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7/25/2019 Trigonometrijski obrasci
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. 2 ( 18)
.. .............................................. ...................................
!. "#$#% ". 2& '##
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!. "#$#% ". 1
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.
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. ? ( 28)
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KL KD
KL KD
KL KD
KL KD
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. A ( ??)
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. & ( ?&)
T'# !0# ; 7'##
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ijklmjmno GpioE9pempq lVFeJj p eGRQenjmnj pU WijJelVFpq EeeiGpmjQj
1 2 4 ? @ A &
QbQf
GY_Yc.aPcbcbg
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d^YcY
GY_Yc.aPcbcbgYb
YaY=YbYa
X+Y
Xb
Xa
X=XbX
a
XY
log (X+Y)log (XY)
log tg (14 +ba)1
4+a
b
logY
logX
log tg ab
ab
logsinab
logcosab
log dd
7/25/2019 Trigonometrijski obrasci
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. 8 ( A2)
!. "#$#% ". 2@ '##
r6
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. 11 ( &@)
". 18W 7'## EF
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. 12 ( &8)
/7-# !0# ; 7'##
+ '#,- + 7'# + 7'# + 7'# + 7'#
". 1 ". 18 ". 2& "#7%# VWR ". 1
7/25/2019 Trigonometrijski obrasci
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&' &
() . = *&
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,, ,0m m m
l
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. 14 ( 1C?)
. ........
r
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. 1? ( 1C@)
!. "#$#% ". 22 '##
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G G M\H M\ 37 /5
> tGMu : M v ;
1 2 4 ? @ A & 8 B 1C 11
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(z^;z{)2
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rD '
3-;5##
30#0##5 d
Gz\ +
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+7- '#,- # #5+
oa
Dyn=od
n+au
Dxn=ad
nou
o=
( y zyp )d
a=(xzx p )d
Dyn=odnDx
n=ad
n
D=
(y
z
yp )
2+
(x
z
xp )
2
D= (yzyp) o+(xzxp ) aD=Dd
ya=y
n1+Dyn
xa=xn1+Dxn
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7/25/2019 Trigonometrijski obrasci
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. 1@ ( 1C@)
!. "#$#% ". 22j '##
d
G G 37 /5 b dv ; a d
1 2 4 ? @ A & 8 B 1C 11
ab
cN ZhaP Z
rD '
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Gz\
Gy\ +
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M\H M\+#,#D#s3#%7$'
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35'.7'#'#-
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35'.7'#'#-
3-%+#
+7- '#,- # #5+
M\
z\
y\
Dyn=od
n+au
Dxn=ad
nou
o=
(yzyp )d
a=(xzxp )d
Dyn=odn y n=yn1+DynDx
n=ad
nx
n=x
n1+Dxn
(x zx p)
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. 1A ( 1C
!. "#$#% ". 24 '##
: :
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Qf
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b| P]\ \
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bf
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; zb)2 M
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b)2 M
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f
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bf
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b
f f b
f b
= +
=
2 2
cos =d2+da
2db2
2dda
=B
2dda
cos=d2+d
b2d
a2
2dd b=
B
2ddb
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@)
b
f
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. 1& ( 1
!. "#$#% ". 24j '##
QbL
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M M
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P]\ b P]\ f
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bQ Gy
f: M
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bv Gz
b: z
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f z
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b: y
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f
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b) \
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b)
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P]\ \bf \
bQ
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b
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bM
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f
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b) haP \
bQ haP \
fQ
Gzb
Gzf
cN2 b2 cN2 f2 Gyb Gy f
cN b2
cN f2
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f... P ; M
b
P ; M fb
2f
2
zb
zb
(zf; z
b)
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b)
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bz
f; z
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bf
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P]\ \bf
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P]\ \bQ P]\ \
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b
(P ; Mb) (P ; M
b) M
bM
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f
1(P ; Mf) 1(P ; Mf) haP \bQ haP \fQGz
bGz
f
cN2 b2
cN2 b2
Gyb
Gyf
cN b2
cN b2
Mb... M
f... P ; M
b
P ; Mf
b2
b2
z: zQ2; zQ1:
Q:
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y: y
Q2; y
Q1:
fQ
fb
Qf
M
bf Mf
bQ Q
M
b
Qb
tg ab=
ybya
xbx
a
d=ybyasina
b =
xbxacos a
b
tg
2=
( sd )(sd a )s ( sd b )
tg
2=
( sd )( sd b )s ( sd a)
s=d+d a+d b2
aT= a
b+
aT= a
b180
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@)
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#,-# -+# 7 3}+ L
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ya
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2
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d : G ; Md j1 ~1
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av M y :
S:
u '"# 3# "3 +3#- v
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3#-/7$'
30
ax + by = d
#'+#
+1.26
a=Dx
o
do
b=Dyo
do
Dyn=yy
n
Dxn=xxn
Dn
mo=[ pvv ]n2 ==
m100=
m
p100==
pv
p ap bp
B
1
!1
B2
!1
$y=m
o
B2#= =, $x=$y
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!#; * 57'- 7#
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7# / $# - /5 - 37 357'-# # b / 2. !D
7'; #3!. 57. '#,-# EF v EG $'# $'# 73#
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$! EF ; EG /
] w H I H I H I H I H I
1 2 4 ? @ A & 8 B
MH v GG
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'#%# 0# 5$ '# 3#x# hcN U
7# #5#% R:-7# G:. U cN b7'. %3#D 77- # haP U
7-%# 3'. 7 37# 7 /5# b P]\ b
#'! H I K hK hK H I >1 2 4 ? @ A &
OK
$#
#
7#
!#-
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. 2C ( 14&)
7'##L P]\ U $'#/. 77-# #$/-# #3!7-#
haP b 37'+#D Gq : G hacNU 57# / G:RP]\U :G cN b !3"#
G:RhaPb :R P]\b !
> G:R ; [ > Gq8 B 1C 11 12 14 1? 1@
Gq : Gq v ] ; OK ] ; OK
[ : 2P P]\2 b2
qQ:q
RvGq
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r p r pp /0#+ 7'# /0#+ #'x-.
'#D '#D $3$#D #'x- 3"# -#5#D#.
# -# # -# #,#5# 37,#-# 30#
3 #'% 30# #'% 30# 577- #$/-# (7# $3$#D!).
!3" (- + !s #3%s
! ! 3#'!s 5!7- /- 7'#/).
1 2 4 ? @ A & 8 B 1C12
*
$
3
45
657-8
9:;$ B> . C C (0
? 2 3
-
b
c
? ?
sin
sin cos
sin cos
b cos
c cos
c
a
b
@
2
*
c
a
b
c
a
b
c
a
b
c
a
b
m=a
sin a
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. 22 ( 14B)
. .............
a
b
c
b cos
+ c cos = a
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. 24 ( 14B)
!. "#$#% ". 1? 'b#
#,#D '/# $ 35 7'# $#5# /#
H I f ...
...
v f;h ; ...
; fvh v ...
h ...
...
... b
H I f ...
...
v f;h ; ...
; fvh v ...
h ...
... ... b
H I f ...
...
v f;h ; ...
; fvh v ...
h ...
...
... b
H I f ...
...
v f;h ; ...
; fvh v ...
h ...
...
... b
3 "+ #,;#D# 7-%#'/#
'# /;5 7$'L
12 P]\
12(v) P]\
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cN12(;)
12 P]\
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hcN 12
cN12(;)
12 P]\
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hcN 12
cN12(;)
12 P]\
12(v) P]\
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hcN 12
cN1
2(;)
f h
b
1
2 (+g )=
1
2p
1
2
tg1
2(g )=b&
b+& &tg
1
2
=
1
2 (+g )+1
2 (g )
g=1
2(+g )
1
2(g )
a=bsin
sin
=&sin
sing
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'#x 5/!# 3 3
'#%# $#
1 2 4 ? @ A & 837'+#D
!'!#
'#D# /'5
5.7'#'#-
77-##$/-#
#3!7-#57#5$
5.7'#'#-
#3!7-#57#
5.7'#'#-
5.7'#'#-
"#
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/ 0 # + 3 " #
57'-
'#%# p pp 2 (q. J.) : 57. #
$# /5 37 : E. F. v E. G.
#'! ,#7 '#,-# E. F. E. G. H I H I H I > H
1 2 4 ? @ A &
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#$/-#$!}
#+5#+!#D
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2(q.
J.)
7/25/2019 Trigonometrijski obrasci
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. 2A ( 1?4)
!. "#$#% ". 28 '##
rD #3 3 j 3 ~ rD #$#3 ~ 3 j
3#% 7 7. # 7. #
3 j $' / $. 3#x# / $. 3#x# ] ; OK
3 ~ 0# / / K
G + 5/#-#
K > H I > K > H I > K >
1 2 4 ? @ A & 8 B
:(q~j
);
M P]\ j~
qHj~
M P]\ ~j
qH~j
;(qj~
)
cN j~
] ; OK
cN ~j (q|3$5.)
j~
/ Uj~
(qj~
) j~
/ Uj~
(q~j
)
hcN Uj~
haP Uj~
hcN U~j
haP U~j
qHj~
qH~j
5'.7'#'#-
5'.7'
#'#-
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"#$#% II
+5 77- #$/- #5- #5; #3!;
'#,#-# 5 x 57. 7-
!D !D 3# #$/- 57
3 qK
D D > D D D
1 2 3 ? @ A & 8 B 1C 11
( qb) ( q
f) ( q )
3#%7$
'
5'.7'#'#-
5'.7'#'#-
5'.7'#'#-
#$/-357'
-
!D#=(qf);(qb)
r#-7!#/
3$5x#$/-
5'.7'#'#-
5'.7'#
'#-
37'+#D#
v'=
%
[ d ] d '
D%'=
(%D')+v'
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. 2& ( 1?A)
'##
!3"#
12
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. 28 ( 1?A)
5/!#7- "#$#% " . ? ' .
$#5#5#D #3!7- 57 ,5 # L
+5 7/' 77- "/0 "#
# 57 #$/- #7/' 03"5 57 { { .
5/#D!
> >
1 2 4 ? @ A & 8 B 1C 11 12 14
$#,#5#D #3!7- 57 ,5 # L
1 2 4 ? @ A & 8 B 1C 11 12 14
{q { . 2
i\ \ O\7/'57
$'7
5',
7'#'#-
5',
7'#'#-
5',
7'#'#-
+
5/#-#
3#-/7
$'
577-#$/-
'0
D% #=%,%
0
m0=
[ p
(
]n1= $%=
m
0
[ p ]
m0=
[ p ( ]
n1=
$%=
m0
[ p ]
7/25/2019 Trigonometrijski obrasci
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. . " " ##
'## L
" /5# /## 37'#D#
7'## MtMu
1B EK > H I > hK > hK > hK
z y Ki
M+5/
#-#
#'
+#
+5$
/./
5#,'#
$#5S#'#,-# 5/#-#
'##"#7%#
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. 2B ( 1?&)
!3"#
7/25/2019 Trigonometrijski obrasci
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. 4C ( 1?8)
S'#
..
'##
# -+! 7'##!# 7 #/#$ #/ 3#% "#7%!#
2& 1 18 18
6. .
$#-# "+'#,-
#7%
+-'
#-'
-+ !3"#
$ "
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/! -%;
+ 5$
'#,- /5 /5 0#
M\
!'##
H I H I v ; v ;1 2 4 ? @ A & 8 B 1C 11
Q\
+5/#-#s
33-7
M\|P]\
\
M\|haP
\
3#%
7$'L
#'+#7'#
5'.
7'#'#-
P]\
\
M\
ha
P\
E
F;8 E
GF E
+F;8
+ GF
90 +
+ GF
F;8
180 +
F;8
GF
270 +
GF
+F;8
..
..
..
n
n=
n1+n180o
Dy#
n=d nsin n Dx#n=d ncos n
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7/25/2019 Trigonometrijski obrasci
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. 42 ( 1@2)
!'+7- "#$#% ". 21 '##
L
1. L 4. .
0#
0 W
1(M)
/
H HH > > >1 2 4 ? @ A & 8 B 1C 11 12 14 1? 1@ 1A
t{u:
2.
3#' 3#' '#,#-# 3#' 0#
"5 -3. #/-# {1(M)
{ { a a / hK hK hK hK> > > hK > hK > hK > hK
1& 18 1B 2C 21 22 24 2? 2@ 2A 2& 28 2B 4C
a a
:
H^s
as
as
^{ H
:
^;
^H {
2 {
2
z2
y2 {
z2 {
y2
121
= t{ H
] t{
] t{
2]
a:
^ :
{z
H {y
H
H:{vtzu H:{vtyu { z { y
tHzu tHyu H:avz H:avy 12
,'#
'#,-#5/#-#
+5/#-#
+/!
5$/5#\ -%
/5 #5-#
f'=H^;
#'+
(M)
gK
fy'=Y
'-Y
o
fx'=X
'-X
o
fy=
Y
'-Y
fx=
X'-
X
p=1
n
do=[p ]
[p]
m=[ p
2]
)1= =mDy=
[ py2]
)1= =$y= mDy[ p ]==
m=m
[ p ]=
=mDx= [ px2 ])1 ==$x= mDx[ p ]==
d=[p # y]
[p ]* dx=
[p # x][p ]