3 2. - .............. .. ........... .. 9 3. ............................................17 4. .........................................35 5. ....... . ................... .... .........53 6. ..... . .... . ..... . .. .. ..... .57 7. .....................................69 8. ... . .......... .. ... . .....78 9. ... .... ......80 . 10. .. . .... ...... ... ... 105 11 . ............................113 12. ........................................121 13. ...................................... .. 140 14. ....... . ........... 154 15. ............................................157 . . .. . .. .. ...... . . ... .. .. ..... .. ... 163 ....... ... ................... ... .......... . . . ..... .. . 164 1: ''1 y = =1,70m). ---: _ ...................... 1 1.70m z 2: 3: 2 I - ( = ( , ( = ( ( = ( 1,
12cm = 153' = = __ 12 = 12 5 3 1. 2. 1 =% = 4 cm = 5 cm. (.) ' = ==> = 8 =4 cm = 60 ==> =1 8 =4v'3cm 2. =8 60 - 4 45 + 2 45 +21 =4-4+12=12 5 1. 3. (Newton) 30 F = = ( = F = 9 = 9 13= 93 4. + = + 1 + =_ 1_2_ + =v +
+
1 1 +
= 1 + - - == -- 6 1. u co u co u 15 cm 2. =1 = 185. 3. =1=30 =55cm 4. 5. 3m =60 1, 7 m, 6. F= (Newton) =30 F Fx Fy 7 1. 7. . )(1- )(1+ )= 2 ) = 1 ) 2 2+ 2= 1 ) + = .1 u v x 8 - . , , . . . 08. , . , .( ) . , . 2. (1) (1rad) . (1) 1/360 60 (60') 1' 60 (60"). D (rad) R = rad = 360 = 1rad = 3600 = 1801radz57o 17' 44,8" ' 360=rad = 1 = _1n_= _n__ rad 360180 D1rad =
10- d - 1800ra 1. 150 45 1230' 150 =150_n_rad= rad 1806 45 =
rad=1804 1230' =12 5=
=
=
'1801800. 72 2. rad
rad 218 =n 180 =90 22 = .180 = 1818 = ,1800 =5400""7708'34" 777 2. 30456090120135150180210225240270300315330 1ln 64323466432346 1. 15,75, aoo,100, 105,250,310,22 30', 2. 3 +30 +30 30), 30 -30 30 12 360 -, 13 2. 1. 45, -120, -5;rad, CO) -"v -180 ,=>>. ........... .......... . .... ....... ........... ........... ) ........... 2. 3 3 3. >
= =1 =>\ =.q\ = 213 3 =2 =13 3 =13 = 1 = 1 = = 45 32 + = => =1 => (or)2 ==>( or) =[J; == 1 => = = = 1 = = = 1 = = 12 2 = = 12 2 L = 12 =12 =h =12 =1 =1 90 180 270 360 180 (-1,0) 270 (0,-1) == ==1, == ==-1, == 180 ==-1, == == ==1, == 304560 goo 180270360 112./3 1-1 222 1 ./3121 -11 222 ./3 1./3 3 ./31 ./3 3 90 33 3. 90 90 34 1. = - +.f3 = + = -13 2. go + 3. = = = = = 1 =_1_8, =_L8 + = (08)2 + =1I
+
= 1 I(1) I
=1
(2)I
=1
(3) ;e (1)
+ =_ 1_ + 1 = (4) 35 4. ;e (1)
(5) =
=1 -1-36 = =-1- -1- =1 + =1 =1
= 1 1 + 1 - 1 + 1) = 153 90 180 90 36362 8 =6064 3cm + =- - 6. + -'--;,-----,----'----,- = + _ + - + + - + - + + - = = + - = +
= = 1. =2,=2J3,=30 =5,=5=5J3 2. 2J3, 212 / 16 + 12. 3. 120,
= +
4. + _ =____g.:__ 5. = 6. = .
= 7. 0,9km 1,1 km. =70 65 6. (i) i)uv= ii)uv= uv= = Uv = = = =1 = 1 =1 . 222 - 1. =45 =8/2 cm =6 cm = =8/2.6 I[ =4 6 =24 cm2 2. =119cm, =5 cm, =3 cm. 1 2 - 52+32-(119)2- 25+9 -19 _15- 1 - 2 5 3- 30- 30- 2 -66 =1==60 2 111315 13 =_ . 3. 5. _vv=_ v_v 1 2 2224 3. =
. . (R = = 2R 2R =2R2 4. 713cm2, 7cm 4cm =1 . . . = = = 2 =7:/13 =13= =13= 22 i) = = 2R 2R 67 6. 1. 68 5/3 cm, 60' 8cm 6 cm 30iii) 4 cm, 2cm 2/3 cm 2. =2/3 cm2,=4 cm =2/3 cm. 3. 4R + y2_ - 4. = = =13==> =13 2 360. 1 60 . =60. =120 = =1 60, 120 + 1 = + 1==> =-1==> =70 '1'' - 2 -30 360 330 (360 -30) -30 =330,=210. 2. + !3 = + !3 == = - !3= =- !3 2 = 13 2 180 30 360 - !3 2 150 = 150. =210. _v3 - - -2-71 7. = =_13 2 = 150 '= 21 3. + = 13 = 13 = =13= = 13 3 0 360 13 3 30. =30. = 210 = =13 3 = 30 = 21 72 360 1. -/2 +13 = =
1 + 1) 2) 2. + 1 = + 12 = =
- 1 =- 3)
+
= 3. 1 = + 1 + 13 = - + 13 =
- = 1. = + - 4 + + = + + 2.
. -
. = - (1- + =2(1- . (1+ 3. =
(3 -
+ (3.-
4. = 90 + + = 4R2 (R 73 7. 5. 1 6. + =13,
+
=1) 7. y=
+ -
8.
= 1 +
= 1 +
1 + 1 + 9. Ua+ +Uv=2R + + _1+_1+_1= + + _ U0 Uy2R - + 10. y CRAMER + = =
gg2 +y3=74 1. = 90 1 +t2 ( 1 + t2) =2t t2 - 2t+ =
2 =-1 +
1-
=-1 +
95 9. t=2 J4 -
=1 J1 -
2 _1 J1 -
3. = + =.1 - +. 1 +
1 +
1-2 __ 1 + 1 +
+ + +_
+ _ + - + - + 4. + =3, 4 . - + 3 1 - =3 1 +
1 +
+3 -
=3+
=0 = 4) == = 4 == =.4. 3 96 5. =t 2 -
+ =(t- 1)2 2- ( 1 + ( +1 - )= 1 +
1 +
= 2 - (yvf) ' 2t + 1 - t2 = = 2 - 2t - 1 + t2 = 1. = _L_2 1 + = _1_2 + + (45 + = 2. + = 2, = 1 +
1-
= 1 +
3.
=
- 1, =1 + 4. =12- 1 5. = t 2- 1 + t2 + + 1- 97 9. =1 -
I = 1 -
I(1), = :t J 1 -
- 1, I =1 + I(2), =J1 + (1) (2) =1 - 1 + (3), =:t J 1 - 1 + + - 1. 30' 12 =1- = 1- 2= 2-12 224 = 98 2.
+ = 3 +
+ =
+
=( 1 - )2 +( 1 + )2 _1 +
- +1 +
+ - 4 _2 +
_2 +1 + - 4- 4 _3 + - 4 3. = = =
+
+
= 1 + + = 1- +1- +1- 2221 + 1 + 1 + - - 1 1
= + _ - 99 9. 4. + 3; =i (1+1)2(1-1)2 =4+4
44- - 4- -4 8 = 1 + 2 1.
=1 +
+ +
= 1 + 3.
+
= 2 3 + 1- 100 4. + =884 n + =884 = + = . + . =
+ (1-
= . (1-
+
=
+
= -
= -
=
- 101 9. , , 3 -
= 1 _
=
-
1. = -
+
= + = + =
+
=
+
+
=
+
+ (1-
(1-
=
+
+
+
+
=
+
2. + = (1+ + = -
+
- =
=
+ +
- = (1+ - 102 = (4+ 3) = (1+ 1. + = -
+ = 2. 18.)= 18.), 3. = + 103 9. 1. 1 + + =
+ ( + )2 + ( )2 =
; 2.
= =() 3. + =120 = 4. =i ( + + 3) 5 1 + , 2. = , 6. + () _ 1 )2
+ + 1) + + _, + - -1 ,2 ()- ' 7. + +
+
+
= [ )3 = -
_1 + + - 12 8. = -
+
+ Sx+ 1 = 9. = + 1 + 1 + 1 g_= 1 - 22448888 104 8 = + = + = = += (1) (2) - = 2 (1) (2) + = = 8 = = + + = 105 10. - = = =2 + = 2 = - + - 22 =
+. 22 I = I = = =----';----'= 106 =---:;-----;;.. =
=- 22 = - 1. + + = 750+150 . 750- 150 22 + = = =2 12 . 13=16 222 22 2. =
22 = = =
=- 3. + + = + 90 - =
+90- . 90 22 = +45 45 4. + + = 90 - = 90 - + 22 107 10. + + _ + + _ + = - -_ + ( +1)_ - - - +_ 22 = - 1 +1 = = 5. 1 + 1 + = + == _ 12 -
22 =
- 1 = + = = J2 2 1. + - - + + - 2. 00 - + + 3. 108 + _ - - 2 =
+ + + - + + = + + - + - + - 4. + + 5. + + = + + = 1 + 8 + = + - = (1) = + + I (2) = + + (3) = I 1) = + +- = + =( = _ + + - _ + - 2- 2 109 10. 2.
165+75) 165 -75) = 2=2 - + - 1800+600)-- 2- 2-1 _- _2_1 - 2--2--4 3. ;3= = ( ; 3 = =
= 33333 = 9.
= 333 = [ - (- =
+ = = 9.
+ = + ill-+ 3334. - = = =
=- =- 1 332 = + , + + + + = = 2 -2 = _ + + - I - + - -- 2 - 2- 2- = = . += 22 = 22 11 ---------------------------------------5 ' '' - 1 .
+
+.. . +
- =_ _n_= n 1818g + +...+ \7;(1). ( 1) f8 2. _2.2 . 2 . - +.. + + .... f8 = 1 - =1 +1=2- 1 18 - 1. 2. + - = 15 15- 1 +
=.f3 111 10. 3. + = - = . + - . + 28)= - . + 4. - = 1 5. + + = 1) . ''2 6. + + + ...+ = 112 W 1. _g_ = =l=2R(1) (R (1) =_g_ 2R (2) =1_ 2R 2R 2. = +
(3) = +
(4) = +2
(5) = +
3. + + 1ao. + = = + = =- + = =- 113 11. = 180; -:L) = 90 _ = ) = 180;- )= 90 -= ) = 180;- )= 90 -= = + + 2 4. = =1 =1 222 =
= = =2R2 = 2R2 I 4R 114 J : 4R- 'JW =2R2 = 0JR I = J - - - 1. = + + - . [= 22 15 11. ++ + + _ 22-2R( + + () 2 =R + + 2 2 r.LA-B.... .V _ _ ')_,..(,- .. - -y - /"'' . 2. = _ + = =>/ZV => = + _ => = > > =>= => =2 (R., => = + + - => = + - => - ==>- ==>==> 180 180+ 116 2R = + 22 A_t= 22 = 22 + _ =- = + 3. = = = = = =180- +=180 + =90 = 4. R = = =++ =5+6 +7=
22 =J - - - E= J 9(9-5)(9-6)(9-7)= / 9 4 32 4R 616 = 5 46R7R == 35/6 41624 5. = =
117 = 2.2=2.2 __ = +_ _ _ +
= + +
_ =
= 1. - 2 - 2 +28) = t 38) =_ 38 + = - t = 2. _ + = =
= +
3. = 4. + = 5. + + = 118 6. 7. = 8. =
2 9. + = 10. =
119 11. 1.
+ = 2. = . 22 3. = + 60" 4. = -
+
=2 + 22 5. + + + + + = 6. - = - 7.
+
+ -1 2y) 120
=1 -
+ =4. + =1 5; = = + ={0,1,2,3, ... } 1. = = =- = = 121 12. = +}, , = 360 + 180 - - 1) = 2 =- J3 2 = = + +6;, = _ J3 = _ = 233 = } = + (- ) 122 3' = +
3' = = + J3 32 - = 2== =2 2. y = y = .. = = - - 6- , - 6- , 6' 6 3. -= +f8) (-= +ra) = - = ++fs , = + +f8 ) =- = ++f8, = +fs
1
