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18.03.2013

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sin3x cos8x = 1.

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:

(sin(3x + 8x) + sin(3x - 8x))/2 = 1;

sin11x - sin5x = 2.

, |sin11x| ≤ 1 |sin5x| ≤ 1, 2 , sin11x = 1 sin5x = -1.

sin11x = 1 x = π/22 + 2πn/11 (nZ).

sin5x = -1 x = -π/10 + 2πm/5 (mZ).

, , ..

π/22 + 2πn/11 = -π/10 + 2πm/5;

(4n + 1)π/22 = (4m - 1)π/10;

20n = 44m - 16;

5n = 11m - 4 (n, mZ).

: m = 4 + 5t, n = 8 + 11t (n, t, mZ).

x = -π/10 + 2πm/5 = -π/10 + 2π(4 + 5t)/5 = 3π/2 + 2πt (tZ).

: x = 3π/2 + 2πt.


ctg2x = cos22x - 1.

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cos22x - 1 = -sin22x :

ctg2x = -sin22x.

, ctg2x ≥ 0, -sin22x ≤ 0. , ctg2x = 0 sin22x = 0.

ctg2x = 0 x = π/2 + πn (nZ).

sin22x = 0 x = πm/2 (mZ).

:

π/2 + πn = πm/2;

n - 2m = 1.

n = 3 + 2t, m = 1 + t (m, n, tZ).

x = πm/2 = (1 + t)π/2 = 3π/2 + πt (tZ).

: x = 3π/2 + πt.


sin3x cos5x = 1.

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:

(sin8x - sin2x)/2 = 1;

sin8x - sin2x = 2.

, sin8x = 1, sin2x = -1.

sin8x = 1 x = π/16 + πn/4 (nZ) (*).

sin2x = -1 x = -π/4 + πm (mZ) (**).

, :

π/16 + πn/4 = -π/4 + πm;

16m - 4n = 5.

4, - . . (*) (**) .

: .


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