§ 32: 32.8 (1)

32.8 Розв’яжіть рівняння:

1) сos²x – sin²2x + cos²3x = 12; cos²x – 12 – sin²2x + cos²x = 0;

12cos2x – 1-cos4x2 + 1+cos6x2 = 0;

cos2x – 1 + cos4x + 1 + cos6x = 0; cos2x + cos6x + cos4x = 0;

2cos4x • cos(–2x) + cos4x = 0; cos4x(3cos2x + 1) = 0;

cos4x = 0; 4x = π2 + πk, k ∈ Z; x = π8 + πk4, k ∈ Z;

cos2x = –12; 2x = ± 2π3 + 2πn, n ∈ Z; x = ± π3 + πn, n ∈ Z.

Відповідь: x = π8 + πk4, k ∈ Z; x = ± π3 + πn, n ∈ Z.

2) sin2x + cos2x = 2sinx;

2sinx • cosx – 2sinx + cos2x = 0; 2sinx(cosx – 22) + 2(cos²x – 12) = 0; 2sinx • (cosx – 22) + 2(cosx – 22)(cosx + 22) = 0;

2(cosx – 22)(sinx + cosx + 22) = 0; 2(cosx – 22) • (cosx – cos(π2 + x) + 22) = 0; 2(cosx – 22) • (–2sin(π4 + x) • sin(–π4) + 22) = 0;

2(cos x – 22) • (2 • sin(π4 + x) + 22) = 0;

cosx = 22; x = ± arccos22 + 2πn, n ∈ Z.

sin(π4 + x) = –12; π4 + x = (–1)^k • arcsin (–12) + πk, k ∈ Z;

x = ± π4 + 2πn, n ∈ Z.

x = –π4 + (–1)^k+1 • π6 + πk, k ∈ Z.

Відповідь: x = ± π4 + 2πn, n ∈ Z; x = –π4 + (–1)^k+1 • π6 + πk, k ∈ Z.

3) cos²x + cos²2x = cos²3x + cos²4x;

1+cos2x2 + 1+cos4x2 = 1+cos6x2 + 1+cos8x2;

cos2x + cos4x = cos 6x + cos8x;

cos2x – cos8x +cos4x – cos6x = 0;

–2sin5x • sin(–3x) – 2sin5x • sin(–x) = 0;

2sin5x(sin3x + sinx) = 0; 2sin5x • 2sin2x • cosx = 0;

4sin5x • cosx • sin2x = 0;

sin5x = 0; 5x = πn, n ∈ Z; x = πn5, n ∈ Z;

cosx = 0; x = π2 + πk, k ∈ Z; x = π2 + πk, k ∈ Z;

sin2x = 0; 2x = πl, l ∈ Z; x = πl2, l ∈ Z;

x = πn5, n ∈ Z;

x = πl2, l ∈ Z.

Відповідь: x = πn5, n ∈ Z; x = πl2, l ∈ Z.

4) sin6x = 2cos(3π2 + 2x); sin6x = 2sin2x;

sin6x – sin2x – sin2x = 0; 2 sin2x • cos4x – sin2x = 0;

sin2x(2cos4x – 1) = 0;

sin2x = 0; 2x = πn, n ∈ Z;

cos4x = 12; 4x = ± arccos12 + 2πk, k ∈ Z;

x = πn2, n ∈ Z; x = πn2, n ∈ Z;

4x = ± π3 + 2πk, k ∈ Z; x = ± π12 + πk2, k ∈ Z.

Відповідь: x = πn2, n ∈ Z; x = ± π12 + πk2, k ∈ Z.