3/truong hop nay trong tam giac khong vuong tanA+tanB+tanC=tanA.tanB.tanC.
Trang 1¦ng dông cña lîng gi¸c.
D¹ng 1:Bµi to¸n tam gi¸c
VD1:cmr:mäi tam gi¸c ta lu«ncã:
1/SinA+sinB+sinC=4cos
2 cos 2 cos 2
C B A
Giai:
2sin
2
cos 2
(cos 2 sin 2 2 cos 2 sin 2 2 cos
.
2
B A B
A B A B
A B A B
A B
=4sin
2 sin 2 sin 2 sin 4 cos
cos
.
.
2 2 2 2 2 2 2
C B A B
A+ A−B+A+B A−B+A+B =
2/cosA²+cosB²+cosC²=1-2cosA.cosB.cosC
Giai:
2
2 cos 1 2
2
cos
1
B A B
2
2 cos 2
cos
B A B
=1+cos(A+B)([cos(A−B) + cos(A+B)]
=1-2.cosA.cosB.cosC=vp
3/ sin2A+sin2B+sin2C=4sinA.sinB.sinC
Vi:sin2C=sin2(π-(A+B))=sin(2π-2(A+B))=-sin2(A+B)
⇔
VT:2sin(A+B)(cos(A-B)-cos(A+B))=2sinC.(-2)sinA.sin(-B)=4sinAsinBsinC.=vp
3/(truong hop nay trong tam giac khong vuong)
tanA+tanB+tanC=tanA.tanB.tanC
−
+
B A
B A
tan tan 1
tan tan
dpcm
2
tan 2
tan 2
tan 2
tan 2
tan
2 B + B C + A C =
A
Giai:
taco:tan( ) tan(
2
2 +B =
A
−
+
=
=
B A
B A
C
C
tan tan 1
2
tan 2 tan
2 tan
1 )
5/
Trang 2) 2 cot 2 cot 2
cot 2
tan 2
tan 2
(tan 2
1 sin
1 sin
1
sin
C B
ta di CM:cot
2 cot 2 cot 2
cot 2
cot 2
cot 2
C B A C
B
) 2
cot 2
cot 2
cot 2
tan 2
tan 2
(tan
2
1
⇔
2
cot 2
(tan 2
1 ) 2
cot 2
(tan 2
1 ) 2
cot
2
(tan
2
sin
1 sin
1 sin
1 2 cos 2 sin
2
2 1 2 cos 2 sin
2 2
1 2 cos
.
2
sin
2
.
2
1
2 2 cos 2 cos
2 sin
2 cos 2 cos
2 sin
2
cos
.
2
cos
2
sin
= +
+
B A
C C
A
B C
B
A
6/
ta co:
2 cos 2 cos 2 cos
2 cos 2
sin ) 2 cos 2 cos 2 cos
sin sin
( 2 1 2 cos 2 cos
2 sin
2 cos
.
2
cos
2 sin
C B A
B A B A C
B A
B A
C A
B C
B
=
+
= +
=
2 tan 2 tan 1 2
cos 2
cos
2 sin 2
sin 2
cos
.
2
B A
B A B
A
+
= +
2 tan 2 tan 1 2 cos 2 cos
2
B A
C
dpcm
BAI TAP VAN DUNG:
1/ CM trong moi tam giac ta luon co :
.
sin
2
tan 2
tan 2
tan 2
tan 2
tan 2
tan 2
sin 2
sin 2
sin 2
cos 2
cos 2
sin 2
cos 2
cos 2
sin 2 cos
.
2
cos
.
2
C A C
B B
A C
B A B
A C C
A B C
B
A
+ +
+
= +
+
2/ cho tam giac ABC cm:
Trang 3^
2
^ 2
^ sin
)
sin(
c
b a C
B
A− = −
3/ Trong tam giac ABC biet tan
2
tan 2
B A
2
1 : 3
1
b a c
4/ CM trong moi tam giac ta co:
.cotA +cotB+cotC=
S
c b a
4
2
^ 2
^ 2
^ + +
5/ cm rang trong moi tam giac bat ki ta co :
tan
a a
a a
a
h r
h h
r h C
B
+
=
−
=
2
2 2
tan
2
6/ cho tam giac ABC cm:
2 cot 2 cot
C ab
B ac
7/ cmr:voi moi tam giac ABC ta luon co:
R
r
cos cos
cos + +
=
8/ CMR voi moi tam giac ta luon co:
a.cotA+b.cotB+c.cotC=2(r+R)
9/ cmr voi moi tam giac ABC ta luon co:
cosAcosBcosC≤
8 1 10/ cmr voi tam giac ABC bat ki:1<cosA+cosB+cosC≤3/2
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