§ 44: 44.75 (1)

44.75 Розв’яжіть систему рівнянь:

1) x – y = π3; x – y = π3;

cos²x – cos²y = –34; (cos x – cos y)(cos x + cos y) = –34;

x – y = π3;

–2sinx+y2 sinx-y2 • 2cosx+y2cosx-y2 = –34;

x – y = π3;

–4 • sinx+y2 • sinπ6cosx+y2 • cosπ6 = –34;

x – y = π3;

–4sinx+y2 • 12 • cosx+y2 • 32 = –34;

x – y = π3; x – y = π3;

–32 • sin (x + y) = – 34; sin(x + y) = 32;

x – y = π3;

x + y = (–1)ⁿπ3 + πn, n ∈ Z.

2x = π3 + (–1)ⁿπ3 + πn; x = π6 + (–1)ⁿπ6 + πn2.

y = π6 + (–1)ⁿπ6 + πn2 – π3; y = –π6 + (–1)ⁿπ6 + πn2, n ∈ Z.

2) x – y = π3; x – y = π3;

cosxcos y = 12; 12(cos(x – y) + cos(x + y)) = 12;

x – y = π3; x – y = π3;

cos π3 + cos(x + y) = 1; cos(x + y) = 1 – 12;

x – y = π3; x – y = π3;

cos(x + y) = 12; x + y = ± π3 + 2πn, n ∈ Z.

2x = π3 ± π3 + 2πn; x = π6 ± π6 + πn, n ∈ Z.

y = π6 ± π6 + πn – π3; y = –π6 ± π6 + πn.

x = π6 + π6 + πn; x = π3 + πn;

y = –π6 + π6 + πn; y = πn;

x = π6 – π6 + πn; x = πn;

y = –π6 – π6 + πn; y = –π3 + πn.

(π3 + πn; πn) і (πn; –π3 + πn), n ∈ Z.

3) x + y = 13; x + y = 13;

sinπx ± sinπy = 1; 2sinπx+ πy2cosπx- πy2 = 1;

x + y = 13; x + y = 13;

2sin π • 132cosπ(x-y)2 = 1; 2sinπ6cos π(x-y)2 = 1;

x + y = 13; x + y = 13; x + y = 13;

2 • 12cosπ(x-y)2 = 1; cosπ(x-y)2 = 1; π(x-y)2 = 2πn, n ∈ Z;

x + y = 13; 2x = 4n + 13;

x – y = 4n;

x = 16 + 2πn, n ∈ Z; y = 16 – 2n.

Відповідь: (16 + 2n і 16 – 2n), n ∈ Z.

4) x + y = π3; x + y = π3;

sinxsiny = 0,25; 12(cos(x – y) – cos(x + y)) = 0,25;

x + y = π3; x + y = π3;

cos(x – y) – cos π3 = 0,5; cos(x – y) – 12 = 0,5;

x + y = π3; x + y = π3;

cos(x – y) = 1; x – y = 2πn;

2x = π3 + 2πn; x = π6 + πn; y = π6 – πn, n ∈ Z.

Відповідь: (π6 + πn; π6 – πn), n ∈ Z .