2) Решите уравнения! a) cos(x + π/6)-1=0 б) 2 cosx = 1 в) 2 cos²x - 5cosx + 2 = 0

Решим тригонометрические уравнения.

а) cos(x + π/6)-1=0

cos(x + π/6) = 1

x + π/6 = 2πn, n ∈ Z

x = -π/6 + 2πn, n ∈ Z

Ответ: x = -π/6 + 2πn, n ∈ Z

б) 2 cosx = 1

cosx = 1/2

x = ±arccos(1/2) + 2πn, n ∈ Z

x = ±π/3 + 2πn, n ∈ Z

Ответ: x = ±π/3 + 2πn, n ∈ Z

в) 2 cos²x - 5cosx + 2 = 0

Пусть cosx = t, тогда

2t² - 5t + 2 = 0

D = (-5)² - 4 * 2 * 2 = 25 - 16 = 9

t₁ = (5 + √9) / (2 * 2) = (5 + 3) / 4 = 8 / 4 = 2

t₂ = (5 - √9) / (2 * 2) = (5 - 3) / 4 = 2 / 4 = 1/2

cosx = 2 (не имеет решений, так как |cosx| ≤ 1)

cosx = 1/2

x = ±arccos(1/2) + 2πn, n ∈ Z

x = ±π/3 + 2πn, n ∈ Z

Ответ: x = ±π/3 + 2πn, n ∈ Z