§ 28: 28.9

28.9 Розв’яжіть рівняння:

1) 3sinx + cosx = 2|:2; 32 • sinx + 12 cos x = 1;

sinx • cosπ6 + cosx • sinπ6 = 1; sin(x + π6) = 1; x + π6 = π2 + πn, n ∈ Z;

x = π2 – π6 + πn, n ∈ Z; x = π3 + πn, n ∈ Z;

2) 2sinx – 2cosx = 1|:2; sinx • 22 – cosx • 22 = 12;

sinxcosπ4 – cosxsinπ4 = 12; sin(x – π4) = 12; x – π4 = (–1)ⁿarcsin12 + πn,

n ∈ Z;

x = (–1)ⁿ π6 + π4 + πn, n ∈ Z;

3) 3sinx2 + 3cosx2 = 3| : 23; 32sinx2 + 12cosx2 = 32;

sinx2cosπ6 + cosx2sinπ6 = 32; sin(x2 + π6) = 32;

x2 + π6 = (–1)ⁿarcsin32 + πn, n ∈ Z; x2 = (–1)ⁿ π3 – π6+ πn, n ∈ Z;

x = (–1)ⁿ 2π3 – π3 + 2πn, n ∈ Z.