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H
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1/1000000
1/200 000
1/1000000
1/200 000
1/5000
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2
360
400
2
90 100
64
1
1
1
1
1
1
0, 1
1
/2 /2
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P
O
1
cos =
P
sin =
P
tg = sin
cos cotg =
cos
sin =
1
tg
tg
T
cotg
C
cos2 + sin2 = 1
P
P
x
cos() = cos sin() = sin
tg() = tg
P P y
cos( ) = cos sin( ) = sin tg( ) = tg
+
P
P
cos(+ ) = cos sin(+ ) = sin
tg(+ ) = tg
2 P P
cos(
2 ) = sin sin(
2 ) = cos tg(
2 ) = cotg
cos(a b) = cos a cos b sin a sin bsin(a b) = sin a cos b sin b cos a
tg(a b) = tgatgb1tga.tgb
cos2 a= 12
(1 + cos 2a)sin2 a= 1
2(1 cos2a)
cos a + cos b= 2 cos a+b2 cosab2
cos a cos b= 2sina+b2 sin
ab2
sin a sin b= 2 sin ab2 cos ab2
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cos a cos b= 12
(cos(a + b) + cos(a b))sin a sin b= 12(cos(a b) cos(a + b))sin a cos b= 1
2(sin(a + b) + sin(a b))
t= tg a2 sin a=
2t1+t2
cos a= 1t2
1+t2 tga= 2t
1t2
ABC
A
a,b,c
A, B
C.
A + B+ C= 180 (
)
a2 =b2 + c2 (
)
sin =
b= a sin Bc= a sin C
cos =
b= a cos Cc= a cos B
=
b= c Bc= b C
ABC
A + B+ C= 180
a2 =b2 + c2 2 bc cos A( )
a
sin A =
b
sin B =
c
sin C (
)
0 = (x0, y0) r
(x x0)2 + (y y0)2 =r2
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()
()
A
B
AB
A
B
[AB]
[M N]
AB = 2R sin
2M N = R (1 cos
2)
AB = R rad= R
180deg= R
200gon
961846
53, 476 gon
0, 375 rad
45
15
A
B
C
BC
2/3
AB
AC
5/4
BC
ABC.
13
0, 35 m
26
0, 75 m
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cos(x ) , sin(x +32
) , tg (x 7) , sin(n+ x)
n N
cos(a b) cos(a + b) sin(a b) sin(a + b) = cos 2a (cos a sin a)(cos 2a sin2a) = cos a sin3a tg (a b) + tg b
1 tg (a b) tg b = tg a
1 + tg 2a tg a= 1cos2a
tg a + tg btg a tg b =
sin(a + b)
sin(a b)
AB
AB
AB
B
AB
A
B
AB
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F
F
2a
2a > d(F, F)
[F F]
2c
x2
a2+
y2
b2 = 1
b2 =a2
c2
y= 0
x= a
(a, 0)
(a, 0)
x
x= 0
y= b (0, b) (0,b) y
a
b
A =ab
x2
a2+
y2
b2 = 1
cos2 + sin2 = 1
P = (x, y)
], ]
xa
= cos
yb
= sin 1 x
a 1 1 y
b 1
x= a cos y= b sin , ], ]
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a b
OA
a cos
OB
a sin
], ]
E
E : x2
a2 +
y2
b2 = 1
C : x2
a2 +
y2
a2 = 1
E
A
yE = b
a
a2 x2
yC =
a2 x2
yEb
=yC
a
yEyC
= b
a
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f
f= a b
a = 1 b
a
e
e2 =a2 b2
a2 = 1 b
2
a2
e
e2 =a2 b2
b2
ba
= 1 f= 1 e2e2 = 2f f2
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a b a
f = aba
b
a
a b
r
1/r
a
1/f
3
X, Y, Z
X, Y, Z
S
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a
b
1/f
6378388, 0 6 356 911, 946 297, 000000
6378388, 0 6 356 911, 946 297, 000000
6378137, 0 6 356 752, 314 298, 257222
6378137, 0 6 356 752, 314 298, 257222
297
296
297 296297
= 1
297
296
297
E
OE
E
tg = b
atg =
b2
a2tg
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O,X ,Y,Z
O
OZ
OX Y
OX Z
P P
E
E
E
0
180
E
nE E
0 90
P P
h
nA
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6370
CE
0.08
D= 1
1.25
D= 4
31.40
D= 20
(R + E)2 =R2 + D2 R2 + 2RE+ E2 =R2 + D2
2RE
D2 (
E2
E D22R
BC = D
BE
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B A C
BAC
AB AC
BAC, B1AC1, B2AC2, B3AC3
BAC
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X
Y
X
Y
X
Y
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D
Y
P
{ X=D sin CY =D cos C
C=
C= 200
C=
200
C= 400
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1.
P A = 31.7843
P A = 42.365
X= 20.28359119 (X=P A sin C)
Y = 37, 19367088 (Y =P A cos C)
2.
P B = 175.291
P B = 127.238
X = 48.1540386
Y = 117.7739265
3.
P C = 210
4923
P C = 12.456
X = 6.382310734Y = 10.69663712
X
Y =
(
AB)
AB = X
sin(
AB)
XA= 2864.76 XB = 4560.87
YA= 1802.32 YB = 3370.24
AB
2309.797017
AB
52.49894669
AB
252.49894669
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XA= 2864.76 XC= 1572.33
YA= 1802.32 YC= 2854.17
AC
4832.52274
AC
217.2358185
AC
17.2358185
XA= 123.456 AP = 34.256
YA= 67.890 AP = 41.123
X=AP sin(
AP) = 144.5314775
Y =AP cos( AP) = 103.2018306
XA= 1 XB = 8 AP = 50
YA= 1 YB = 2 BP= 350
P
XP = 5 YP = 5
A= (1, 1) B = (8, 2)
AP = 50
BP= 350
P
AB
sinp =
P B
sin a=
AP
sin b=
AB
sin(a + b)
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A.B.
A
B
50
2
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A
B
2
3
A
A
B
C
C
D
D
C
5
10 20
0.3
0.4
10
50
10
15
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20
6
15
20
2
1/80000 12 106
10
1
100
1/80000 70 100
30 106
1.2 106
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3
10
2.5 %
DH = 61.25 cos5 = 61.01692526
DH = 502 2.262= 49.9488789
100
= 0.5
0.2
100
=
2 4
S
D= S
D= S
2 (/2)
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F
d
f =
G
s
d = f
s G
D = f
s G + q
D
fs
=k
k= 100
G
q
D= k G + q
D= k G + q
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OC
i
OM
ON
M N=G
C
M1N1 OC. M1CM = N1CN =i
CM1N CN1N M1 N1
D= OC+ q= k M1N1+ q
M1C = M C cos i N1C = N C cos i M1N1 =M Ncos i= Gcos i D= kGcos i +q DH=D cos i
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80
20
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40
50
10
15
45
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8
45
45
7
P
AB
P
AB
AB
45
45
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D
AB
AB
D
0
360
5
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n
n 1
100
12
0.873
1/n
n
n
d
5 30 +
5 38
d, 2d, 3d, ..., nd
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ABC
B
A
C
45
AC
B
BC
BA
AE
BC
DE
45
CD
CD
M
AB
i1
AE
CD
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A
B
R
P
AB
M
R
A
B
P
A
B
R
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4
4
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4.5
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C
T T
C
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R
B
B
R
B
R
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20
1
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P
AB
P = 20 +
25 20AB
P B
P = 25 25 20AB
P A
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A
B
A
B
Dh
A
B
Dh = la lb
lA A lB B
AB = lA lB
A
B
A
B
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lA lB = AB
C
S1 S2
l1A H1B B
l1A H1B = ABl1B > H1B l
1A l1B AB
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180
90
A= 100
A
B
AB
100 + 1.10 = 101.10
B
101.10 0.5 = 100.60
+ = B100 + 1.10 0.5 = 100.60
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A
100
100 1.5 = 101.5
B : 101.5 1.10 = 100.4
C : 101.5 0.4 = 101.1
D : 101.5 0.7 = 100.8
ab : 100 +1.10 = 101.10
B : 101.10 +1.30 = 102.40
B :
+C Ar +C Av
100
R= 102.340
R
1
R= 2125
1 = 1641
1
2
1 = 1836
2 = 1520
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+
R 2125 102.340 (1)1 1836 1641 484 +484 102.8242 525 1520 316 +800 103.140
3 1920 2142 1617 817 101.5234 1436 732 1188 +371 102.7115 1631 195 +176 102.516 (2)
7842 7666 1988 18127666 1812+176 +176 (2) (1) = +176
R
R
R
2
= 3
R 21251 1836 16412 525 15203 1920 21424 1436 7325 946 16316 1827 13207 1142 6398 1345 12429 926 1311
184414028 14022
6
E= 2.5 emqL= 2.5 3 2 = 10.6
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3
+
1 1422
100.0002 1643 1839 417 417 99.583
1636 101.226 99.590
1524 99.7023
1
10.5
1
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100
.
10.5
1.36
1
1
2
4
ACB
a
CA CB
CA
CB
i
CA
z
CA
i + z= 90
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360
60
60
60
60
268 37 42 132 48 47 =?
267 96 102132 48 47
135 48 55
400
10
1
= 100
1000
325.698
152.274 =?
325.698
152.274 173.424
V V
N
HH
OO
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0 200
1
2
240.3604
40.3606
240.3605
54.6823
254.6821
54.6822
185.6783
315.7453
115.7455315.7454
53.3646
253.3644
53.3645
262.3809
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200
.
105.4530
305.4532
385.8432
185.8430
400.0000385.9414
14.0598
+105.3521 19.4107
300
P
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HB = HA + dH
dH = hi + BCH
hi
A
H
B
ABC
BC = D h
dH = hi + D h
H
AB
H = hi
dH=D h
H=hi
dh= AB sin h
H=hi
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h2 c h = h2 c
dh= AB sin h
c
ACC
sin c
CC =
sin ACC
AC =
cos h2AC
ACC= 90 h2
AC= AB CC =H hi
sin c=
(H
hi) cos h2
AB
c h2 sin c
c sin(1) cos h2
c= H hiAB sin(1)
c
0200
z
C G =z
C D = 400 z
400
z+ 400 z= 400
88.4510
311.5542
400.0052
52 : 2 = 26
88.4484
311.5516
400.0000
z = 88.4484
i = 100 88.4484 = 11.5516
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A
B
s
R
CA CB
a
A
AS
s
A
BM A
AT,
A
B
T
h
SAT
S
AS
Ab
AS=Ab = D
ST =D h
dH, ST
c=
r=
d H=ST+ (c r)
c
D2 = c (c + 2R)
D2 = c2 + 2R c
2R c
c
2
D2 = 2R c
c= D2
2R
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R= 6370
50
c= 0.195
r
r= n D2
R
n
n,
n= 0.08
D= 50
r= 0.026
N = c r= K D2
K= 7 108
( )
Dh= D h + K D2
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Emoyenne = Emq
n
Emax= 2.5 Emq
5
Emoyenne = 5
4= 2.5
Emax= 2.5 5 = 12.5
12.5
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n
n
400
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A, B, C, ...
M
N
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M A
N A
M
N
a
b
M A
sin b = M N
sin(a + b) M A=
M N
sin b
sin(a + b)
abcd
A
M A
N A
Ez
M A
A
A1. AA1
AA1sin Ez
= M A1
sin M AN
AA1= M A1sin Ez
sin M AN
AA1 M A1 Ez
AA1 M AN AA1 sin M AN= 1, M AN= 100
Ez A
AA1 M AN
M
N
R
P
P
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P
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P
d N P
e
sin e= d
N P
e
x
A 322.531
198.273 a 62.542
B
552.861
301.200 b 75.126
C 125.637
362.025 c 71.851
10
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CA
144.1661
AB
73.2463
AP
10.7043
CA
256.0897
AB 252.2813
BP
348.3691
P A 369.5645
475.3552
X : 0.03861
Y : 0.0179
AP
10.7043
AP 281
281
e
e=
0.01
281 = 0.008
e= 0.00226
23
e= 2.5 102
= 17.7
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S= (7 2) (2 ) = 10 = 1000
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P
P
P
Ez
Ez P
P
P
a1= r1 Ez (Ez )
Ez P
a2 = r2
Ez
a3 = r3 Eza4 = r4 Ez
P
A=
a21+ a22+ a
23+ a
24= Ez
S r2
r
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E1
P
1 =
E1
1
2 (1
2
)
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2 = E1
2 33 = E1
3 4
4 =
E1
4
P
P
L=
21+ 22+
23+
24= E1
D
D
A
L
I= A2 + L2
2.5 52
5 = 19.75
Ez = 2.5 52
= 8.84
2 /100
E1= 2.5 2 = 5
15< 19.75
A =
0.0009
2452 + 2672 + 2282 + 2602 + 2362 = 0.008L = 0.05
12.36 = 0.17
I =
0.0082 + 0.172 = 0.17
(
0.014< 0.17)
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A= (1, 1) B = (8, 2)
AP = 50
BP= 350
P
ABsinp
= P Bsin a
= APsin b
= ABsin(a + b)
X
Y =
AB =
7
1 7 = 90.9665
AB =
X
sin( AB) =
7
sin 90.96= 7.071
a
b
a= 90.96 50 = 40.96
BA =
AB = 90.96 + 200 = 290.96
b= 350 290.96 = 59.04
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AP
sin(59.04) =
AB
sin(40.96 + 59.04) = AP = AB sin(59.04)
sin(100) = 5.66
Y = 5.66 cos(50) = 4
X = 5.66 sin(50) = 4
P
PX= 4 + XA= 5
PY = 4 + YA= 5
P = (5, 5)
A = (x1, y1)
m(= YX
)y y1= m(x x1)
AP
BP
AP = m1= (100 AP) = 50 = 1
BP = m2= (100 BP) = (250) = 1
AP :y 1 = 1(x 1) y = x
BP :y 2 = 1(x 8) y= x + 10
P
x= x + 10 = x= 5
y = x = x + 10 = 5
P = (5, 5)
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H
A
=
+
2 =
1.00 + 1.00
2 = 1.00
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=
+
2 =
2.00 + 1.00
2 = 1.50
+
2 =
2.00 + 1.00
2 = 1.50
M
A
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AM
AB = 100a= 80
b= 60
AB
sin(A + B) =
AM
sin b
AM= AB sin bsin(A + B)
= 100
h
h = 10
10
) = 0.15838
H= 0.158 100 = 15.8
M
100 + hi + 15.8 = 115.8 + hi
hi= 1.6
115.8 + hi= 117.4
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