HÀM SỐ LƯỢNG GIÁC
[Pages:15]Truy cp website: ti t?i liu thi min ph?
H?M S LNG GI?C
A. T?M TT L? THUYT
I. C?c c?ng thc lng gi?c
1. C?c hng ng thc:
* sin2 + cos2 = 1 vi mi
* tan .cot = 1
vi mi k 2
*
1+
tan2
=
1 cos2
vi mi k2
*
1+
cot2
=
1 sin2
vi mi k
2. H thc c?c cung c bit
a.Hai cung i nhau: v? -
cos(-) = cos
sin(-) = -sin
tan(-) = - tan
cot(-) = -cot
b. Hai cung ph nhau: v? - 2
cos( - ) = sin 2
tan( - ) = cot 2
c. Hai cung b? nhau: v? - sin( - ) = sin
sin( - ) = cos 2
cot( - ) = tan 2
cos( - ) = - cos
tan( - ) = - tan
cot( - ) = -cot
d) Hai cung hn k?m nhau : v? + sin( + ) = -sin
cos( + ) = - cos
tan( + ) = tan
cot( + ) = cot
3. C?c c?ng thc lng gi?c a. C?ng thc cng cos(a b) = cosa.cos b sina.sin b
sin(a b) = sina.cos b cosa.sin b
tan(a b) = tan a tan b 1 tan a.tan b
b) C?ng thc nh?n
sin 2a = 2 sina cosa
cos 2a = cos2 a - sin2 a = 1 - 2sin2 a = 2cos2 a - 1
sin 3a = 3sina - 4sin3 a
cos3a = 4cos3 a - 3cosa
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c. C?ng thc h bc
sin2 a = 1 - cos 2a 2
cos2 a = 1 + cos 2a 2
tan2 a = 1 - cos 2a 1 + cos 2a
d. C?ng thc bin i t?ch th?nh tng
cosa.cos b = 1[cos(a - b) + cos(a + b)] 2
sina.sin b = 1 [cos(a - b) - cos(a + b)] 2
sina.cos b = 1[sin(a - b) + sin(a + b)] . 2
e. C?ng thc bin i tng th?nh t?ch
cosa + cos b = 2cos a + b .cos a - b
2
2
cosa - cos b = -2sin a + b .sin a - b
2
2
sina + sin b = 2sin a + b .cos a - b
2
2
sina - sin b = 2cos a + b .sin a - b
2
2
tana + tan b = sin(a + b) cosa cos b
tana - tan b = sin(a - b) . cosa cos b
II. T?nh tun ho?n ca h?m s
nh ngha: H?m s y = f(x) x?c nh tr?n tp D c gi l? h?m s tun
ho?n nu c? s T 0 sao cho vi mi x D ta c? x T D v? f(x + T) = f(x) .
Nu c? s T dng nh nht tha m?n c?c iu kin tr?n th? h?m s ? c gi l? h?m s tun ho?n vi chu k? T . III. C?c h?m s lng gi?c 1. H?m s y = sin x
? Tp x?c nh: D = R ? Tp gi?c tr: [ - 1;1] , tc l? -1 sin x 1 x R
? H?m s ng bin tr?n mi khong (- + k2; + k2) , nghch bin tr?n
2
2
mi khong ( + k2; 3 + k2) .
2
2
? H?m s y = sin x l? h?m s l n?n th h?m s nhn gc ta O l?m
t?m i xng. ? H?m s y = sin x l? h?m s tun ho?n vi chu k? T = 2 .
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? th h?m s y = sin x .
y
-5
-
2
-2
- 2
-3
-3
2
1 O
2
2
3
2 2
3x 5 2
2. H?m s y = cos x ? Tp x?c nh: D = R ? Tp gi?c tr: [ - 1;1] , tc l? -1 cos x 1 x R ? H?m s y = cos x nghch bin tr?n mi khong (k2; + k2) , ng bin tr?n mi khong (- + k2; k2) . ? H?m s y = cos x l? h?m s chn n?n th h?m s nhn trc Oy l?m trc i xng. ? H?m s y = cos x l? h?m s tun ho?n vi chu k? T = 2 . ? th h?m s y = cos x . th h?m s y = cos x bng c?ch tnh tin th h?m s y = sin x theo v?c t v = (- ; 0) .
2
y
-5
-
2
-2
- 2
-3
-3
2
1
O 2
3
2 2
3x 5 2
3. H?m s y = tanx
? Tp x?c nh : D =
\
2
+
k,
k
? Tp gi? tr: ? L? h?m s l ? L? h?m s tun ho?n vi chu k? T =
?
H?m
ng
bin
tr?n
mi
khong
-
2
+
k;
2
+
k
? th nhn mi ng thng x = + k, k l?m mt ng tim cn. 2
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Truy cp website: ti t?i liu thi min ph?
? th
y
-
-2
-
2
2
-5
-3
O
2
2
3
5
2 2
2x
4. H?m s y = cot x
? Tp x?c nh : D = \k, k
? Tp gi? tr: ? L? h?m s l ? L? h?m s tun ho?n vi chu k? T =
? H?m nghch bin tr?n mi khong (k; + k)
? th nhn mi ng thng x = k, k l?m mt ng tim cn. ? th
y
-
-2
-
2
2
-5
-3
O
2
2
3
5
2 2
2x
B.PHNG PH?P GII TO?N.
Vn 1. Tp x?c nh v? tp gi? tr ca h?m s
Phng ph?p . ? H?m s y = f(x) c? ngha f(x) 0 v? f(x) tn ti ? H?m s y = 1 c? ngha f(x) 0 v? f(x) tn ti.
f(x)
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? sin u(x) 0 u(x) k, k
? cos u(x) 0 u(x) + k, k . 2
? -1 sinx, cosx 1. C?c v? d
V? d 1. T?m tp x?c nh ca h?m s sau: 1. y = tan(x - )
6
2. y = cot2( 2 - 3x) 3
Li gii.
1. iu kin: cos(x - ) 0 x - + k x 2 + k
6
62
3
TX: D =
\
2 3
+
k,
k
.
2. iu kin: sin( 2 - 3x) 0 2 - 3x k x 2 - k
3
3
93
TX: D =
\
2 9
+
k
3
,
k
.
V? d 2. T?m tp x?c nh ca h?m s sau:
1. y = tan 2x + cot(3x + )
sin x + 1
6
2. y = tan 5x sin 4x - cos 3x
Li gii.
1.
iu
kin:
sin x -1
sin(3x
+
) 6
0
x x
- + k2 2
- + k 18 3
Vy TX: D =
\ -
2
+
k2,
-
18
+
n 3
;
k,
n
2.
Ta
c?:
sin 4x
-
cos 3x
=
sin 4x
-
sin
2
-
3x
=
2
cos
x 2
+
4
sin
7x 2
-
4
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cos 5x 0
iu
kin:
cos
x 2
+
4
0
x
10
+
k
5
x
2
+
k2
sin
7x 2
+
4
0
x
-
14
+
k2 7
Vy TX: D =
\
10
+
k 5
,
2
+
n2,
-
14
+
2m 7
.
C?C B?I TO?N LUYN TP B?i 1 T?m tp x?c nh ca h?m s sau:
1. y = 1 - sin 2x cos 3x - 1
3. y = tan(2x - ) 4
2. y = 1 - cos 3x 1 + sin 4x
4. y = 1 + cot2 x 1 - sin 3x
B?i 2 T?m tp x?c nh ca h?m s sau:
1. y =
1
sin 2x - cos 3x
2. y =
tan 2x
3 sin 2x - cos 2x
3. y = cot x 2 sin x - 1
4. y = tan(x - ).cot(x - )
4
3
B?i 3 T?m tp x?c nh ca h?m s sau:
1. y = tan(2x + ) 3
3. y =
2 + sin x tan2 x
5. y = sin 3x sin 8x - sin 5x
2. y = tan 3x.cot 5x
4. y = tan 3x + cot(x + ) 3
6. y = tan 4x cos 4x + sin 3x
Vn 2. T?nh cht ca h?m s v? th h?m s
Phng ph?p . Cho h?m s y = f(x) tun ho?n vi chu k? T * kho s?t s bin thi?n v? v th ca h?m s, ta ch cn kho s?t v? v th h?m s tr?n mt on c? d?i bng T sau ? ta tnh tin theo c?c v?c
t k.v (vi v = (T; 0), k ) ta c to?n b th ca h?m s. * S nghim ca phng tr?nh f(x) = k , (vi k l? hng s) ch?nh bng s giao im ca hai th y = f(x) v? y = k .
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* Nghim ca bt phng tr?nh f(x) 0 l? min x m? th h?m s y = f(x)
nm tr?n trc Ox . Ch? ?: ? H?m s f(x) = a sin ux + bcos vx + c ( vi u,v ) l? h?m s tun ho?n vi
chu k? T = 2 ( (u,v) l? c chung ln nht). (u, v)
? H?m s f(x) = a.tan ux + b.cot vx + c (vi u,v ) l? h?m tun ho?n vi
chu k? T = . (u, v)
C?c v? d
V? d 1. X?t t?nh tun ho?n v? t?m chu k? c s ca c?c h?m s : f(x) = cos 3x .cos x
22
Li gii.
Ta c?
f(x) = 1 (cos x + cos 2x)
2
h?m s tun ho?n vi chu k? c s
T0
= 2 .
V? d 2. X?t t?nh tun ho?n v? t?m chu k? c s (nu c?) ca c?c h?m s sau.
( ) 1. f(x) = cos x + cos 3.x
2. f(x) = sin x2
Li gii. 1. Gi s h?m s ? cho tun ho?n c? s thc dng T tha
f(x + T) = f(x) cos(x + T) + cos 3(x + T) = cos x + cos 3x
Cho x = 0 cos T + cos
3T
=
2
cos cos
T=1 3T =
1
T
=
2n
3T = 2m
3 = m v? l?, do m,n n
m l? s hu t. n
Vy h?m s ? cho kh?ng tun ho?n.
2. Gi s h?m s ? cho l? h?m s tun ho?n
T 0 : f(x + T) = f(x) sin(x + T)2 = sin x2 x
Cho x = 0 sinT2 = 0 T2 = k T = k
f(x + k) = f(x) x .
( )2
Cho x = 2k ta c?: f( 2k) = sin k2 = sin(k2) = 0 .
( ) ( ) 2
f(x + k) = sin k2 + k = sin 3k + 2k 2 = sin(2k 2)
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f(x + k) 0 . Vy h?m s ? cho kh?ng phi l? h?m s tun ho?n.
V? d 3. Cho a,b,c,d l? c?c s thc kh?c 0. Chng minh rng h?m s f(x) = a sin cx + bcosdx l? h?m s tun ho?n khi v? ch khi c l? s hu t.
d Li gii. * Gi s f(x) l? h?m s tun ho?n T 0 : f(x + T) = f(x) x
a sin cT + b cosdT = b cosdT = 1 Cho x = 0,x = -T -a sin cT + b cosdT = b sin cT = 0
dT cT
= 2n = m
c d
=
m 2n
.
* Gi s c k,l : c = k . t T = 2k = 2l
d
dl
cd
Ta c?: f(x + T) = f(x) x f(x) l? h?m s tun ho?n vi chu k?
T = 2k = 2l . cd
V? d 4. Cho h?m s y = f(x) v? y = g(x) l? hai h?m s tun ho?n vi chu k
ln lt l? T1 ,T2 . Chng minh rng nu
T1 T 2
l? s hu t th? c?c h?m s
f(x) g(x); f(x).g(x) l? nhng h?m s tun ho?n.
Li gii.
V? T1 l? s hu t n?n tn ti hai s nguy?n m,n; n 0 sao cho T 2
T1 T2
=
m n
nT1
= mT2
= T
Khi ? f(x + T) = f(x + nT1) = f(x) v? g(x + T) = g(x + mT2 ) = g(x)
Suy ra f(x + T) g(x + T) = f(x) g(x) v? f(x + T).g(x + T) = f(x).g(x) ,
f(x + T) = f(x) . T ? ta c? iu phi chng minh. g(x + T) g(x)
Nhn x?t: 1. H?m s f(x) = a sin ux + bcos vx + c ( vi u,v chu k? T = 2 ( (u,v) l? c chung ln nht).
(u, v)
) l? h?m s tun ho?n vi
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