Bồi dưỡng học sinh giỏi lượng giác Tài liệu dùng cho học sinh chuyên Toán và học sinh giỏi

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5NG TRUNG HOC QUdC GIA CHU VA»
PHAN HUY
KHAI
(Chu
bien)
CHU"
XUAN DUNG - HOANG VAN PHU - CU PHUONG ANH
HOCSmUGWl
c
NHA
XUAT BAN DAI HOC
QUOC
GIA HA NOI
I Ha
NQll
" •
^-^VryTRLTONG
TRUNG HOC
QU6c
GIA CHU
VAN
AN
PHAN HUY KHAI
(Chu bien)
ChCr XUAN DUNG
-
HOANG VAN PHU
- CU
PHUONG

ANH
-a
Boitftmig
Hocsimcwi
llJdNG GlAC
Tdi
lieu
dCing
cho hoc
sinh
chuyfin
Todn
hoc
sinh
gidi.
'HI/
VI!:N
1
NHA
XUi^T
BAN BAI
HOC
QUOC
GIA HA
Nfil
Lcdnoiddu
Boi dUdng
hoc
sink gidi lM(/ng gidc
la

quyen sach
md dau
trong
bo
sach viet
cho
hoc
sinh chuyen Toan
va boi
diTdng
hoc
sinh gioi
ve mon
Toan
cua
nhom giao
vien trirdng Trung
hoc
Quoe
gia Chu Van An.
Quycn sach
gom 5
chiTdng:
- Chirring 1: Dang IhiJc liTdnggiae.
- ChU'ring
2:
Baft dang Ihu'e lu'ring giac.
•
- ChU'ring
3:

Ding thite lUdng giac trong
lam
giae.
- ChU'ring
4: Bat
dang thiJc trong tarn giac.
- Chirring
5: Vai
ifng diing
cua
li/dng giac trong
vi^c
giai
cdc b^i to^n sd cap.
Cuon sach
nay sc
cung
ca'p cho ban doc mot so
lifdng raft
Idn c^c bai
toan chon
loc
gom du the
loai.
Moi
phan
cua
cuon sach
c6 the xem
nhiT

mot
chuyen
de
rieng
trinh
bay
tron
ven mot van de mot
each hoan chinh (rieng phan phu'dng trinh lifdng
giac
se
du'dc
de cap
trong cuon khac).
Ngoai vi$c
he
thong
va
phan loai
bai tap,
chiing
toi
luon
chu
y ve
vice phat
trien
mot bai tap
theo
cac

hifdng tong quat
hoa, dac
bi§t
hoa va so
sanh binh luan
cac phiTdng phap giai khac nhau. Trong
mpl
chijfng mifc nha'l dinh chung
loi
manh
dan
di^a
ra
cac
giai phap
ve
khia canh
sif
pham
de
dung cuon sach
nay. Vi
le do
cuon sach
nay se
la
mot tai
lieu
tot de cac ban
giao vien lham khao

vao muc
dich
day
cho hoc
sinh
mot
each tifduy
mot bai
loan (chuf khong
ddn
ihuan giang
day cho
hoc sinh hicu
mot bai
loan
cu the).
Can nhan manh them rang trong cuon sach
nay
danh
mot
phan
de
trinh
bay
each
thic't
lap cac he
thuTc li^dng giac trong
lam
giac diTa

vao moi
quan
he
giiJa
cac yeu to'
cua
mot lam
giac
vdi cac
nghiem
cua mot
phu'dng trinh
bac
ba
Wring iJng. Theo
quan niem ciia chiing
loi day la mot
trong nhCfng phan
dac sac cua
cuon sach.
Mac
dij hel
siJc
c6'
gang trong
qua
trinh bien soan cuon sach, nhi/ng
vdi mot
dung lu'ring
qua

Irin
can
truyen tiii
cho ban doc nen
cuon sach khong
the
tranh khoi
nhffng khic'm khuyet.
l/t
>
Chiing
loi rat
mong nhan diTric
sif gop
y
cua ban doc de
cuon sach hoan ihien
hrin
nCa
trong
cac Ian tai ban sau.
Thi^
lij'
gop y xin
gufi
ve
Phan
Huy
Khiii
Tri/dng Trung

hoc
Quoc
gia Chu Van An.
10
-
pho
Thuy Khuc
-
quan
Tay Ho
-
Ha
NQI
Xin chan lhanh
cam
dn!
^
Cty
Timn
MTV
DVVH Kbang
VIft
OAINC
THU'C
LirCfNC
GIAC
§
1.
CAC
CONG THirC LUONQ GIAC

CO BAN
1. Cac
thuTc Irfcjfng giac cd ban
sin^a + cos'a = 1
cos a
cota
=
sma
1
+
tan^a
=
1
cos^
a
sin
a
tana
=
cos a
tana.cota = 1
2^
_
1
+
cot
a =
sin^
a
2.

Cac c6ng thrfc
CQng
cung:
sin(a + P)
=
sinacosP + sinPcosa
sin(a
-
P) = sinacosP
-
sinPcosa
cos(a + P) = cosacosP
-
sinasinP
cos(a
-
P) = cosacosP + sinasinP
tana + tanP
tan(a + P)
=
tan(a
-
p)
=
1
-
tan
a
tan p
tana

-
tanp
1
+ tan
a
tan
P
3.
Cac cong thitc nhan cung:
sin2a = 2sinacosa
cos2a = cos^a
-
sin'a
=
2cos^a
- ] =
1
-
2sin'^a
,
„
2tana
tan2a
= —
1-tan^a
sin3a = 3sina
-
4sin'a
cos3a = 4cos^a
-

3cosa
tan3a
=
3tana-lan'^a
l-3tan^a
4.
Cac cong thtfc blen tong thanh tich
.
„ ^ . a
+ p
a-p
sina + sinP = 2.sin
^cos
2
2
.
„ „ a+p . a-P
sina
-
sinP = 2 cos
^-sin
2
2 „ - a+p a-P
cosa + cosP
-
2cos
!-cos
2
2
^SlduSng

hfc
sinb gioi Lupag gldc
-
rhan
Huy
Kbal
^
„
sin(a
+ P)
tana
+
tanp
= -
cos a cos (3
sin(a - P)
tana
-
tanp
=
cota
+
cotp
=
cota
-
cotp
=
cosa cosp
sin(a+p)

sin a sin
P
sin(p-a)
sin a sin
p
5.
Cac
cong thitc bi6n tich thanh
ts'ng
sin(a +
P)
+ sin(a - p)
sinacosp
= ^—^ —
_ cos(a + P) + cos(a
-
P)
cosacosP
= — —
2
.
„
cos(a -
P)
- cos(a + P)
sinasinP
= — —.
2
6.
Gia tri

Itf^ng giac
cua cac goc
(cung) c6 lidn quan
dac bi^t
-
Hai goc doi
nhau:
sin(-a)
=
-sina
cos(-a)
=
cosa
tan(-a)
=
-tana
cot(-a)
=
-cota.
-
Hai goc
bu nhau
sin(7t
- a) = sina
cos(7t
- a) = -cosa
tan(7i
- a) = -tana
cot(7t
- a) =

-cota.
-
Hai goc
phu nhau
sin
cos
a
2
tan
71
—
a
u
a
7C
cot
a
= cosa
= sina
= cota
= tana.
- Hai
goc
hdn nhau
TI
.sin(7i
+ a) =
-sina
cos(7i
+ a) =

-cosa
tan(7t
+ a) =
tana
cot(7:
+ a) =
cota.
' i:Rfti
t>jj;if(
;>!
^i,•'
.
, .1.
,,,,:,<
§
2. DANG THirC LUONQ
QiAc
KH6NG DIEU
KI^N
Cdc
bai
toan trong
muc nay co
dang
sau
day: ChiJug minh
cac he
thiJc ii/cJng
giac khong
CO

kern theo dieu kien gi.
-^ui
PhiTctng phap giai
cac bai
toan
n^y
thuin tiiy dura
vao cac
phep bien
doi
lu'cJng giac.
^, ^
Bai
1.
1. ChiJug minh
r^ng
sin
18"
=
^^^zl.
4
2.
Chtyng minh sinl"
la
s6'
v6
ti.
Giai
1. Taco: sin54"
=

cos.36"
3 sin
18"
-
4sin^
18"
=
1
-
2sin^
1S"
o 4sin'
18"
-
2sin^
18"
-
3sin
18"
+
1
= 0
(sinl8"-I)(4sin^l8"
+
2sinl8"-1)
=
0.
(1)
Do 0"
<

18"
<
90" =>
0 <
sinl8"
<
1,
nen
(l)o4sin'l8"
+
2sinl8"-1
=0. (2)
Lai tuf
0 <
sinlS"
<
1,
nen
tir (2)
suy ra
.sinl8"
=
^^^^-^^ => dpcm.
4
2.
Ap
dung cong thuTc sin3a
=
3sina
-

4sin^a,
va gia
thiet phan chiJng sinl" 1^
so hffu
ti, khi do
theo tinh cha't ciia
cac
phep tinh
vdi
so' hi?u
ti suy ra
sin3",
sin9",
sin27", sinSl"
la
sohifuti.
Do sin8l"
=
cos9"
va
.sin
18"
=
2sin9"cos9",
nen suy ra
sin
18"
la so
hffu
ti. TiT

phan
1/ta
c6:
la so
hiJu ti, tiTc
la ^/izl
=
£ vdi p,q e N
4
4 q
=>
N/5
= 4-+
1,
vay
N/S
la so
hffu
ti.
4
Do
la
dieu
vo li vi
N/5
nhtf
da
biet
la so v6 ti. Vay gia
thiet phan chi?ng

la
sai,
neri sinl"
la
so'v6
ti
=> dpcm.
>. '
Nhdn xet:
1.
Bkng
each
suf dung cong thiJc:
cos3a
=
4cos''a
-
3cosa,
Vdi
phep
giai
tu'cfng
tU",
ta chuTng minh diTdc cosl" la so v6 ti.
2.
Bay
gicf
xet bai toan sau:
ChiJng minh rang cos20'^ la so v6 ti. Khi do earh
giai

hoan toan
khac
each
giai
tren. '
Ap
dung cong thufc:
cos3a
=
4cos^a
-
3cosa
^
8cos'20
-
6cos20
-1=0.
Thay a = 20", ta co: ^ = 4cos^20" - 3cos20"
^
8cos^20
-
6cos20
-1=0.
Vay
cos20" la nghiem cua phu'dng
trinh
Sx"*
- 6x - 1 = 0. (3)
Ta
CO

ket qua quen bie't sau trong li thuye't da thiJc.
Xet
phiTdng
trinh
da
thuTc:
anx" +
an.ix""'+

+
aix'+a„
= 0, (4)
Trong
do a; la so'nguycn vdi moi i = 0, 1, 2, ,n.
Goi
P la tap hdp tat ca cac \idc cua ao con Q la tap hdp tat ca cac \idc cUa ap.
Khi
do ne'u (4) co nghiem hiJu
ti,
thi nghiem do phai c6 dang
x
=
i^,
vdip e P, q e Q.
q
Ap
dung vao (3), ta thay moi nghiem hSu ti
ciia
(3) ne'u c6, thi chung deu
thuoc vao tap hcfp sau:

Q=
{±1/8;
± 1/4;
±1/2;
± 1}.
Tuy
nhien bang
each
thuf
trifc
tie'p ta thay moi phan
tii"
ciia
Q deu khong phai
la
nghiem
ciia
(3). Noi
each
khac
moi nghiem
ciia
(3) deu la so' v6 ti. Vi
eos20"
la mot nghiem
ciia
(3) nen
eos20"la
so v6
ti.

Do la dpem.
Bai
2. ChiJng
minh:
4cos36" + cot7"3()' + ^ + + S + S +
Giai
Theobai 1, ta
CO
sinl8"=
=>cos36"=l-2sin^l8"=l-
2-^"^"'^'
16
•4cos36" = 4- =S + \. (1)
Ap dung cong thiJc: Neu 0 < a < 90", Ihi coty = cola +
Vl
+ col^a, ta c6:
cot7"30' =
col
15"+
Vl + col^ 15" .
(2)
Vi
col
15
= .
Vl-cos30"
-
^773
=
^^"^^ = 2 + \/3, nen thay vao (2) va

co: cos7"30' = 2 + 73 +
Vl+4
+
3
+ 4V3 = 2 + ^3 + 78 + 4^3
=
2+
V3+^(V2+
^)'
=
^/2+73
+
^/4+76.
(3)
Tir(l),
(3) suy ra:
4cos36" + cot7"30' = V[
+
V2+>/3 + 74 +
V5
+
76 => dpem.
Chu
y: De thay:
,
2 «
r
'i cos
cola + V1 + cot a =
——

cos a 1
-
+
sma
sin
a
1
+
cos a
2
cos'
sma
a a
2
sm cos -
Bai
3. Chu'ng minh
rting
tan-10" + tan-50" + tan-70" = 9
Ap
dung cong thu'c
1
+
tan'a
=
dUctng
vdi dang thu'c sau:
Giai
2
h->.,y'

cos a
thi
dang thu'c can chu'ng minh tu'dng
(1)
eos^lO" ' cos-50" ' eos^ 70
^
cos^ 50" cos' 70" + cos^ lOcos^ 70" + cos^ lOcos^ 50 _
^
cos2l0"cos-5)"cos2 70" ~ •
Goi
A va B lu'dng vtng la
tii'
so' va mau so
ciia
vc'
trai
ciia
(1), la co:
A
= cos'(60" - 10")cos'(60" + lO") + eos'10"[cos'(60" + lO") + cos'(60" - lO")]
=
[(cos60eoslO + sin60sinl0)(cos60eosl0- sin60sinlO)]^ ',
+
cos-10"[(cos60cosl0- sin60sinl0)' + (cos60cosl0 + sin60sinlO)^]
=
(cos'60eos-10
- sin^60sin^l0)- + co,s^lO"(2cosYiO"eos'lO" + 2sin'60sin^l0")
n^-'
(
1 T

-cos-10 sin^lO
4 4
•co.sMo"
-cos-l()
+ -sin^lO
cos^lO
+
cos' lO"
cos^lO"
2
_9_
l6
(2)
BOI auOng
aye
ainti glol Lupng gUc - rhmn Huy KluU
Mat khac tiTcfng tiT nhtfbien doi A, ta cung c6:
B
=
cos^lO'W(60"-
10)cos^(60+
10)
=
cos^l0"(cos^60cos^l0-
sin^60sin^l0)^
( 3^
2 /-
=
COSMIC
COSMIC

—
=
I 4^
V
cos''10 cos
10 =
—(4cos''lO"-3coslO")^
16^ /
1
=
—(cos30*')^
(do
cos3a
=
4cos'a
-
3cosa)
16
1
3
16 4 64
(3)
9
TCf(l)(2)(3)
suy ra:
VT(1)=
64
12
Vay (1) dung => dpcm
Nh^n xet:

Xet cac
each
giai khac thi du tren sau day:
^'u^ k A .w. , -> 3tana-lanV ^ ,
Lack
2: Ap dung
cong
thi/c: tan3a = , ta co:
l-3tan''a
tan"
3a =
9
tan^
a - 6
tan"*
a + tan^ a
l-6tan'a + 9tan^a
(4)
Vi
tanl30 = tan^l50 = tan^210 = ^, nen ti!f (4) suy ra khi thay a = 10*', ta c6:
1
_9tan^l0"-6tan''l0" + tan'^10"
3~ l-6tan^l0 + 9tan^l0"
hay Stan'-lO" - 27lan''lO" + 33tan^lO" -1=0. (5) '
Tif (5) suy ra tan^lO" la nghiem cua phi/dng
trinh:
3x'- 27x^ + 33x - 1=0. (6) ;
Tu'dng tiT tan^'>0, tan'70 cfing la nghiem cua (6)
Mat khac de tha'y tan-lO", tanl^O", tan^70 la ba so khac nhau
(cu the ta CO tan^lO < tanl^O < tan^70), nen tan'10, lan-50, tan'70 la ba

nghiem khac nhau cua (6).
Theo
dinh H Viet vdi phU'rtng
trinh
bac ba, ta c6:
/ 27^
tan^l()" + tanl'50 + tan^70 =
= 9 => dpcm.
Chu y: Xin
nhSc
lai vdi phu'dng
trinh
bac ba:
ax' +bx^ + cx + d = 0 (a ;^ 0)
ta
CO
dinh li Vict sau:
Ncu
X|,
X2,
Xi la ba
nghiem
ciia
no, thl ta c6:
b
X| + X^ + Xi = —
a
C A :
X1X2 + X2X, + X,X| =. —
a .

d
X|X2X,=
.
a
Cdch
3: Dang
thuTc
csln
chifng
minh tU'rtng dU'dng vdi
dang
thiJe
sau: .
tan-10"
+ tan'5()" + tan'70 = 3tan'60"
o (lan-60 -
tan^
10) + (lan'60 - tan^50) + (tan'60 - tan-70") = 0 (7)
Ap dung
cong
thiJc: •
2
, 2n
sin(a-P)sin(a
+ P)
tan
a - tan P = j- ——,
cos a
cos'P
va chu y

rang
cos-6()"
= —, nen dc
thay
4
4sin50"sin70" 4sin lO" sin
1
lO" 4sin(-10")sin 130" ^
(7) o + —n + ^ -7, = 0
cos-10
cos-50" cos70"
o 4sin.'50"sin70"cos50"cos7()" + 4sinl0"sinl
10"cos-10cos^70"
'
- 4sinlO'sin
13()"cos'l()cosl*i0
= 0
e>
sinlOO'sinl4()"cos50cos70
+ sin20sinl40"cosl0cos70
-
sin20sinlOOcosl()cos5()
= 0
(do
sinl
10 = sin 70; sin 130 = sin50 va sin2a =
2sinacosa)
«>
cosl0cos70cos-50
+

coslOcos5()cos^70
-
cos50cos7()cos-10
= 0 (8)
(do sinlOO =
coslO;
sinl40 =
cos5();
sin20 =
cos70;
sinl40 =
cos50; )
Do
cosl()cos70cos50
^ 0, nen
(8)
ocos50
+
cos70-cos
10 = 0
o
2cos60coslO
-
coslO
= 0
ocoslO-cos
10 = 0. (9)
Vi
(9) diing nen (7) diing => dpcm.
Cdch

4: Ap dung
cong
IhiJc:
tan'a
= 1 - thi dang thiJc can
chuTng
minh
tan
2a
• 'ft.
tiTdng diTctng vc'Ji dang thiJc sau:
tan
10 ^ tan 50 ^ tan 70 ^
Ian20^unl00^tanl40~~
^ t"n50 tan70 tan 10 ,
Rd rang (10) <=> + = 3
tan
80
tan
40 tan 20
<=> Ian50tan4()tan2()
+
tan7()tan8()tan2()
-
tanl()ian8()tan40
=
3tan8()tan40tan20
<=> lan20
+
tan80

-
tan
40 =
3tan2()lan4()tan80
(do Ian50lan40
=
tan70tan2()
=
tanl0tan8()
=1) '
<=>
tan2()(
1
-
tan40tan8())
+
tan8()(l
-
tan20lan40)
- tan40(l+tan20tan80)
= (). (11) i
i
. ^ , . r.
I'ln
oc
+
tan
p ^
Ap dung cong thiTc: tan(a
+ P) = , nen

1-tan
a
tan
P
tan80
+tan40 Ian4'0
+
tan20
(11) CO tan20
+
tan 80 tan 40(1
+
tan 20
tan
80)
^ 0
tan
120
tan
60
c:> —^-—
tan
4()(tan 80
-
tan 20)
=
tan 40(1
+
tan 80
tan

20)
tan
60
<=>
—5—(ian8()-tan2())
=
1
+
tan80tan20
tan
60
tan
80-tan 20
<=>
=
tan
60
1
+tan 80
tan
20
CO
tan60
=
lan60.
(12)
TiT
(12)
suy
ra (11)

dung
=>
dpcm.
Binh luan:
Trong
4
each giai,
c6 3
each su" dung
thuan
tiiy
bien ddi lu'dng
giac,
eon
1
each
ket
http
vciii
cac
kicn
thiJe
vc
tinh
chat nghiem ciia
mot
phu'dng
trinh
dai so' (cu
the

sii'
dung
dinh
li
Viet trong thi du
nay).
Bai
4. Chu'ng
minh
rang
—^— + —+ —^— = 4.
71
371 571
COS cos COS
7
7 7
Giai
7t
371 571 ^ , , , ,
Vi
—; —; —
nam trong
so cac
nghiem cua phu'cfng
trinh:
3x
+ 4x =
(2k
+
1)71, vdi

k G Z
TiT
do suy ra xet
phu'cfng
trinh
sau
day:
eos3x
-
-cos4x
hay
eos3x
+
eos4x
=
0.
(1)
Ta
eo:
cos3x
=
4eos
x -
3cosx
cos4x
=
2cos"2x
- 1
=
2(2eos^x

- 1)' -
1
=
8cos^x
-
8eos'x
+ 1.
Vi
the
(1) CO
4COS-X
-
3cosx
+
Scos'x
-
8eos^x
+1=0
CO
8cos'*x
+
4eos\ 8eos\
3cosx
+1=0
CO
(cosx
+
l)(8eos''x
-
4eos'x

-
4cosx
+ 1) = 0 (2)
.
f
71
37: 57:
1
,, , ,^ ^
Khi
X 6
<^
—; —; — \i
cosx
+
1
0,
nen suy
ra
l 7 7 7 I
),n«
371
571
7
8x'
-
4x'
- 4x +
1
= 0

n
7
X|
+X2 +X3 =
X|X2
+
X2X3
+ X3X1 =
Pos—
COS
—
,
cos—
la ba
nghiem phan biet ciia phu'cfng
trinh:
7
7 7
Dat
X|
=
cos-y; X2
= eos^;
X3
-
cos-^,
ta c6
theo
dinh
li

Viet:
• *
/
4^
1
2
-4
_
"~~~2
,s.
-
8
(4)
X1X2X3
-
-1
8
Ta
eo:
—^-—
+ —^
1
71
37: 57:
cos
- cos cos
111
X|X2
+X2X3
+X3X1

XI
X 2 X 3
Thay
(4) vao
(5)
va c6:
1
1 1
— +
—
+
—
X|
X2 X3
(5)
_ 1
+
=
——
= 4
=0 dpcm.
3n
57: 1
cos
cos COS
7
7 7
8
Nhdn
xet: Xet

each giai khac bang each
thuan
luy bien ddi lu'dng
giac
nhu'sau:
Ta
co:
1
1 1
+
^
+
-
7: 371 57:
cos
cos COS
7
7 7
7: 37: 37: 57: . 57: 71
COS COS +COS COS +COS -cos-
_7__7____7___7L___^-__
7t
37: 57:
cos
COS cos
7.
7 7
(6)
71
37:

Dat
Si =
cos—
+
cos—
+ cos
7
7
55
1
71
37: 37: 57: 57: 7:
S2 -
COS—COS
—+
COS—cos—+
cos
—cos
—
71
37: 5TC
SI
= cos —COS—COS—,
111
11
IS,
thi
(6) CO + r- +
n
37: 57: S,

COS COS COS
7
7 7
(7)
Do
sin —
;^ 0, nen ta
c6:
7
2S,
sin
—
= 2sin—cos
—
+
2sin
—cos—- + 2sin—cos
5n
1
.In
. 4n .lit .6% .An
= sin
— +
sin sin
— +
sin sin
—
11111
.
6TZ . n

- sin—=:sin
—.
7
7
Do sin
—5^0,
=>S|
=
—
7
2
(8)
Ta
c6:
7t
371 571^
COS
—
+
COS
+
COS
—
7
7 7
2
/
2
"
COS

— +
COS
7
7
2
37C 2
+
COS
—
1
+
COS
27t
671
IOTC
1
+
COS—
1 + cos -~
2
' 4
3
+
2
l()7:~
27t 671
COS—
+
COS
— + cos

7
7 7
(9)
rr
• 27t 57t 67t 7t
IOTI
37t . .
Ta
CO
cos— = -cos—; cos— = -cos—; cos^^ = -cos—, nen
tif
(9), ta co:
S2-^Sf-^l3-S,|. (10)
2
4
Thay(8)vao(l())vac6:
82=
+ - = (II)
^
8 4 8 2
Ta
lai c6:
371
57t 1
Si
= cos —cos—cos^^— = —cos —
7
7 7
1
7C 1

=
COS"
—+ —
2
7 4
37r 7t
cos +
COS
—
7
7
871
27t
cos +
COS
—
7
7
ir,
27r
1
+ cos
—
1
+
—
4
37t
7t
cos +

COS
—
7
7
-
_1 1
~
4 4
57t
371 7t
COS
+
COS
+
COS
—
7
7 7
4
4 '
(12)
Thay(ll)(12)
vao (7) va c6:
1
en '
1
1
+
-
71 37t

COS
-
COS COS
7
7 7
571
1
(Ipcm.
8
Cty
Train
nrV D
VVH
lUiang
Vift
Bai
5.
Chufng minh rang:
4
7t 4 37t 4 571
cos +
COS
+
COS
14 14 14 ^3
.
2 n 2 37t 2 571
4COS
- COS -cos
14 14 14

Giai
Taco:
-,f
cos7x = cos6xcosx -
sin6xsinx
=
cosx(4cos^2x - 3cos2x) - 2sinxsin3xcos3x
=
cosx[4(2cos\
l)"*
- 3(2cos^x - 1)] - 2sinx(3sinx - 4sin\)(4cos^x - 3cos)
=
cosx|4(8cos\ 12cos''x + 6cos'x -
I)
- 6cos^x + 3J
-
2sin'x(3 - 4sin^x)(4cos\ 3cosx)l
=
eosx(64cos''x - 112cos''x + 56cos-x - 7). (1)
Trong(l)thay
x-—, va do cos—= cos—= 0, cos—^ 0, nen
tijf
(1) suy ra
14 14 2 14 •
phircfng
trinh:
64x' - 112x' + 56x - 7 = 0 (2)
nhan
x, =cos'— lii
nghicm. Tifcfng

M
Xj
=:cos^—, x^ =cos^—
cung
I^
14 • 14 • 14
nghicm
cua (1). R6
rang
X|, X:,
X3
la ba
nghicm
khac
nhau
cua (2), nen theo
djnh
li
Viet,
ta c6:
112
XI
+ X2 + x, =
64
56
X1X2
4-
X2X3
+ X3X1 —
X1X2X3

=
64
7
64
4
7t 4 371 4 571
COS
+
COS
+
COS
2 2 2
Taco:
14 14
14^Xi+X2+X3
.
2 ^ 2 371 2 571 4XIX',XT
4cOS
-
COS
COS
^'^l'^2'^3
14 14 14
_
(X|
+X2
+X3)^
-2(X|X2
+X2X3
+X3X1

4X1X2X3
112
64
64
2
56 56
'64_64t ,
64 16
•
dpcm.