Ответ:
Пошаговое объяснение:
1) √3cosx + sin2x = 0; √3cosx + 2sinx · cosx = 0; cosx · (√3 + 2sinx) = 0; a) cosx = 0 ⇒ x = π/2 + πk, k ∈ Z. b) √3 + 2sinx = 0; 2sinx = - √3; sinx = - √3/2 ⇒ x = - π/3 + 2πn, n ∈ Z; x = - 2π/3 + 2πm, m ∈ Z. Ответ: x = π/2 + πk, k ∈ Z; - π/3 + 2πn, n ∈ Z; - 2π/3 + 2πm, m ∈ Z 2) cosx + cos2x = sinx; cosx - sinx + cos²x - sin²x = 0; (cosx - sinx) + (cosx + sinx)(cosx - sinx) = 0; (cosx - sinx) ( 1 +cosx + sinx) = 0 ⇒ a) cosx - sinx = 0; cosx ≠o ⇒ 1 - tgx = 0; tgx = 1; x = π/4 + πk, k ∈ Z; b) 1 +cosx + sinx =0; cosx + cos(90° - x) = - 1; 2cos (x + 90° - x)/2 · cos (x - 90° + x)/2 = - 1; 2cos45° · cos(x - 45°) = -1; cos(x - 45°) = - √2/2 ⇒ x - 45° = ±(π - 45°) + 360°n, n ∈ Z; x = 45° ± (π - 45°) + 360°n, n ∈ Z. Ответ: x = π/4 + πk, k ∈ Z; x = 45° ± (π - 45°) + 360°n, n ∈ Z. 3)cos³xsinx - sin³xcosx = √2/8; cosxsinx (cos²x - sin²x) · 2 = 2 · √2/8 ⇒ 2sin2x · cos2x = 2 · √2/4; sin4x = √2/2 ⇒ 4x = π/4 + 2πm, m ∈ Z, x = π/16 + π/2 m, m ∈ Z ; 4x = 3π/4 + 2πk, k ∈Z, x = 3π/16 + πk/2, k ∈Z. Ответ: x = π/16 + π/2 m, m ∈ Z ; x = 3π/16 + πk/2, k ∈Z.