PREFACE

107

1
1 5
5
11
1 17

2 22
22
29
2 37

2
3 43
43
49
54
59
3 65

4 71
71
77
84
4 90

96
96

97

3
-
-

-
-

-
-
-
-

-
-
-

4
1
a1
a2 an a1 a2 an a1
an n n
an

an
1 d a2 a1 a3 a2 an an 1

2 an a1 n 1 d

an
a2 a3 an
1 r
a1 a2 an 1
n 1
2 an a1r a1 0 r 0

a1 a
1
an an 1 d n2
a1 a
2 a1 0 r 0
an ran 1 n2

n
1 n 1
2 n k n k 1
n

5
6

1. an

1 2 4 6 8 10 an 2
2 1 4 9 16 25 an 2
1 2 3 4 5
3 an 3
2 3 4 5 6
4 1 3 7 15 31 an 3
x 1 c2 1 2 2 2 3 2 4 2 5
an 2n
c1
2 2 2 2 2
2 2 3 4 5
2
an n
n
3 1 an
n 1
c2 1
1 2 3 4 5
4 2 1 2 1 2 1 2 1
n
an 2 1

2. an 3 2
1 n an 4
2 73 3
3 an 100 n 3
x 1 an 3 2 n 1 2n 1
2 1 2n 1 73 n 36 73 36
1
3 an 2n 1 100 c n 49
2
n 50
 7

3. 1 an a2 6 a4 24 a10 an
6
2
2 an an an 1 an 1 n2 a3 4 a7 64
r a1 4
x 1 a1 r r 0
a2 a1r 6 1
3
a4 a1r 24 2
2 2
r 4 r 0 r 2 1 a1 3
1
9 9 n 1 n 1
a10 a1r 3 2 1536 an a1r 3 2
2 an
2
an 1 an 1 c an 1 an an 1 r
a7 a3 r
7 3
c 64 4 r c 16
4
r cr
4
2
a3 a1 r
3 1
c4 a1 2 c a1 1
2

r 2 a1 1

4. 3 x y 18 x y
10
x 3 x y x y 18
x y 2
x 3y 1
3 x
2y x 18 2
y x 18 y
1 2 2
1 y x 2 2x 3x 54 0
3
9
x 6 x x 6
2
1 y 12 x 6 y 12
8

5. 2
1 400 1 200
10 10
x 2 400 200 600
10 10000 10000 600 16 400
2 16 1 33 10

6. 4 4 4

1 2 3
an n
1 a1 a2 a3 3
2 an 3
3 an 4
x 1 a1 1 a2 4 a3 16
2 n n 1 4
a1 1
an
an 4an 1 n2
3 an 1 4
n 1 n 1
an 1 4 4
 9

7. an
n a1 5 a2 12
a3 22
1 a4 3 1 2 3
2 an 3
3 an 4
x 1 4
4 3 3 4 1
a4 a3 3 4 1 22 13 35
2
n n 1 3n 1
a1 5
an
an an 1 3n 1 n2
3 a1 5
a2 a1 3 2 1
a3 a2 3 3 1

an 1 an 2 3 n 1 1
 an an 1 3n 1
n 1 3n 2
c an 5 3 2 3 n n 1
2
2 5 n 1 3n 2
n 1 a1 5 an
2 2

8. an 19 d 0 a5 a15 0
10
A a1 0 B a19 0 C a10 0
D a6 a14 0 E a9 a10 a11 0
x a1 d 0
a5 a15 0 c a1 4d a1 14d 0 c 2a1 18d 0
c a1 9d 0 c a10 0
A a1 a10 9d 9d 0
B a19 a10 9d 9d 0
C
D a6 a14 a5 d a15 d a5 a15 0
E a9 a10 a11 0 d 0 0 d 0
BCE
10

a1 3
9. an
an 2an 1 1 n2
1 a2 a3 a4 3
2 an n 2
3 2 5
x 1 a2 2 a1 1 2 3 1 5
a3 2 a2 1 2 5 1 9
a4 2 a3 1 2 9 1 17
n
2 1 an 2 1
1
3 1 n 1 a1 2 1 3
k
2 n k ak 2 1
n k 1
k k 1
ak 1 2 ak 1 2 2 1 1 2 1
n
n an 2 1

n n 1 n 2
10. 1 2 2 3 3 4 n n 1 n
3
10
v 1 2 3
1 n 1 1 2 2 2
3
2 n k
k k 1 k 2
1 2 2 3 3 4 k k 1
3
n k 1
1 2 2 3 3 4 k k 1 k 1 k 2
k k 1 k 2
k 1 k 2
3
k 1 k 2 k 3
3
n k 1
n
 11

a1 a2 an a1 a2 an
n a1
Sn a1 a2 an

a1 d n
Sn a1 a1 d a1 n 1 d
n 2a1 n 1 d
2
n a1 an
2

a1 0 r 0 n
na1 r 1
2 n 1 n
Sn a1 a1r a1r a1r a1 1 r
r 1
1 r

n Sn an
a1 S1
an Sn Sn 1 n2

n n 1
1 1 2 3 n
2
2 2 2 2 n n 1 2n 1
2 1 2 3 n
6
2
3 3 3 3 n n 1
3 1 2 3 n
2

a r
1 n a 1 nr
n
2 n a 1 r
12

1. 1 12 15 18 99 5
2 4 8 64
2 5
3 9 27 729
x 1 12 3 n
99 12 3 n 1 cn 30
30 12 99 30 111
1665
2 2
n 1
2 2 2 2
2 an
3 3 3 3
n 1 6
64 2 2 64 2
n 6 n 6
729 3 3 729 3
6
2 2
1
3 3 266
6
2 729
1
3

2. 102 398 n 2000

n 10
x n 2
n 2 102 398
2000
2
n 6

3.
P1 1 0
P2 1 2
2 P3 P4
P5 P8 10
x P8 4 4
2 4
2 4 6 1 1 2
P8 x 1 2 2 2 2 85
1 2
2 4
3 5 7 2 1 2
P8 y 2 2 2 2 2 170
1 2
P8 85 170
 13

4. Byte B 1 KB
10
2 B 1 MB
10
2 KB
10 10
1 GB 2 MB 1 TB 2 GB 1 KB 1 MB 1 GB
1 TB 10
A 1,000,000 Bytes B 1,000,000,000 Bytes
C 1,000,000,000,000 Bytes D 1,000,000,000,000,000 Bytes
E 1,000,000,000,000,000,000 Bytes
10 10 10 10 10 10 10 10 10 10
x 2 2 2 2 2 2 2 2 2 2
10 10 4
2 2 1
10
2 1
10 4 4
~ 2 1024
4
~1000 1,000,000,000,000 Bytes
C

5.
3 3 3 3
1 1 2 3 20 5
2 2 2 2
2 9 10 11 20 5
2 2
x 3 3 3 3 20 20 1 20 21
1 1 2 3 20 44100
2 2
2 2 2 2 2 2 2 2 2 2 2 2
2 9 10 11 20 1 2 3 20 1 2 3 8
2 2 2 2 20 21 41 8 9 17
2 9 10 11 20 2870 204 2666
6 6

6. 3
10000
1 1 5
7
2 7 1.03 ~1.2299
5
x 1 1 10000 1 3 10300
2 7
7 6
10000 1 3 10000 1 3 10000 1 3
2 6 7
10000 1.03 10000 1.03 10000 1.03 10000 1.03
7
10000 1.03 1 1.03
~78932
1 1.03
78932
14

7. 1 4 2 5 3 6 n n 3 10
x 1 4 2 5 3 6 n n 3
1 1 3 2 2 3 3 3 3 n n 3
2 2 2 2
1 3 2 2 3 3 3 3 n n 3
2 2 2 2
1 2 3 n 1 2 3 n 3
n n 1 2n 1 n n 1
3
6 2
n n 1 n 5
3

8. O1 O1 O2
O2 O3
1 O2 5
2 O1 O 2 O3 O 4 5
x 1 O1 2
2
O2
2
2
2 O1 O2
2
2
O2 O3
2
2 2 2
2 3
O1 O2 O3 O4 1
2 2 2
2 2 2
2 3
2L 1 2L 2L 2L
2 2 2
2 2 2
2 3
2L 1
2 2 2
6 32
L
2
 15

1
9. an n Sn n an
2
n a1
2
1 n2 an an 1 5
2 an 5
2 2
x 1 n2 Sn n an Sn 1 n 1 an 1
2 2
an n an n 1 an 1

c n
2 2
1 an n 1 an 1

n 1
c an an 1
n 1
1
2 1 a1
2
1
琭
a1
2
1
a琭
2 琭
a1
3
2
琭
a3 琭
a2
4
3
琭
a4 琭
a3
5
㜱
n 2
琭
an 1 琭
an 2
n
n 1
an 琭
an 1
n 1
1 1 2 3 n 2 n 1
c an
2 3 4 5 n n 1
1
n n 1
1 1 1
n 1 a1 an
1 2 2 n n 1
16

10. 1 4
1 1 1 4
2

1 2 3
1 n 5
2 n 5
2 1
x 1
n
n 1 3 3 3
n n
1 1 3 3 1
1 3 2
2 n an
1 1 1 3 1 1 3 1 3 3
a1 a2 a3
4 4 4 4 4 4 4 4 4 4
2 n 1
1 1 3 1 3 1 3
an
4 4 4 4 4 4 4
n
1 3
1 n
4 4 3
1
3 4
1
4
 17

1
8 8
a
C 1 an d bn 2n bn
A
B
C
D
E
dc2n
a a d a d
x an an 1 2n 1
2n 1
2
d
bn bn 1 2
d d
bn 2 2 0
C

10 6 2 20
AB 2 an a4 7 a10 5
E 1 5
A B a20
3 3
C 24 D n n 23
E n 60 n 40 9
a1 8
a1 4 1 d 7
x A a1 d c 1
a1 10 1 d 5 d
3
1 5
B a20 a1 20 1 d 8 19
3 3
1 1 1
C an 8 n 1 0c8 n 0cn 25
3 3 3
26
1
D C 26 a25 8 25 1 0
3
n n 24 25
1
n 16 n 1
3 1 1
E Sn 60 c n 16 n 120


2 3 3
c n2 49n 360 0 c n 9 n 40 0cn 9 40
ABE
18

AC 3
a
DE A an 3n
2 2
B an d an d
C an an
D an r an
2
r
E an r 2an
r
a
3n a an d a
x A an d an 3n 1
3 3n
3 1

2 2
B an an 1 an an 1 an an 1 d an an 1
2
an
C an r an ran c an ran r an
an r
1 1 1

2
a2n 2 a2n r 2 2
D r an r
a2n a2n
2an 2 an 1 r
E r 2an r
2an 1 2 an 1

ACDE

8 48
3
4 an a1 1 an an 1 n n2 a10 3025
x
琭
a1 1
3
琭
a2 a1 2
琭
3
琭
a3 a2 3
琭

3
a9 10
a10 琭
2
10 11
c a10 1 2
3
3
3
10
3
55
2
3025
2
 19

1 1 1 1
5 Sn 2 3 n n 1 Sn 0.0001 n
2 2 2 2
14
n
1 1
1 n
1 1 1 1 2 2 1
x Sn 2 3 n 1
2 2 2 2 1 2
1
2
n
1 1
1 Sn 0.0001 c c 2n 10000
2 10000
n
13 10 3
2 2 2 1024 8 8192 10000
14 10 4
2 2 2 1024 16 16384 10000
n 14

6 30 206 769
x an
30 30
1 7 11 13 17 19 23 29 8
8 30 30
1 30 8 30
206 8 25 6 a206 30 25 a6
a1 1 a2 7 a3 11 a4 13 a5 17 a6 19
a206 30 25 19 769

20

n
7 1 r n Sn S2n 2 1 Sn
n
Sn 2 1
1 Sn c S2n
n n
x S2n 2 Sn 2 Sn
c r Sn 2 Sn
n n

cr 2
n n

cr 2
n
1 1 2 n
Sn 2 1
1 2

8 1 2 3
an 81 3810
2
x 81 9
81
2 2 2 2
1 2 3 90 1 2 3 9
90 91 9 10 19
3810
2 6

9 an n Sn 2an 2020 n an 5
S5 62620
x a1 S1 2a1 2020 a1 2020
n2 an Sn Sn 1 2an 2020 2an 1 2020 2an 2an 1

an
c 2an 1 an c 2
an 1

an 2
5
2020 1 2
S5 62620
1 2
 21

24
0 3

an n
1 an 8
2 100 8
x 1 a1 1 a2 a1 1 2 4 a3 a2 1 2 3 10
n n 1
an an 1 1 2 3 n an 1
2
a1 1
an n n 1
an an 1 n2
2
2
4 5 5 6
a4 a3 10 10 20 a5 a4 20 15 35
2 2
6 7 7 8
a6 a5 35 21 56 a7 a6 56 28 84
2 2
100

n 1
q n an 4 8 3 7 8
0
v 1 n 1 a1 4 8 3 7 7
k 1
2 n k ak 4 8 3 7
n k 1
k 1 1 k
ak 1 4 8 3 4 8 3
k 1
4 8 8 3
k 1 k 1
4 8 3 28 8
k 1 k 1
4 8 3 7 28 8 7
ak 1 7

n 1
n an 4 8 3 7

2
1
x1 x2 xn
x1 x2 xn I
n
2
n x1 x2 xn x1 w1
w1 w2 wn x2 w2
x1w1 x2w2 xnwn
W
w1 w2 wn
3 xn wn
1 x1 x2 xn

x1 x2
1
n
G xn x1 x2 xn n

2
n ri i 1 2 n

 1 
r
n
r r1 1 r2 1 rn 1

1
m Pm m 1m99
m Pm 100 m Pm
P50

2 n m Pm
1 n
m
2 I n
100
1 I I M Pm M
2 I Pm I I 1

22
 23

x1 x2 xn I
1
1 O2 x1 I
2
x2 I 2
xn I 2

n
1 2 2 2
x1 x2 xn I2
n

O (& x2& I &

1 2 2 2
2 x1 I xn I
n

(& n &
1 2 2 2
x1 x2 xn I2

1
xi a yi axi
2
xi b yi xi b
3 I O
1
xi a b yi axi b
2 x1 x2 xn Ix Ox
yi axi b a 0 i 1 2 n
1 yi Iy aIx b
2 yi Oy a Ox

1
x1 x2 xn I O
xi I
Xi i 1 2 n x1 x2 xn X1 X2
O
Xn
2
0 1
24

1. 40 40 20 20 80 100

1 3
2 4
3 3
x 1
40 40 20 20 80 100
50
6
1
2 O2 40 50
2
40 50
2
20 50
2
20 50
2
80 50
2

6
2
100 50 900
3 O 900 30

2. 30
38 38 45 47 50 52 55 56 58 60
60 62 64 65 68 70 72 73 75 77
78 80 81 83 88 90 91 93 94 96

1 40 P40 5
2 75 P75 5
x 40
1 40 30 12 12 13
100
62 64
P40 63
2
75
2 75 30 22.5 23
100
P75 81
 25

3. 20 20
40
10

80 75 65 90
20 20 20 40
x W 80 0.2 75 0.2 65 0.2 90 0.4
80

4. 1 3 49 63 5
2 60 20 10
5
3
3
x 1 G 49 63 21
2 r
3
1 1 60 1 20 1 10
r  1
r
60  20 
3
1 1 10 1 0.2 20
20
26

5. 10
8 9 10 11 12 13 8 9 10 11 12 13
A B
8 6 7 7 6 8 18 2 1 1 2 18
8 9 10 11 12 13 8 9 10 11 12 13
C D
1 2 18 18 2 1 10 10 1 1 10 10
8 9 10 11 12 13
E
7 7 7 7 7 7
x 10.5
C
C

6. 300

1 85 3
2 50 3
3 90
4
x 1 80
2 40
3 95
 27

7. 270C 230C 190C 210C 250C 290C 310C

1 0C 5
2 0F
9
0F 0F 0C 32 5
5
x 1 xi
27 23 19 21 25 29 31
Ix 25 0C
7

(& 
2 2 2 2 2 2 2
27 25 23 25 19 25 21 25 25 25 29 25 31 25
Ox
7
4 0C
250C 40C
9
2 yi yi xi 32
5

9 9
Iy Ix 32 25 32 77 0F
5 5
9 9
Oy Ox 4 7.2 0F
5 5
770F 7.20F

8. 8 2.5
64 6
1 7 60
5
2
2 5
3
x 7 8 2 60 64 2
1
2.5 5 6 3
2 2
3 5
2 x
x 64 2
cx 68
6 3
68
28

9. x y 1 4 5 5 2 0.5 1.5 a
b c x y a b c 10
x x y 1 4 5 x y 10
x y 1 4 5 5
5 5
x y 10
x
5
x 0.5 c 4x y 5 1
2
x y 10
y
5
y 1.5 c x 4y 25 2
2
1 2 x 3 y 7 5 4
1 4 4 4
1 a 1.5 4 b 0
2 2
5 4
5 c 0.5
2

10. 15 70
8 25 62 4 40
10
x 15 70 25 62
65
40

(& 15 x1 x2
1 2 2 2 2
8 x15 70 8

c x1
2 2 2
x2 x15 74460

(& 25 y1 y2
1 2 2 2 2
4 y25 62 4

c y1
2 2 2
y2 y25 96500

(& 40 74460 "49

1 2
96500 65 7
 29

1
xi yi i 1 2 n

1
2

3
4
5

1
xi yi i 1 2 n xi Ix
Ox yi Iy Oy
1 xi yi Xi Yi i 1 2 n
xi Ix yi Iy
Xi Yi
Ox Oy
X1Y1 X2Y2 XnYn
2 r
n
2
x1 y1 x2 y2 xn yn xi yi
Sxy
"Sxx "Syy
Ix Iy r

Sxy x1 Ix y1 Iy xn Ix yn Iy
2 2
Sxx x1 Ix xn Ix
2 2
Syy y1 Iy yn Iy
3
1 r 0 r 0 r 0
2 1r1
3 r 1 r 1
4 r
30

4
xi yi r xi' axi b yi' cyi d a c
0 xi' yi' r a c

1
xi yi Xi Yi Y X
Y rX r
2
xi yi i 1 2 n Ix Iy
Oy
r y x
Ox
Oy
y Iy r x Ix r
Ox
Sxy
y Iy x Ix
Sxx
x1 y1 x2 y2 xn yn nIxIy
y Iy 2 2 2 2 x Ix
x1 x2 xn nIx
 31

1. A F x y 10
A B C

D E F

x Ar 1
Br 1
Cr 0
D 1 r 0
E0 r 1
Fr 0
B D C F E A

2. 10
A B C

D E

x 9 xi yi i 1 2 9 =
A xi 2yi 1 =
B 6 xi yi =
C xi 4 yi =
D 6 xi 4 yi =
E yi xi =
ADE
32

3. xi yi Xi Yi
xi Ix
Xi 1.5 1.5 1 0.5 1 0.5 0
Ox
yi Iy
Yi 1 1.5 0 1 1.5 0.5 0.5
Oy
1 Xi Yi 5
2 Y X 5
x 1 Xi Yi
X1 Y1 X2 Y2 X7 Y7
r
7
1.5 1 1.5 1.5 1 0 0.5 1 1 1.5 0.5 0.5 0 0.5
7
6
7
6
2 Y X Y rX Y X
7

4. 200C 360C
0.99
0C 22 24 26 28 30 32
135 203 279 361 437 512
330C
10
A 530 B 550 C 570 D 590 E 620
x 1
330C y
y 512 512 135
cy 549.7~550
33 32 32 22
B
 33

5. 1 xi yi r 0.8 2xi 3 1 4yi r1

5
2 xi 1 yi r 0.8 2 3xi 4yi 5
r2 5
x
1 xi yi r 0.8
2xi 3 1 4yi xi yi
r1 r 0.8
2 xi 1 yi r 0.8 xi yi r0 0.8
2 3xi 4yi 5 xi yi
r2 r0 0.8

6. xi yi i 1 2 n
1 Ix 10 Iy 20 Ox 2 Oy 4 x y r 0.8
y x 5
2 Ix 5 Iy 2 Ox 3 Oy 8 y x 3 6
x y 5
x Oy
1 y x y Iy r x Ix
Ox
4
y 20 0.8 x 10 c y 1.6x 4
2
Oy
2 y x y Iy r x Ix
Ox
8
y 2 r x 5 3 6
3
8 3
6 2 r 3 5 cr
3 4
3
x y
4
34

7. 1 A x C y 2 B x C y
A x B y
10

1 2
x

A B C
1 8.5 5
2 7.5 12
3 6.5 15
4 5.5 18
5 4.5 25
1 2 6 3.5 32
A x B y 7 2.5 35
8 1.5 38
9 0.5 45
 35

8. 20 xi yi i 1 2 20 Ix 3 Iy 4 x y
r 0.8 y x 2 0
10
Ay x 3 4
By x y 4x 8
Cx y
D x 4 y 8
E 2xi 3 4 5yi 0.8
x A Ix Iy 3 4
Oy Oy
B y Iy r x Ix c y 4 m x 3 m r
Ox Ox
2 0 m 4 y 4x 8
Oy Oy 4
C B r mc 5 c O y Ox
Ox Ox 0.8
D x 4 y 4x 8 y 8
E 2xi 3 4 5yi 0.8
AD

9. x y 160 164 168 172 176

48 46 50 54 52
1 3
2 3
3 173 4
x x y x Ix y Iy x Ix
2
y Iy 2
x Ix y Iy
160 48 8 2 64 4 16
164 46 4 4 16 16 16
168 50 0 0 0 0 0
172 54 4 4 16 16 16
176 52 8 2 64 4 16
Ix 168 Iy 50 160 40 64
64
160 "40
1 r 0.8

64
2 m
160
64
y 50 x 168 c y 0.4x 17.2
160
y 0.4x 17.2
3 x 173 y 0.4x 17.2 y 52
36

10. ai bi r 0.9 ai Ia 3
Oa 1 bi Ib 2 Ob 2 xi 3ai 1 yi 3 2bi
1 xi Ix Ox 2
2 yi Iy Oy 2
3 xi yi R 3
4 y x 3
x 12
Ix 3Ia 1 8 Ox 3 Oa 3
Iy 3 2Ib 1 Oy 2 Ob 4
3 xi 3ai 1 yi 3 2bi ai bi
R r 0.9
Oy
4 y x y Iy R x Ix
Ox
4
y 1 0.9 x 8 cy 1.2x 8.6
3
 37

2
8 24
C 1 5 A B C D E
4
AA
BB
CC
DD
EE
x C 4
C

D 2 55 1 1 2
2 10 10
70
A6
B7
C8
D9
E 10
x 70 55 70 38.5
P70 39
1 2 8 36
1 2 8 9 45
9
D

38

D 3 y 3 x 1 2 3 2
2 x y y 1 2 a b
r
25
Aa 3 Ba 4 Cb 5 Dr Er 2
5
1 2 a b
x Iy 3 c 3ca b 9 b 3 9 a 3 6 a
4
2
x y x Ix y Iy x Ix x Ix y Iy
1 1 1 2 1 2
2 2 0 1 0 0
3 a 1 a 3 1 a 3
2 b 0 6 a 0 0
Ix 2 Iy 3 2 a 1
a 1
2ca 5 b 4
2

 2 
4 25
2
r 2 2 2 2
1 2 1 5
D

8 5 2 16
AC 4 x 55 8 y
DE 50 10 0.8
0.8 16 10 100

A 60 B 10
C 60 D 10
5
E y x y x 15
4
x x' 0.8x 16 y' y 10
AB Ix' 0.8 Ix 16 44 16 60 Ox' 0.8 Ox 6.4
CD Iy' Iy 10 50 10 60 Oy' Oy 10
E r' r 0.8

Oy'
y' Iy' r' x' Ix'
Ox'
10 5
y' 60 0.8 x' 60 c y' x' 15
6.4 4
ACDE
 39

AC 5 300 x y
DE 0 100
0.016
Ax y
B x a by x y a b
b 0
Cx 5 y 5 0.016
D 10x 10y 0.016
x Ix y Iy
E x y Ix I y x y Ox Oy
Ox Oy
x y x y 0.016
x A
B
CDE xi yi r
xi' axi b yi' cyi d xi yi
ac
r
ac
ACDE

8 40
6
88 x 70 y
x y 90 60
x 88 90 70 60
x y 90 60

7 20 20 30
30 60 54 51
73 60
x x 60
60 0.2 54 0.2 51 0.3 x 0.360 c 0.3x21.9 c x73
73 60

40

8 1 3 4 5 7 O 2
1 11 13 14 15 17 Ox
2 10 30 40 50 70 Oy
3 1 1 3 3 4 4 5 5 7 7 Oz
Ox Oy Oz 2 20 2
x ai 1 3 4 5 7 i 1 2 3 4 5
1 xi 11 13 14 15 17 xi ai 10 Ox O 2
2 yi 10 30 40 50 70 yi 10ai
Oy 10 O 20
1 3 4 5 7
3 1 3 4 5 7 I 4
5

(&
2 2 2 2 2
1 3 4 5 7 2
O 4 2
5
2 1 3 4 5 7
Iz 4
10

(& (&
2 2 2 2 2 2 2 2 2 2
2 1 3 4 5 7 2 1 3 4 5 7 2
Oz 4 4 2
10 5
Ox 2 Oy 20 Oz 2 O x Oy Oz 2 20 2

q 60 40 20 80
20
x
 1 60
r
1 20
4
r 1 40 1 80 1 0.2 20
20
 41

0 80 90 100 110 120

1100 1200 1000 800 900

85 1120
x x y
2 2
x y x Ix y Iy x Ix y Iy x Ix y Iy
80 1100 20 100 400 10000 2000
90 1200 10 200 100 40000 2000
100 1000 0 0 0 0 0
110 800 10 200 100 40000 2000
120 900 20 100 400 10000 2000
Ix 100 Iy 1000 1000 100000 8000
y x
8000
y 1000 x 100 c y 8x 1800
1000
85 x 85 y 8 85 1800 1120

10 20
q 20 50
450 40 25
85

x xi yi
xi 50 5 45
yi 25 85 xi
20 4 2
5 45 5 45
Iy Ix 450 585
4 2 4 2
5 5
Oy Ox 40 50
4 4
585 50

42

w 40 40 60
Ix 25 Iy 40 Ox 5 Oy 10
14
x xi yi zi xi yi
x1 x2 x40
25 25 c x1 x2 x40 1000
40
y1 y2 y40
40 40 c y1 y2 y40 1600
40
z1 z 2 z40 x1 x2 x40 y1 y2 y40
Iz
40 40
1000 1600
65
40
5

(& 40 x1 &
1
5 c x1
2 2 2 2 2 2 2 2 2
x2 x40 25 x2 x40 40 5 25 26000

10

(& 40 y1 &
1
10 c y1
2 2 2 2 2 2 2 2 2
y2 y40 40 y2 y40 40 10 40 68000

14

(& 40 z1 z2 &
1
14 c z1
2 2 2 2 2 2 2 2 2
z40 65 z2 z40 40 14 65 176840
2 2 2 2 2 2
z1 z2 z40 x1 y1 x2 y2 x40 y40
2 2 2
x1 x2 x40 2 x1 y1 x2 y2 x40 y40
2 2 2
y1 y2 y40
94000 2 x1 y1 x2 y2 x40 y40 176840
x1 y1 x2 y2 x40 y40 41420
x1 y1 x2 y2 x40 y40 40IxIy 41420 40 25 40
r 0.71
40OxOy 40 5 10

x1 Ix y1 Iy x2 Ix y2 Iy xn Ix yn Iy
x1 y1 x2 y2 xn yn x1 x2 xn Iy y1 y2 yn Ix nIx Iy
x1 y1 x2 y2 xn yn nIx Iy nIx Iy nIx Iy
x1 y1 x2 y2 xn yn nIx Iy
3
1
1 P Q PcQ P Q Q P
2 P Q Q P PvQ P Q

2
A B A
A A A B A B A B
A B

1
x A
x A x.A x A xA
2 ‡ ‡
3 n S S S
S 2 3 5 7 n S 4
4
A B A B A B
5
A S A S A)S
A S S'A S A

1
1 A B A-B x x.A x.B
2 A B A,B x x.A x.B
2
1 U
2 A B A B x x.A xB
3 A A x x.U xA U A
3
1 A,B A -B
2 A-B A ,B

43
44

2
k 1
n1 2 n2 k nk
n1 n2 nk
3
k 1 n1
2 n2 k nk
n1 n2 nk
4
A B C
1 n A-B n A n B n A,B

2 n A-B-C n A n B n C n A,B n B,C n A,C

n A,B,C
 45

1. 10
A 5 3 5 3
B 5 3 5 3
C x 2 y 3 x 2 y 3
D x 2 y 3 x 2 y 3
E 6 6
x ADE

2. 1 2 3 4 1
3 2 4 10
x

14

3. 50 10 5 60 10
x 50 x 10 y 5 z
50x 10y 5z 60 c 10x 2y z 12
1 x 0 2y z 12
y 6 5 4 3 2 1 0
z 0 2 4 6 8 10 12
2 x 1 2y z 2
y 1 0
z 0 2
2 7 9
46

4. 18
10
x
1 1 18
2 2 8
3 3 2
18 8 2 28

5. 2 5 3 4

10
x 4
2 5 3 4 120

6.
8

10
x
8 8
8
2 2 2 2 2 2 2 2 1 2 1 255
8
255
 47

7. 1 360 5
2 360 15 5
3 2
x 360
a b c
2 3 5 360 2 3 5
1 a 0 1 2 3 4 b 0 1 2 3
c 0 1 2
360 4 3 2 24
2 15 3 5 15 3 5
a 0 1 2 3 4 b 1 2 2 c 1 1
4 2 1 8 15

8. 1 1 200 2 3 5
2 1 200 3 5 5
x U 1 200
A B C 2 3 5
A,B 6 B,C 15
1 n A-B n A n B n A,B
1 n A-B 100 66 33 133

2 n B',C' n B-C
2 n B',C' n U n B-C
2 n B',C' 200 n B n C n B,C
2 n B',C' 200 66 40 13 107
48

9.

30 30 35 35 35 40 40 40

10
x

21

10. 40 25 23 22
15 11 12
5 10
x A B C
n A-B-C 40 5 35

n A-B-C n A n B n C n A,B n B,C n C,A n A,B,C

c 35 25 23 22 15 11 12 n A,B,C c n A,B,C 3
3
 49

n
n n! n n 1 n 2 2 1

n k
n
P k n k 0kn
n n!
Pk n n 1 n k 1
n k !

n k m1 m 2 mk
n!
m1 m2 mk n n
m1!m2! mk!

k
n k k n

1. P3
n
P3
n 2
5 12 n 10
x n 1 n 2 5
12 c
n n 2
P3 P3 5
n 2 n 1 12
c 12n
2 2
36n 24 5n 15n 10
2
c 7n2 51n 14 0cn 7
7
n 7
50

2. 5 4

1 5
2 5
5
x 1 P 4 5 4 3 2 120
4
2 P3 4 3 2 24

3. 0 1 2 3 4 5 4
1 5
2 1 5
x 1
0 5 5 4 3 300
2 3
0
2
4

5 4 3 4 4 3 4 4 3 156
 51

4. A B C D E F G
1 A B C 5
2 A B C 5
x 1 A B C

5!
A B C 3!
5! 3! 720
2 A B C
D E F G 4!

5
A B C 5 3 P3
5
4! P 3 24 60 1440

5.
1 3
2 3
3 4
x 1
5! 4! 96
2

5! 4! 4! 3!
120 24 24 6 78
3
3 2 3 3
3
P 2 3! 3 2 6 36
52

6. 1 5 5 6 6 8 8 8 5
2
5
x 7!
1 210
2!2!3!
2
6!
15
4!2!

7. A B
1 3
2 C 3
3 C D 4
x 1 A B 5 4
9!
126
5!4!
4! 5!
2 A C B 60
2!2! 3!2!
3 C D C C D
4! 3! 2!
60
2!2! 2!1! 1!1!
60 6 3 2 24
24
8. 0 9
1 168 5
2 A 9 5
x 1
26 26 10 6760
2
A A 9
4
1 26 10 1 26 10 10 9 9
49400
 53

9. P O 2 3
14 10
x 2 x 3 y
2x 3y 14 x y
x y 3
x 7 4 1
y 0 2 4
7! 6! 5!
1 15 5 21
7! 4!2! 1!4!

10.
1 5
2 5
x 1 4

6!
360
2!
2 3

6!
2! 240
3!
54

n
Ck n k 0kn
n n!
Ck
k! n k !

n
Ck
n n
1 C0 1 Cn 1
n n
2 C k C n k 0kn

n n n 0 n n 1 1 n n k k n 0 n
n x y C 0x y C 1x y C kx y C nx y
n n k k
C kx y k 0 1 2 n

n n 1 n 1
Ck Ck 1 Ck 1kn 1

1. 12 9
1 5
2 2 1 5
x 12 12 12 11 10
1 12 9 C9 C3 220
3 2 1
2 2 1 10 8
2 10
C 1 C 8 2 45 90
 55

2. 7 6 13 4
10
x
13 7
C4 C4
715 35 680

3. 10
10
x 10 5
8

8
5 8
C1 C1 5 8 40

4. 9
1 3 5
2 3 5
x 1 3 6 3 3
9 6 3
C 3 C 3 C 3 84 20 1 1680
9 6 3
C 3 C 3 C 3
2 280
3!
56

5. 0.99
5
10
5
x 1 0.01
5 0 5 1 5 2 5 3 5 4
C 0 0.01 C1 0.01 C2 0.01 C3 0.01 C4 0.01
5 5
C5 0.01
~1 5 0.01 10 0.0001 10 0.000001
0.95099
~0.951

6. 1 C0
10
C1
10
C2
10
C 10
10
5
4 5 6 13
2 C 4 C 4 C 4 C 4 5
x 1
10 10 10 10 10
1 1 C0 C1 C2 C 10
c C0 C1 C2
10 10 10 10 10
C 10 2 1024
4 5 5 5 6 6 6 7
2 C4 C4 C5 C4 C 5 C 5 C4 C5
4 5 6 13
C4 C4 C4 C4
5 5 6 13
C5 C4 C4 C4
6 6 13
C5 C4 C4
14
C5 2002
 57

7. 1 1 2 3 4 5 6 2 4
5
2 6 A B 4 5
x 1
6 4
1 2 4 C 2 C 4 15
6 3
C3 C3
2 3 3 10
2
15 10 25
2 A B
1 A4 B2 2 A3 B3 3 A2 B4
6 2 6 3 6 4
C 4 C 2 C 3 C 3 C 2 C 4 50

8. 10 3 7
10
x 10 3 7

7 8

3 3
1 3 7 1 10
1 4 9
8
C 3 56
58

9.
1 4 3
2 4 3
3 4 4
x 1
1
5 2 2 2 2 5 4
C 4 C 1 C 1 C1 C1 C4 2 80
2
1
5 4 2 2
C 1 C 2 C 1 C 1 120
3
10
C4 80 210 80 130

10. 10 1 2 3 4 10
1 4 4 5 5
2 4 4 2
1 2 4 7 3 5 7 8 2 5 6 9 5
x 1 5
5 1 5 2 5
4 5
C 1 C 2 40
2

7 1 6 2
1 10
7 6
C 1 C 2 105
 59

1
2

1 S S
2 S A B
1
2
3 S
4 ‡
5 A-B A B
6 A,B A B
7 A,B ‡ A B A B

8 A' S A A A' A

1
S
n A
A P A n A n S
n S
A S
2
S A B
1 0P A 1
2 P S 1 P ‡ 0
3 P A 1 P A
4 P A-B P A P B P A,B
A B A,B ‡ P A-B P A P B
5 P A B P A P A,B
6 P A ,B 1 P A-B
7 P A-B-C P A P B P C P A,B P A,C P B,C
P A,B,C
60

S A1 A2 An S

1 A1-A2- -An S
2 i j Ai , Aj ‡
Ai pi i 1 2 n Ai mi
i 1 2 n
m1 p1 m2 p2 mn pn

1.
S A B C

1 A B 5
2 A C 5
x x y
x y
S 1 2 3 4 5 6
S 1 2 3 4 5 6
A 1 2 3 4 5 6
B 2 4 6 2 4 6
C 2 3 5 2 3 5
1 A-B 1 2 3 4 5 6 2
2 A-B 4 6
2 A,C 2 3 5
 61

2. S A B P A-B 0.7 P A,B 0.2

P A,B' 0.3
1 P A 5
2 P B 5
x 1 P A,B' P A P A,B c 0.3 P A 0.2
P A 0.5
2 P A-B P A P B P A,B c 0.7 0.5 P B 0.2
P B 0.4

3. 5 1
10
x

15 9 3 3
1 3 3 1
8 8 8 8
1 3 3 1
15 9 3 3 6
8 8 8 8

4. 1 7
A B
A B 10
7
x S n S C 3 35
A 3 2 1
B 3 2 1 1 2
3 4
A,B 2 1 n A,B C 2 C 1 12
12
P A,B
35
62

5. 3 7 10
3
x S n S 6 216
A 3 7
3!
5 1 1c 3
2!
4 2 1 c 3! 6
3!
3 3 1c 3
2!
3!
3 2 2c 3
2!
n A 15 5
n A 3 6 3 3 15 P A
n S 216 72

6. 10

1 5
2 5
5
x S n S 10
10
1 A 5 n A P5
10
n A P5 189
P A 5 0.3024
n S 10 625
10 5 9
2 B 5 n B C1 C2 P3
10 5 9
n B C1 C2 P3 63
P B 5 0.504
n S 10 125
 63

7.
1 5
2 5
3
x S n S 3 27
1 A

3 n A 9 1
n A C1 3 9 P A
n S 27 3
2 B
n B 3 3! 9
n B 9 1
P B
n S 27 3

8. 25
10 6
10
x 0
x

25 10 6 x
1 15 75 125
216 216 216 216
1 15 75 125
25 10 6 x 0
216 216 216 216
x 5 5
64

9. 10
2
10
3
10
x S n S C2 45
A A
x
x n A C 2
x
n A C2 2
P A 1 P A' 1 1 x 6 5
n S 45 3
6 4

10. 13 52

1 5
2 5
52
x S n S C 4 270725
1 A
13 4 4 13 4 4
n A P2 C3 C1 C2 C 2 C 2 5304
n A 5304 24
P A
n S 270725 1225
2 B
4 13
n B C4 P4 17160
n B 17160 264
P B
n S 270725 4165
 65

3
8 24
D 1 1 1000
A4 B5 C6 D7 E8
3 3 3 3 3 3 3 3 3 3
x 1 1000 1 2 3 4 5 6 7 8 9 10
3 3 3
1 4 9
10 3 7
D

E 2 O 0
9 4 AAA1234 AAB4434
AOA1234 ABO3444 AAA4444

3 4 3 3 3 4
A 25 10 B 25 9 10 C 25 9
3 3
D 25 9980 E 25 9981
x -
3 4 3
25 10 19 25 9981
E

66

C 3
10 20
A 90 B 100 C 135 D 145 E 190
x

10 9
10 9 45 90 135
2
C

20 10
C2 10 C2
190 10 45
135
C

10 6 2 20
BC 4 S A B P A 0.8 P B 0.5
D
A P A,B P A B B P A,B P B A
CP A B P B A DP A B P A',B'
EP B A P A',B'
x P A 0.8 P B 0.5
0.3P A,B 0.5 1
P A B P A P A,B 0.8 P A,B
0.3P A B 0.5 2
P B A P B P A,B 0.5 P A,B
0P B A 0.2 3
P A',B' 1 P A-B 1 P A P B P A,B P A,B 0.3
0P A',B' 0.2 4
1 2 3 4 BCD
 67

CE 5 1 2 3 4 5 6

3 1
A 8 B 5
5 5
2 2
C D
5 5
1
E
5
x A 3 6 4 5 4 6 5 6 4
4 4
6
C 2 15
B 1 2 1 3 2
2 2
6
C 2 15
C
3 3
C2 C2 6 2
6
C 2 15 5
D 1 1
3 3
C1 C1 9 3
6
C 2 15 5
E 2
3
C2 3 1
6
C 2 15 5
CE

9 36
6 50 37 30 25
20 13 17
5 7
x A B C

37 15 5 12 30 15 5 8 25 12 5 8
5 2 0 7

68

7 6 6 5

2520
x
6 5 4 3 2 720
5
3 C 4 5 4 3 2 1800
720 1800 2520

8 1 2 3
4 5 6
5
x

5
 69

4 5
504
x 1 2 1 1 5! 120

1 2 1 4 5! 4! 384

1
120 384 504

10 20
0 a1 a2 a10 1 1
1 a1 a2 a10 5
2 a1 a2 a10 4 5
x a1 a2 a10 1 1
1 a1 a2 a10 10 1 1
10 1 0 1 10
9 1 1 1 8
8 1 2 1 6
a1 a2 a10
10 8 6 4 2 0 2 4 6 8 10
11
2 a1 a2 a10 4 10 7 1 3 1
7 1 3 1 a1 a2 a10
10!
120
7!3!

70

q
12 600 1500 6

0.48
100 1 600 1500
101
x 1 n 0.48
12
0.48 cn 2500
n
2 100 2400 600 5
1500 6
600 1500 0
5 6 2389
2400 2400 2400
5 6 2389
600 1500 0 5
2400 2400 2400
101 5
4
M M 900 M 900

M
sin M M

M
cos M M

M
tan M M
M

sin M
00 M 900 tan M
cos M

2 2
00 M 900 sin M cos M 1

00 M 900 sin 900 M cos M cos 900 M sin M

300 450 600

M
300 450 600

1 2 3
sin M
2 2 2

3 2 1
cos M
2 2 2

3
tan M 1 3
3

M sin M
tan M cos M

71
72

1
2
3
4

1.
1 cos 300 tan 300 sin 600 tan 600 5
2 2 2
2 cos 450 tan 450 sin 450 5
x 1 cos 300 tan 300 sin 600 tan 600
3 3
3
1 1 3
3
2
2 2 2 2
2 2 2
2 cos 450 tan 450 sin 450
2 2
2 2 2
1 1
2 2

2
2. 1 M sin M cos M tan M 5
3
sin M 5 cos M
2 M tan M 2 5
3 sin M cos M
x 1
M 32 22 5
5 2 25
5
cos M tan M
3 5
sin M
2 tan M cos M
cos M
sin M 5 cos M
sin M 5 cos M cos M cos M tan M 5 2 5 7
3 sin M cos M 3 sin M cos M 3 tan M 1 3 2 1 5
cos M cos M
 73

1
3. 1 sin 200 cos 500 5
2
2 sin 650 cos 650 tan 650 5
x 1
1 sin 300 cos 500 sin 400
2
00 M B 900 sin M sin B
sin 400 sin 300 sin 200
1
cos 500 sin 200
2
2 sin 650 sin 250 cos 650
tan 650 tan 450 1 sin 650 cos 650
M sin M 1
tan 650 sin 650 cos 650

4.
2 2 2 2 2 2 2 2
1 sin 100 sin 200 sin 300 sin 400 sin 500 sin 600 sin 700 sin 800 5
2 tan 330 tan 570 5
x 1
2 2 2 2 2 2 2 2
sin 100 sin 200 sin 300 sin 400 cos 400 cos 300 cos 200 cos 100
2 2 2 2 2 2 2 2
sin 100 cos 100 sin 200 cos 200 sin 300 cos 300 sin 400 cos 400
1 1 1 1
4
2 330 570 900

a b
tan 330 tan 570
b a
tan 330 tan 570 1
74

1 sin M cos M k
5. 00 M 900 k 10
cos M 1 sin M cos M
2 2
x 1 sin M cos M 1 sin M cos M
cos M 1 sin M cos M 1 sin M
2 2
1 2 sin M sin M cos M
cos M 1 sin M
2 2 sin M
cos M 1 sin M
2 1 sin M
cos M 1 sin M
2
cos M
k 2

6. ABCD AB 15 BC 9 P CD
PAD = PBC > tan = tan >
10
x DP CP DP CP
tan = tan >
AD BC 9 9
DP CP CD
9 9
15 5
9 3
 75

7. ABC AD BC AC 15
3 3
tan B sin C AB 10
2 5
x ACD
AD AD 3
sin C
AC 15 5
c AD 9
ABD
AD 9 3
tan B
BD BD 2
c BD 6
AB AD 92 62 117
2 2
BD 3"13

8. A P B Q 4 AB AB 10
3
sin M AQ BQ 10
5
x AB
AQB 900
1
ABQ AQ M
2
ABQ
AQ AQ 3 3
sin ABQ sin M c AQ 10 6

c BQ 10 6 8
AB 10 5 5
2 2

AQ BQ 6 8 14
76

9. AD 16
AB 6 BCF 450 A
AH 10
x A E
ABE BCF 450
AH EB BF
AB cos 450 BC sin 450
2 2
E

6 16
2 2
112

5
10. M sin M cos M
4
1 sin M cos M 5
3 3
2 sin M cos M 5
x 1 sin M cos M 1 c sin M cos M
2 2 2
2 sin M cos M 1
25 9 9
c 2 sin M cos M 1 c 2 sin M cos M c sin M cos M
16 16 32
3 3 2 2
2 sin M cos M sin M cos M sin M sin M cos M cos M
5 9 5 23 115
1
4 32 4 32 128
 77

00 1800

M B 3600 M B 3600 n n
M B

1 x

OP x
2 M M
2 2
P x y y r r 0
y
sin M
r
x
cos M
r
y
tan M x 0
x

M
2 2
1 sin M cos M 1
sin M
2 tan M M y
cos M
3 M sin M sin M
cos M cos M
tan M tan M M y
4 1800 M sin 1800 M sin M
cos 1800 M cos M
tan 1800 M tan M M y
5 900 M sin 900 M cos M
cos 900 M sin M
78

P x y O r
x y r cos M r sin M 00M 3600 M x
OP

L M L L tan M

O O L
O P OP r L OP
M [r M] P
P[r M] r 0 O L
b O x

1 P [r M] x y r cos M r sin M
2 P x y [r M ]

x2 y2 cos M
x y
r sin M
r r
 79

1. M 00M 3600
1 3800 2 2 4700 2 3 8400 2
4 3700 2 5 2700 2
x 1 3800 3600 200
200
2 4700 3600 1100
1100
3 8400 3600 3 2400
2400
4 3700 3600 2 3500
3500
5 2700 3600 1 900
900

2. 1 sin M cos M M 5
4
2 M O A 3 y M tan M
3
sin M cos M 5
x 1 sin M cos M
sin M 0 cos M 0
sin M 0 cos M 0
M
y 4
2 tan M y 4


3 3
2 2
r OA 3 4 5
y 4 x 3
sin M cos M
r 5 r 5
1
sin M cos M
5
80

3.
1 sin 3900 3
2 cos 4800 3
3 tan 1350 4
x 1 sin 3900 sin 2 3600 3900
x 1 sin 3900 sin 3300

 3
1
x 1 sin 3900 2 2
1
1
x 1 sin 3900
2
2 cos 4800 cos 3600 1200
cos 1200

  1
1
3
2 2

1
2
3 tan 1350 tan 3600 1350
3 tan 1350 tan 2250
1
3 tan 1350 1
1

1
4. sin M M cos M tan M 10
3
2 2

1 sin2M
x sin M cos M 1
cos M

1 sin M
M cos M 0
2
cos M

(&1
2
1 22
3 3

1
sin M 3 1 2
tan M
cos M 22 22 4
3
 81

5. 1 L 450 1 2 L 5
3
2 1 L y x 1 L L 5
3
x 1 L 450 m tan 450 1
L 1 2 y 2 1 x 1
y x 3
2 L M
3 3
L tan M
3 3
M 300
L L 450 M 450 300 750

4
6. M M tan M
3
sin 1800 M sin M 900 10

3
x M P 3 4
2 2
OP 4 5
4 3
sin M cos M
5 5
sin 1800 M sin M 900
sin M sin 900 M sin M sin 900 M sin M cos M
4 3 7
5 5 5

7. 1 Q [6 1200] 5
2 x P 2 2
00 3600 5
x 1 r 6 M 1200
1 3
x y 6 cos 1200 6 sin 1200 6 6
2 2
3 33
2 P [r M]

r  2
2
2
2
8 22
2 1 2 1
2 2
cos M sin M
22 22
M 1350
P [22 1350]
82

8. M ABC ACB AB 2
BC 3 sin M cos M 10
x
AC 2 3 "13
2 2

AB 2 2"13
"13
c sin M sin 1800 ACB sin ACB
AC 13
BC 3 3"13
"13
cos M cos 1800 ACB cos ACB
AC 13

C 0 0 B 3 0 A 3 2
A M
3 2
x  
x y
cr "13
2 2 2 2
y 3 2
y 2 2"13
"13
c sin M
r 13
x 3 3"13
"13
cos M
r 13
4 3 cos M 1
9. tan M cos M sin M 0 10
3 2 sin M 1
x tan M 0 M
4
1 M tan M
3

r x   3
x y 3 4
2 2 2 2
y 4 5
x 3 y 4 3 4 1
cos M sin M c cos M sin M 0
r 5 r 5 5 5 5
4
2 M tan M
3

r x 3
x y 3 4
2 2 2 2
y 4 5
x 3 y 4 3 4 1
cos M sin M c cos M sin M 0
r 5 r 5 5 5 5
3
3 1
3 cos M 1 5 4
2 sin M 1 4 3
2 1
5
 83

10. 1 00M 3600 2 cos M 3 cos M

2
2 0 M 5
2 2
2 sin M 4 cos M 3 sin M cos M M tan M 5
2
x 1 2 cos M 3 cos M 2 0
c 2 cos M 1 cos M 2 0
1
c cos M 2 cos M 1
2
1
cos 600 cos 3600 600 cos 3000 M 600 3000
2
2
2 2
sin M 4 cos M 3 sin M cos M
c sin M 3 sin M cos M 4 cos M
2 2
0
c sin M cos M sin M 4 cos M 0
c sin M cos M 0 sin M 4 cos M 0
c sin M cos M sin M 4 cos M
sin M
c tan M 1 4
cos M
M tan M 0
tan M 1

1
2
cos M
2 2
sin M 4 cos M 3 sin Mcos M
2 2
cos M cos M
sin M 2 sin M
4 3
cos M cos M
2
tan M 4 3 tan M
c tan M 3 tan M 4 0
2

c tan M 1 tan M 4 0
c tan M 1 4
M tan M 0
tan M 1
84

" M
" cos M

ABC A B C a b c
1 1 1
ABC bc sin A ca sin B ab sin C
2 2 2

ABC A B C a b c ABC
a b c
R 2R
sin A sin B sin C

ABC A B C a b c
2 2 2
a b c 2bc cos A
2 2 2
b c a 2ca cos B
2 2 2
c a b 2ab cos C

2 2 2
b c a
cos A
2bc
2 2 2
c a b
cos B
2ca
2 2 2
a b c
cos C
2ab

ABC a b c ABC s s a s b s c
a b c
s ABC
2
 85

53 25
1. 1 ABC AB 5 AC 6 cos B cos C BC 5
80 32
1
5
2 ABC AB 8 AC 45 cos A ABC 5

x 1 AD BC D

BC
BD CD 5 cos B 6 cos C
53 25
5 6
80 32
8
1 2
5 5
2 cos A sin A

1
ABC AB AC sin A
2
1 2
5
8 45 32
2

2. ABC AB 2 BC 1 A 300 C 10
x BC AB
sin A sin C
1 2
sin 300 sin C
sin C 2 sin 300=

2
1
2
2
2
C 450 1350
86

3. 1 ABC BC 7 AC 5 C 1200 AB 5
2 ABC AB 3 AC 4 BC "13 A 5
2 2 2
x 1 AB 5 7 2 5 7 cos 1200
25 49 35
109
AB 109
"13
2 2 2
4 3
2 cos A
2 4 3
12 1
24 2
A 600

4. ABCD DAC 300

BAC 450 BC 32 CD 10
x ACD ABC R
CD BC
2R 2R
sin DAC sin BAC
CD 32
sin 300 sin 450
32 32 1
CD sin 300 3
sin 450 1 2
2
 87

5. ABC 10 D AB BDC 450 CD

6 2
BCD sin 150 10
4
x ADC
CD 10
sin 600 sin 450
10
CD sin 600
sin 450
3 10
56
2 1
2
BCD 1800 CBD BDC 1800 1200 450 150
1 1
BCD CB CD sin BCD 10 56 sin 150=
2 2
1 6 2 75 253
10 56
2 4 2

6. O ABCD AB 5 BC 4 CD 1 B 600
1 AC 5
2 AD 5
x 1
2 2 2
AC 5 4 2 5 4 cos 600
25 16 20
21
AC "21
2 ACD D 1800 600 1200
AD x
2 2 2
AC x 1 2 x 1 cos 1200
c 21 x x 1
2

c x x 20 0
2

c x 4 x 5 0
cx 4 5
AD 4
88

7. A B C
D CD B ACD 450 ADC 600
10
x R r
AB
AB
ABC 2R
sin 450
AB AB
2
cR
2 sin 450
AB
ABD 2r
sin 600
AB AB
3
cr
2 sin 600
2 2
AB AB 1 1
2 3
LR2 Lr2 L L 3 2
2 3

8. ABC AB 6 BC 7 AC 8 A BC D
AD 10
x AD BAC
BD CD AB AC 6 8 3 4
3 4
c BD 7 3 CD 7 4
3 4 3 4
AD x ABD ABC
2 2 2 2 2 2
3 6 x 7 6 8
cos B
2 3 6 2 7 6
c 7 9 36
2
x 3 49 36 64
c x 36
2

cx 6
AD 6
 89

9. ABC AB 2 AC 3 BAC 600

AB AC ABDE ACFG
AEG 10
x AE 2 AG 3
EAG 1800 BAC 1800 600 1200

2 2 2
EG 2 3 2 2 3 cos 1200
1
4 9 12 19
2
EG "19
"19
2R R AEG
sin 1200
"19 "19 "19 "57
3 3
cR
2 sin 1200 3
2
2

10. 10 A B 600
A B 450 A B
10
x O C
OC 10 CAO 450
CBO 450
AOC
OC
tan 450 1
OA
OA 10
BOC
OB 10
AOB 600

2 2 2
AB 10 10 2 10 10 cos 600
100
AB 10
4
6 18
D 1 M P x y M O OP 3
P y
A sin M B cos M C 3 sin M 1800 D 3 sin M E 3 sin 1800 M
x P x y 3 cos M 3 sin M
P y 3 sin M
D

D 2
A tan 700 B tan 1400 C tan 2100 D tan 2800 E tan 3500
x tan 1400 tan 1800 400 tan 400
tan 2100 tan 3600 1500 tan 1500 tan 300
tan 2800 tan 3600 800 tan 800
tan 3500 tan 3600 100 tan 100
tan 800 tan 400 tan 100 tan 300 tan 700
tan 2800 tan 1400 tan 3500 tan 2100 tan 700
D

90
 91

C 3 ABCD a
AE b EAD M E
Aa b cos M B a b cos M Ca b sin M
Da b sin M E a b tan M
x E AD BC F G
E
EG FG EF
FG AE sin M
a b sin M
C

10 6 2 20
AB 4 M tan M
A tan 3600 M
B tan 1800 M
C tan 3600 M
D tan 1800 M
E tan 900 M
x M tan M 0
A tan 3600 M tan M
B tan 1800 M tan M
C tan 3600 M tan M
D tan 1800 M tan M
E 900 M tan 900 M 0
AB

92

AC 5
D A sin 400 cos 400
B sin 500 cos 500
C sin 500 tan 500
D cos 500 tan 500
E cos 1000 tan 1000
x A sin 400 sin 450 cos 450 cos 400
B sin 500 sin 450 cos 450 cos 500
C sin 500 1 tan 450 tan 500
D cos 500 1 tan 450 tan 500
E cos 1000 cos 800 1 tan 450 tan 800 tan 1000
ACD

7 42
6 L1 y 3 x 1 L2 y x 2 L1 L2 150
x L1 L2 M1 M2
L1 L2 3 1
tan M1 3 tan M2 1 M1 600 M2 450
L1 L2 M1 M2 150
 93

17
7 cos M 5 sin M 5 00 M 900 cos M sin M
13
x cos M 5 sin M 5
c cos M 5 1 sin M
c cos M 25 1 2 sin M
2 2
sin M
c 1 sin M 25 1
2 2
2 sin M sin M
c 26 sin M 50 sin M 24 0
2

c 13 sin M 25 sin M 12 0
2

c 13 sin M 12 sin M 1 0
12
c sin M 1
13
12 12 5
sin M cos M 5 1
13 13 13
5 12 17
cos M sin M
13 13 13

8 ABCD BAC 450 ACD 300

BC 2 AD 2
x ACD ABC R
AD BC
2R 2R
sin ACD sin BAC
AD 2
sin 300 sin 450

2
2 1 2
AD sin 300
sin 450 2 1
2

94

9 O A [3 130] B [6 1330] OAB

"21
x AOB 1330 130 1200
2 2 2
AB 6 3 2 6 3 cos 1200
1
36 9 2 6 3
2
36 9 18
63
c AB "63 37
AB
2R R OAB
sin 1200

"21
AB 37 37
3 3
cR
2 sin 1200
2
2

0 ABC CAB 900 AB 3

AC 4 BC AC BCDE
ACGF
1 CDG 6
"73
x 1 CD BC 3
2 DG
2 2
4 5
CG AC 4
AB 3
sin DCG sin 1800 ACB sin ACB
BC 5
1 1 3
CDG CD CG sin DCG 5 4 6
2 2 5
AC 4
2 cos DCG cos 1800 ACB cos ACB
BC 5
2 2 2 4
CDG DG 5 4 2 5 4 cos DCG 25 16 40
5
25 16 32 73
DG "73
 95

10 20
q A B A
A 50 C A 200
D ACB 600 ADB 300
1 A B 6
2 sin A 4
x 1 AC 50 AD 200 CD 200 50 150
CBD 600 300 300 CDB
BCD
BC CD 150
ABC
2 2 2 2 2 1
AB AC BC 2 AC BC cos 600 50 150 2 50 150 17500
2
AB 17500 507
BC AB
2 ABC
sin A sin 600
BC 150 3 3"21
sin A sin 600
AB 507 2 14

w 100 A B C A
AB AC 750 300 B BA
BC 600 CH
x CH x
ACH 300 600 900
AC 2x
ACB 1800 750 600 450
100 2x
ABC
sin 450 sin 600
3
100
100 sin 600 2 503
2 2
cx 256
2 sin 450
2
2
256

96

ON
AC
COMP Mode 0
DEG Mode 4
RAD Mode 5

--- ---

SHIFT 
log
x
SHIFT 10
y
x
sin sin
cos cos
tan tan
1 1
sin SHIFT sin
1 1
cos SHIFT cos
1 1
tan SHIFT tan
SHIFT