Bt dai so 11

I. H THC C BN 1. nh ngha cc gi tr lng gic: OP = cos a OQ = sin a AT = tan a BT " = cot a Nhn xt: "a, - 1 cos a 1; - 1 sin a 1 tamãng cầu xc nh khi a tang

Tcotang

sinQ B M a T"

O

p

A

cosin

p + kp , k Z , 2

cota xc nh khi a kp , k Z 2. Du ca cc gi tr lng gic: Cung phn t Gi tr lng gic simãng cầu cosa tana cota 3. H thc c bn: + + + + + + + + I II II IV

sin2a + cos2a = 1; tana.cota = 1 1 1 1 + tan2 a = ; 1 + cot 2 a = cos2 a sin 2 a

4. Cung lin kt: Cung i nhau cos(- a) = cos a sin(- a) = - sin a tan(- a) = - tung a cot(- a) = - cot a Cung b nhau sin(p - a) = sin a cos(p - a) = - cos a tan(p - a) = - tan a cot(p - a) = - cot a Trang 1 Cung ph nhau p sin - a = cos a 2 p cos - a = sin a 2 p tan - a = cot a 2 p cot - a = chảy a 2

i s 11

Trn S Tng

Cung hn km p sin(p + a) = - sin a cos(p + a) = - cos a tan(p + a) = tan a cot(p + a) = cot a

Cung hn km

p 2

p sin + a = cos a 2 p cos + a = - sin a 2 p tung + a = - cot a 2 p cot + a = - tan a 2

5. Bng gi tr lng gic ca cc gc (cung) c bit0 00 sin 0

p 6300

p 4450

p 3600

p 2900 1

2p 3

3p 4

p1800 0

3p 2

2p

1200

1350

2700 1

3600 0

1 2 3 2 3 3 3

2 2 2 21

3 2 1 2 3 3 3

3 2 1 2 -

2 2 2 21

cos

1

0

1

0

1

tan

0

- 30

0

0

cotg

1

-

3 3

1

0

II. CNG THC CNG Cng thc cng: sin(a + b) = sin a. cos b + sin b.cos a sin(a - b) = sin a.cos b - sin b.cos a cos(a + b) = cos a. cos b - sin a.sin b cos(a - b) = cos a. cos b + sin a.sin b tan a + tung b 1 - rã a.tan b rã a - tan b tan(a - b) = 1 + tung a.tan b tan(a + b) =

H qu:

p 1 + tung x p 1 - chảy x tung + x = , chảy - x = 4 1 - rã x 4 1 + tung x

Trang 2

Trn S Tng III. CNG THC NHà Nội 1. Cng thc nhn i: sin2a = 2sina.cosa cos 2 a = cos2 a - sin 2 a = 2 cos2 a - 1 = 1 - 2sin 2 a tan 2a = 2. Cng thc h bc: sin 2 a = 1 - cos 2 a 2 1 + cos 2a 2 cos a = 2 1 - cos 2 a 2 tan a = 1 + cos 2 a 2 tan a 1 - rã 2 a ; cot 2 a = cot 2 a - 1 2 cot a 3. Cng thc nhn ba: sin 3a = 3sin a - 4sin3 a cos 3a = 4 cos3 a - 3cos a 3chảy a - tan 3 a tung 3a = 1 - 3rã 2 a

i s 11

4. Cng thc biu din sina, cosa, tana theo t = tan t: t = chảy a 2t (a p + 2 kp ) th: sin a = ; 2 1 + t2

a : 2 cos a = 1 - t2 1 + t2

;

tung a =

2t 1 - t2

IV. CNG THC BIN I 1. Cng thc bin i tng thnh tch: a+b a-b .cos 2 2 a+b a-b sin a - sin b = 2 cos .sin 2 2 a+b a-b cos a + cos b = 2 cos . cos 2 2 a+b a-b cos a - cos b = - 2 sin .sin 2 2 sin a + sin b = 2sin sin(a + b) cos a.cos b sin(a - b) rã a - chảy b = cos a.cos b sin(a + b) cot a + cot b = sin a.sin b sin(b - a) cot a - cot b = sin a.sinb rã a + tung b =

p p sin a + cos a = 2.sin a + = 2.cos a - 4 4 p p sin a - cos a = 2 sin a - = - 2 cos a + 4 4 2. Cng thc bin i tch thnh tng: 1 cos a.cos b = cos(a - b) + cos(a + b) 2 1 sin a.sin b = cos(a - b) - cos(a + b) 2 1 sin a.cos b = sin(a - b) + sin(a + b) 2

Trang 3

i s 11

Trn S Tng

HAM SO LNG GIAC PHNG TRNH LNG GIAC

CHNG I

I. HM S LNG GICVn 1: TP XC NH, TPhường GI TR, TNH CTP Hà Nội L, CHU K

y = sin x : Tp xc nh D = R; tp gi tr T = -1, 1 ; hm l, chu k T0 = 2p . * * y = sin(ax + b) c chu k T0 = 2p a

y = sin(f(x)) xc nh f ( x ) xc nh. y = cos x : Tp xc nh D = R; Tp gi tr T = -1, 1 ; hm chn, chu k T0 = 2p .

* *

y = cos(ax + b) c chu k T0 =

2p a

y = cos(f(x)) xc nh f ( x ) xc nh. p y = chảy x : Tp xc nh D = R + kp , k Z ; tp gi tr T = R, hm l, chu k T0 = p . 2

* *

y = tan(ax + b) c chu k T0 =

p a p + kp (k Z ) 2

y = tan(f(x)) xc nh f ( x )

y = cot x : Tp xc nh D = R kp , k Z ; tp gi tr T = R, hm l, chu k T0 = p . * * * y = cot(ax + b) c chu k T0 =

p a

y = cot(f(x)) xc nh f ( x ) kp (k Z ) . y = f1(x) c chu k T1 ; y = f2(x) c chu k T2

Th hm s y = f1 ( x ) f2 ( x ) c chu k T0 l bi bình thường nh nht ca T1 v T2.

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Trang 4

Trn S TngBai 1. Tm tp xc nh v tp gi tr ca cc hm s sau:

i s 11

2x a/ y = sin x -1 d/ y = 1 - cos2 x

b/ y = sin x e/ y = 1 sin x + 1

c/ y = 2 - sin x p f/ y = rã x - 6 i/ y = 1 tan x - 1

sin x p g/ y = cot x + h/ y = cos( x - p ) 3 Bai 2. Tm gi tr ln nht, gi tr nh nht ca hm s: p a/ y = 2sin x + + 1 4 d/ y = 4sin 2 x - 4 sin x + 3 g/ y = sinx + cosx b/ y = 2 cos x + 1 - 3 e/ y = cos2 x + 2sin x + 2

c/ y = sin x

f/ y = sin 4 x - 2 cos2 x + 1i/ y = sin x + 3 cos x + 3 c/ y = sinx + cosx f/ y = sinx.cosx i/ y = rã x

h/ y = 3 sin 2 x - cos 2 x Bai 3. Xt tnh chn l ca hm s: a/ y = sin2x b/ y = 2sinx + 3 d/ y = tanx + cotx e/ y = sin4 x g/ y = sin x - tung x sin x + cot x h/ y = cos3 x + 1 sin3 x x 3

Bai 4. Tm chu k ca hm s:

a/ y = sin 2 x d/ y = sin 2 x + cos x 2

b/ y = cos

c/ y = sin 2 x f/ y = cos 3x 2x - sin 5 7

e/ y = tung x + cot 3 x h/ y = cos2 4 x c/ p . d/ 4p . e/ p . . f/ 70p .

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g/ y = 2sin x . cos3 x HD: a/ p . b/ 6p .

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i/ y = tan(-3x + 1) g/ p . h/

p . 4

i/

p 3

Vn 2: 1/

TH CA HM S LNG GIC

V th hm s lng gic: Tm tp xc nh D. Tm chu k T0 ca hm s. Xc nh tnh chn l (nu cn). Lp bng bin thin trn mt on c di bng chu k T0 c th chn: T T x 0, T0 hoc x - 0 , 0 . 2 2 V th trn on c di bng chu k.

r r Ri suy ra phn th cn li bng php tnh tin theo vc t v = k .T0 .i v bn tri v r phi tuy vậy tuy vậy vi trc honh Ox (vi i l vc t n v trn trc Ox). Trang 5

i s 11

Trn S Tng

2/ Mt s php bin i th: a/ T th hm s y = f(x), suy ra th hm s y = f(x) + a bng cch tnh tin th y = f(x) ln trn trc honh a n v nu a > 0 v tnh tin xung trộn di trc honh a n v nu a

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