§ 44: 44.64 (1)

44.64 Розв’яжіть рівняння

1) cos²5x + 7sin²5x = 4sin10x;

cos²5x + 7sin²5x – 8sin5xcos5x = 0|: cos²5x ≠ 0;

1 + 7tg²5x – 8tg5x = 0;

7tg²5x – 8tg5x + 1 = 0;

tg5x = t;

7t² – 8t + 1 = 0;

D = 64 – 28 = 36; t₁ = 8+614 = 1; t₂ = 8-614 = 17;

tg 5x = 1 або tg 5x = 17;

5x = π4 + πn, n ∈ Z; x = π20 + πn5, n ∈ Z;

5x = arctg 17 + πn, n ∈ Z; x = 15 arctg 17 + 15 πn, n ∈ Z.

2) 3sin²x – 7sinxcosx + 14cos²x – 2 = 0;

3sin²x – 7sinxcosx + 14cos2x – 2sin²x – 2cos²x = 0;

sin²x – 7sinxcosx + 12cos²x = 0 |: cos²x;

tg²x – 7tgx + 12 = 0;

tgx = t;

t² – 7t + 12 = 0;

t₁ = 3; tg x = 3; x = arctg 3 + πn, n ∈ Z;

t₂ = 4; tg x = 4; x = arctg 4 + πn, n ∈ Z.