1 Exercit˘ii rezolvate - Deliu

[Pages:2]Trigonometrie plana ?si sferica

Ecua?tii trigonometrice

1 Exerci?tii rezolvate

1. cos 2x + 4 sin x - 1 = 0 R: ^Inlocuim ^in ecua?tie cos 2x = 1 - 2 sin2 x ?si ob?tinem

1 - 2 sin2 x + 4 sin x - 1 = 0 2 sin x(2 - sin x) = 0

Cum sin x 1 2 - sin x 0 deci singura solu?tie acceptabila este sin x = 0 de unde ob?tinem

x = k + (-1)k arcsin 0, k Z x = k, k Z 2. 2 cos x + 2 sin2 x + ctg2 x - 3 = 0, x k, k Z R: Folosind formulele care exprima sin2 x ?si ctg2 x ^in func?tie de cos2 x ob?tinem

2

cos

x

+

2(1

-

cos2

x)

+

1

cos2 x - cos2

x

-

3

=

0

Pun^and t = cos x, ^in urma calculelor se ob?tine

2t4 - 2t3 + 2t - 1 = 0 ( 2t - 1)( 2t3 + 1) = 0

t1 = 1 = 2

2 2

x

=

2k

?

;

4

t2

=

-

1 62

x

=

2k

?

arccos

- 1 62

3. 4 sin x + 2 cos x - 3 tg x - 2 = 0

R:

Facem

substitu?tia

t

=

tg

x 2

.

Avem

sin x

=

2t 1 + t2 ,

cos x

=

1 - t2 1 + t2

?si

tg x

=

2t 1 - t2 .

Dupa

efectuarea calculelor se ob?tine ecua?tia

2t4 - 7t3 - 2t2 + t = 0

care

are

radacinile

t1

=

0,

t2

=

-

1 2

,

t3,4

=

2

?

3.

x tg

2

=0

x 2

= k x = 2k,

kZ

tg x = - 1 x = k + arctg - 1 x = 2k + 2 arctg

tg

2 x

2 2 =2+ 3

x

=

2k

2 + 2 arctg

tg

2 x

=2- 3

x

=

2k

+ 2 arctg

2

2+ 3

2- 3

, ,

kZ kZ

-1 2

, kZ

4.

3 sin x + cos R: ^Impar?tind

x=1 prin

3

ob?tinem sin x + 1 cos x =

3

1 .

3

Punem

1 3

=

tg

6

sin

x

+

sin cos

6 6

cos x =

1 3

sin

x

cos

6

+

cos

x

sin

6

=

1 3

cos

6

de

unde

folosind

formula

pentru

sinusul

sumei

gasim

sin

x

+

6

=

1 2

,

a?sadar

x + = k + (-1)k arcsin 1 x = k + (-1)k - 1

6

2

6

5. 3 sin2 x + 2 sin x cos x - cos2 x = 0 R: ^Impar?tind prin cos2 x 0 ?si pun^and t = tg x ob?tinem ecua?tia

3t2 + 2t - 1 = 0

t1

= -1 tg x = -1 x = k + arctg(-1) = k -

4

,

kZ

t2

=

1 3

tg x

=

1 3

x

=

k

+

arctg

1 3

, kZ

1

Trigonometrie plana ?si sferica

Ecua?tii trigonometrice

6. 2 sin4 x - 2 3 sin3 x cos x + 4 sin2 x cos2 x + 2 3 sin x cos3 x - 2 cos4 x = 1

R: ^Inlocuind ^in membrul drept 1 = sin2 x + cos2 x ecua?tia devine:

2 sin4 x - 2 3 sin3 x cos x + 4 sin2 x cos2 x + 2 3 sin x cos3 x - 2 cos4 x = (sin2 x + cos2 x)2

sin4 x - 2 3 sin3 x cos x + 2 sin2 x cos2 x + 2 3 sin x cos3 x - 3 cos4 x = 0

^Impar?tind prin cos4 x ?si not^and t = tg x ob?tinem

t4 - 2 3t3 + 2t2 + 2 3 - 3 = 0 (t2 - 1)(t2 - 2 3 + 3) = 0

t1,2

= ?1 tg x = ?1 x = k ?

4

,

kZ

t3,4 =

3 tg x =

3 x = k +

3

,

kZ

7. 5(sin x + cos x) - 2 sin 2x = 4

R: Facem substitu?tia u = sin x + cos x sin 2x = u2 - 1. Se ob?tine ecua?tia 2u2 - 5u + 2 = 0 cu

radacinile

reale

u1= 2,

u2 =

1 2

.

u = sin x + cos x = 2 cos

x

-

4

u = sin x + cos x =

2 cos

x

-

4

= 2 cos

x

-

4

=

1 2

cos

x

-

4

= 2 > 1 nu exista solu?tii;

=

2 4

x

=

2k

?

arccos

2 4

+

4

.

8. cos2 x + cos2 2x - cos2 3x = 1

R:

1 2

(1

+

cos 2x)

+

1 2

(1

+

cos 4x)

-

1 2

(1

+

cos 6x)

=

1

cos 2x

-

cos

6x

=

1

-

cos 4x.

Folosind

formulele de transformare a diferen?tei ?si sumei ^in produs gasim -2 sin 4x sin 2x = 2 sin2 2x

2 sin 2x(sin 4x + sin 2x) = 0 4 sin 2x sin 3x cos x = 0

sin 2x = 0 2x = k

x=

k 2

,

k

Z

sin 3x = 0 3x cos x = 0 x =

= k (2k + 1)

x

2

= ,

k

k3

,k Z,

Z mul?time

de

solu?tii

care

este

inclusa

^in

prima

mul?time.

9. 2(sin6 x + cos6 x) + sin4 x + cos4 x = 1

R:

Cu

substitu?tia

y

= sin 2x

avem

sin4

x

+

cos4

x

=

1

-

1 2

y2,

sin6

x

+

cos6

x

=

1

-

3 4

y2,

iar

ecua?tia

devine 2 sin 2x = 1

1 -243xy=2

sin 2x = -1 2x

=2+k2k1+--212

y2 = 1 y2 = 2xx==kk+-4

1 cu

, k

4

,

k

radacinile

Z; Z.

y

=

?1.

10. cos x cos 7x = cos 3x cos 5x R: Transform^and cele doua produse ^in sume avem

1 (cos 8x + cos 6x) = 1 (cos 8x + cos 2x) cos 6x - cos 2x = 0 -2 sin 4x sin 2x = 0

2

2

sin 4x = 0 x =

k 4

,

k

Z

sin 2x = 0 x =

k 2

,

k

Z,

mul?time

de

solu?tii

care

este

inclusa ^in

prima

mul?time.

2 Tema

1. cos x + 3 sin x = m; discu?tie dupa m R

2. 2 cos2 x - sin 2x + sin x + cos x = 1

3. cos2 x + 3 sin2 x + 2 3 sin x cos x = 1

4. cos2 x + cos2 2x + cos2 3x + cos2 4x = 2

5.

sin3 x cos 3x + sin 3x cos3 x =

3 4

2

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