DocHdl3OnPRINECT2017tmpTarget .

3
. 0 50 100
5~6 1 .

5 cm
4 cm
2 3 ,
.
.
ABC ADE
x, y

5 cm
x cm
10 cm
A B
!
? .

2.5~20`km .
?
! !
.
.
, .
153
(Aristarchos, B.C.
320?~ B.C. 250?) ,
.

()
ABC ADE ∠A , ∠ABC=∠ADE
ABC»ADE. 12 AEÓ=10`cm.
ABC ()
( ) = ,
()
( ) ?
A B



ABC, AB'C', AB"C", y
.

sin`A
ABÓ ACÓ
cos`A
BCÓ ABÓ
tan`A
.
sin`A, cos`A, tan`A ∠A .
.
| | sin, cos, tan sine, cosine, tangent .
sin`A, cos`A, tan`A A ∠A .
sin, cos, tan
(Euler,

ABC ∠A
= '55
A
C
B2
1
A
C
B
6
5
ABC ∠A ∠B .1
2÷5
A B
156 .
∠B=90ù ABC tan`A='3 , sin`A+cos`A .3
ABC ∠A ∠B .2

A B
ACÓ=4, BCÓ=3, ∠B=90ù
ABC .

= '7 4
'7 4
∠B=90ù ABC sin`A=;4#; , cos`A .2
|
;8%; .
cos`A .
, .

259
1 ABC ∠A
.
.
ABC cos`A=;4#; , x,
y .
1
.
4 ∠B=90ù ABC cos`A=;8%; ,
sin`A tan`A .
6 •
10`%
;1Á0; .
∠A
, ∠A .
158 .
30ù, 45ù, 60ù ?
1 2 ABC,
DEF . .
30ù, 45ù, 60ù .
60 30
45 .
30ù, 45ù, 60ù
30ù, 45ù, 60ù ?
ABC ∠A= , BCÓ=1,
ACÓ="Ã1Û`+1Û`= .
DEF ∠D= , DEÓ=1,
EFÓ="Ã2Û`-1Û`= .
ABC ABÓ=1 , ∠A .
DEF DFÓ=2 , ∠D .
1 2

1 '2 45ù
.
sin`45ù= BCÓ ACÓ
2. 30ù, 45ù, 60ù 159
, 2 '3 60ù, 30ù
.
=;2!;
= '32
= 1 '3 = '33
.
tan`30ù_cos`60ù cos`45ùÖtan`45ù
1
'3 2
.
cos`30ù_tan`60ù
sin`45ù-tan`45ù
tan`30ùÖsin`60ù
sin`30ù+cos`45ù=;2!;+ '22 =1+'2 2
cos`30ù_tan`60ù= '32 _'3=;2#;
sin`45ù-tan`45ù=
=
.
.
?

.

. 100`m 30ù
sin`30ù=;2!; 50`m
.



4÷2
.
2
| 30ù, 45ù, 60ù
A B
=;2!; ;1Ó2;=;2!;
x=6, y=6'3
?

.

.

. 0 50 100
260
1 .
sin`30ù cos`45ù
tan`45ù sin`60ù
3 .
cos`30ù_tan`60ùÖsin`30ù
'3_tan`45ù-2_sin`60ù_tan`30ù
(sin`30ù+cos`30ù)_(cos`60ù-sin`60ù)
4 ABC x, y
.
∠A∠B=12, ∠A∠C=13
, cos`A+tan`B .
6 •
, x, y
.
A
sin`30ù_cos`60ù tan`60ùÖtan`30ù

A B
(Hipparchos, B.C. 190?~B.C. 125?)
.
( , “”)

O 1
AOB, COD .
∠AOD=40ù , .
AOB sin`40ù cos`40ù
.
.
40ù ?
1
.
.
∠AOD=aù , aù
.
sin`aù=ABÓ,
cos`aù=OBÓ,
tan`aù=CDÓ

= ABÓ = , cos`40ù= OBÓ OAÓ
= OBÓ =
= CDÓ =
16020 17010
180 D

40ù
.
16020 17010
‘=’ .
16020 17010
.
OBÓ 1 .
0ù
.
0ù 90ù ?
0ù 90ù

B 1
, CDÓ tan`90ù
.
, CDÓ 0 .
0ù .
tan`0ù=0
, ABÓ 1 OBÓ 0
.
.
B
1
1
C
A
O

.
cos`45ù_sin`90ù cos`0ùÖsin`30ù
2
3
∠A , "Ã(sin`A+sin`90ù)Û`+"Ã(sin`90ù-sin`A)Û` .

0ù !
90ù !
166 .
.
0ù, 1ù, 2ù, y, 89ù, 90ù
.
?

sin`23ù cos`59ù tan`80ù
, x .5
sin`xù=0.2419 cos`xù=0.4540 tan`xù=0.8693
1 ,
ABÓ, OBÓ .
6
.
‘=’ .
. 0 50 100
261
3 , .
sin`10ù
cos`24ù
tan`75ù
ABC
. x$

(sin`90ù+cos`45ù)_(cos`0ù-sin`45ù)
sin`60ùÖtan`30ù+tan`0ù_tan`45ù
5
1
.
sin`xù
sin`yù
cos`xù
cos`yù
tan`xù
sin`zù
168 .
,
.
0.4
0.6
A0.8
y
x1
1
B
0.2
0.4
0.6
A0.8
y
x
1
1
∠B ∠B , ,
.

1 1
A .
B, A x
C .
0.4
0.6
A
»ABC=45$ sin(»ABC)=0.707 cos(»ABC)=0.707 tan(»ABC)=1.0000.8
y
x1
1
B
0.2
0.4
0.6
»ABC=60$ sin(»ABC)=0.866 cos(»ABC)=0.500 tan(»ABC)=1.7320.8
y
x
1
∠B sin`B, cos`B, tan`B
.
, , .
& 169

62ù,
5`m.
ABC BCÓ ABÓ
.

.
BCÓ= `(m).
?
∠B=90ù ABC
∠A ACÓ
sin`A= BCÓ ACÓ
BCÓ=ACÓ_sin`A
cos`A= ABÓ ACÓ
ABÓ=ACÓ_cos`A
.
,
.
.

A B
A, B 38ù,
B A C
90ù. B C
100`m ,
A, B
.
1
ABC tan`38ù= ABÓ BCÓ
= ABÓ 100
ABÓ=100_tan`38ù
ABÓ=100_0.7813=78.13`(m)
78.13`m
1000`m ,

.
51ù.
1.7`m ,
. (, 0.1`m
.)
B, C
. A B
10`km, A C
8'3`km, A B, C
30ù ,
.
8÷3 km A
AC D .
ADB

BDC

2'¶13`km.
2'¶13`km
A, B
, C , ∠A=75ù,
∠B=45ù, ACÓ=30`m. ABÓ
.
3 C
A B
30 m
75$ 45$
20`% BCÓ 30`%
A'BC' . A'BC' ABC

.

A
|
H .
CAH sin`60ù= CHÓ ACÓ

ABC S
S=;2!;_6_2'3=6'3`(cmÛ`)
60$

.
?

ABCD .4
|

ABC
D AC H
DHC sin`30ù= DHÓ CDÓ


DAC
S =ABC+DAC
262
3
1.6`m, 50`m,
30ù ,
.
A, B, C
∠A=27ù, ∠B=90ù,
A, B
.
2 13ù
A, C . A, C
300`m , C A
m .
300 m
A B
.
30ù
AB'C'D'
4. 175

. (, 1`m .)

.
.




100`m B, C
42ù, 58ù.
A .
(, 1`m .)

176 .
.
ABC
01
.
BC, AB .
03 ∠C=90ù ABC ABÓ=3,
BCÓ=2 , ?
cos`A='53 tan`A=2'5 5
sin`B='53 cos`B=;3@;
tan`B=;2#;

05 ?
tan`45ù_cos`60ù='32
sin`30ù_cos`60ù=;4#;
tan`60ù-cos`30ù=-'36
sin`30ù_tan`60ù='32 sin`45ùÖcos`45ù+tan`30ù_cos`30ù
=3'3 2
8 cm
D
A
,
x+y .
A
B
4
1. 01, 02, 03, 04, 10, 11
2. 30ù, 45ù, 60ù 05, 06
3. 07, 08
4. 09, 12, 13
10
3 (-1, 3)
x, y
A, B
. ∠BAO=aù ,
sin`aù_cos`aù
.
12 700`m
A,
B 40ù, 31ù.
A, B . (,
tan`50ù=1.19, tan`59ù=1.66 .)
A B
. .
16 m
sin`xù+cos`xù .
D
E
F
A
‘(clinometer)’ .

.
1 ,
.
2 ,
.
.
.
,
.
60$