A.T.E.I. PEIRAIA

SQOLH TEQNOLOGIKWN EFARMOGWN

TMHMA POLITIKWN MHQANIKWN T.E.

MAJHMATIKA I

Antstrofec trigwnometrikc sunartseic

Gia kje trigwnometrik sunrthsh orzetai antstoiqh antstrofh trigwnometrik sunrthsh.

Oi antstrofec trigwnometrikc sunartseic, orzontai me bsh tmma tou pedou orismo twn trigwnometri-

kn, sto opoo autc enai > (na proc na).Profanc aut to tmma enai tmma peridou.

Gia pardeigma h sunrthsh hmtono enai > sto

[pi2 , pi2 ] kai me bsh aut orzetai h antstrofhthc. Onomzontai kai txo thc trigwnometrikc sunrthshc pou antistoiqon,

giat antistoiqon to dosmno trigwnometrik arijm se antstoiqh gwna,

ra kai sto mkoc tou antstoiqou txou tou monadiaou kklou.

1. H antstrofoc thc sunrthshc y = x enai h y = ox (y = arcsin x)

H y = x qeiR(f) = [1, 1] x

[pi2,pi

2

].

Epeid h sunrthsh aut enai gnhswc axousa sto disthma

[pi2 , pi2 ] ja qei antstrofo thn opoasumbolzoume:

x = oy

an diathrsoume thn anexrthth metablht me:

y = ox (1)

pou qei pedo orismo to [1, 1] kai pedo timn to [pi2 , pi2 ].

x=

1

x=1

x=pi/2

x=pi

/2

y = 1

y = 1

y = pi/2

y = pi/2

xO

y

B

A

y = x

y = ox

1

H y = ox enai gnhswc axousa sto [1, 1], kolh sto [1, 0], kurt sto [0, 1] meymin = pi2 , ymax =

pi

2kai shmeo kampc sto 0. To digramma thc (1) enai summetrik tou diagrmmatoc thc y = x wcproc th diqotmo thc xoy.

Parathrome ti h y = x enai epshc gnhswc fjnousa sto[pi2 ,

3pi2

].

10 5 0 5 103

2

1

0

1

2

3

x=

pi 2

x=

3pi 2

x=

5pi 2

x=

7pi 2

x=pi 2

x=3

pi 2

x=5

pi 2

x=7

pi 2

x

y

O

y = sinx, npi pi/2 x npi + pi/2, n Z

n = 0n = 2 n = 2

n = 1 n = 3n = 1n = 3

H antstrofc thc enai epshc gnhswc fjnousa.

Genik h y = x enai gnhswc axousa sto disthma

pi pi2 x pi + pi

2gia = 0, 2, 4, . . .kai gnhswc fjnousa sto disthma

pi pi2 x pi + pi

2gia = 1, 3, . . .

3 2 1 0 1 2 3

10

5

0

5

10

y = 7pi2

y = 5pi2

y = 3pi2

y = 1pi2

y = 1pi2

y = 3pi2

y = 5pi2

y = 7pi2

x

y

y = arcsin x, npi pi2 y npi + pi2 , n Z

O

n = 0

n = 2

n = 2

n = 1

n = 3

n = 1

n = 3

H y = ox enai gnhswc axousa gnhswc fjnousa antstoiqa. 'Etsi qoume:

y = ox, pi pi2 y pi + pi

2.

2

2. Antstrofh Sunrthsh

'Estw h sunrthsh f : A B me y = f(x). An gia kje y B uprqei na monadik x A ttoioste f(x) = y, tte aut h antistoqish orzei mia na sunrthsh f1 : B A me x = f1(y), h opoaonomzetai antstrofh sunrthsh thc f .

3. Pargwgoc antstrofhc sunrthshc

'Estw ti h sunrthsh y = f(x) enai gnhswc montonh sto [, ], paragwgsimh sto x0 (, ) kaif (x0) 6= 0, tte h antstrofh sunrthsh f1 enai paragwgsimh sto y0 = f(x0) kai isqei.

(f1)(y0) = limxx0

f1(y) f1(y0)y y0 =

1f (x0) yx =

1xy

.

4. Pargwgoi antstrofwn trigwnometrikn sunartsewn

(ox) =1

1 x2 , |x| < 1

(ox) = 11 x2 , |x| < 1

(ox) =1

1 + x2,

(ox) = 11 + x2

,

i. Pargwgoc thcy = ox , y

[pi2,pi

2

].

H sunrthsh x = y qei antstrofh sunrthsh thn y = ox, opte isqei:

yx =1xy (ox) =

1(y)

=1

y.

Epeid y [pi2 , pi2 ] ja enai y 0 kaiy =

1 2y =

1 x2 .

'Ara

(ox) =1

1 x2 , |x| < 1 .

3

ii. H sunrthshx :

[pi2,pi

2

] [1, 1], x y = xenai > afo

d

dx(x) = x > 0, x

(pi2,pi

2

).

'Ara orzetai h sunrthsh

ox : [1,1][pi2,pi

2

], x ox = y

'Etsi qoume

y = ox y = x .

Paragwgzontac kai ta do mlh thc teleutaac isthtac wc proc x kai jewrntac y = y(x)lambnoume gia to anoikt disthma

(pi2 , pi2 ).d

dx(y) = 1 d

dy(y)

dy

dx= 1 (y)dy

dx= 1 dy

dx=

1y

=1

1 2y ,

afo gia kje y (pi2 , pi2 ) isqei ti y > 0.'Ara qoume:

d

dx(ox) =

11 x2 , x (1, 1)

m

(ox) =1

1 x2 , |x| < 1 .

Epkouroc Kajhghtc : Dr. Pappc G. Alxandroc

4