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18.03.2013

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4sin2x - 3sinxcosx + 5cos2x = 3.

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, , . :

sin2x - 3sinxcosx + 2cos2 + 3(sin2x + cos2x) = 3;

sin2x - 3sinxcosx + 2cos2x = 0.

, sin2x ≠ 0 (, sinx = 0, cosx = 0, ).

1 - 3ctgx + 2ctg2x = 0;

2ctg2x - 3ctgx + 1 = 0.

, ctgx = t t:

2t2 - 3t + 1 = 0.

t1 = 1, t2 = 1/2.

x

t1: ctgx = 1, x = π/4 + πn (nZ);

t2: ctgx = 1/2, x = arcctg(1/2) + πm (mZ).

: x = π/4 + πn x = arcctg(1/2) + πm.


sinx + tg(x/2) = 2.

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, π + 2πn (nZ) , tg(x/2) = t. :

2t/(1 + t2) + t = 2;

t3 - 2t2 + 3t - 2 = 0;

t2(t - 1) - (t2 - 3t + 2) = 0;

t2(t - 1) - (t - 2)(t - 1) = 0;

(t - 1)(t2 - t + 2) = 0;

, t = 1. :

tg(x/2) = 1,

x = π/2 + 2πn, nZ.

: x = π/2 + 2πn.


3sinx - 2cosx = 1/2.

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, π + 2πn (nZ) . tg(x/2) = y. :

2y

1 + y2

- 2· 1 - y2

1 + y2

= 1

2

;

3y2 + 12y - 5 = 0.

y = (-6 ± 51)/3. , :

tg(x/2) = (-6 ± 51)/3;

x = 2arctg((-6 ± 51)/3) + 2πn (nZ).

: x = 2arctg((-6 ± 51)/3) + 2πn.


4sinx - 3cosx = 3.

______________________________

tg(x/2) = y. , π + 2πn (nZ) , .

:

2y

1 + y2

- 3· 1 - y2

1 + y2

= 3.

8y = 6;

y = 3/4.

tg(x/2) = 3/4,

x = 2arctg(3/4) + 2πn (nZ).

: x = 2arctg(3/4) + 2πn x = π + 2πn.


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