2) cos x/2 = 1/2 3) cos(2x+π/3) = 1/2 4) 2sin(-x/2)+1=0 Решите уравнения 1) 2 cos²x + 3cosx+1=0 2) 4 sin²x - 8cosx+1=0 =0 3) sin²x+2sinxcosx-3cos²x=0 Решите уравнение 3)*x7 cosx-4sin2x=0

cos(x/2) = 1/2

x/2 = ±arccos(1/2) + 2πk, k ∈ Z

x/2 = ±π/3 + 2πk, k ∈ Z

x = ±2π/3 + 4πk, k ∈ Z

cos(2x + π/3) = 1/2

2x + π/3 = ±arccos(1/2) + 2πk, k ∈ Z

2x + π/3 = ±π/3 + 2πk, k ∈ Z

2x = -π/3 ± π/3 + 2πk, k ∈ Z

x = -π/6 ± π/6 + πk, k ∈ Z

x₁ = -π/6 + π/6 + πk = πk, k ∈ Z

x₂ = -π/6 - π/6 + πk = -π/3 + πk, k ∈ Z

2sin(-x/2) + 1 = 0

sin(-x/2) = -1/2

-x/2 = arcsin(-1/2) + 2πk, k ∈ Z или -x/2 = π - arcsin(-1/2) + 2πk, k ∈ Z

-x/2 = -π/6 + 2πk, k ∈ Z или -x/2 = π + π/6 + 2πk, k ∈ Z

-x/2 = -π/6 + 2πk, k ∈ Z или -x/2 = 7π/6 + 2πk, k ∈ Z

x = π/3 - 4πk, k ∈ Z или x = -7π/3 - 4πk, k ∈ Z

2cos²x + 3cosx + 1 = 0

Пусть cosx = t, тогда 2t² + 3t + 1 = 0

D = 3² - 4 * 2 * 1 = 9 - 8 = 1

t₁ = (-3 + 1) / 4 = -2/4 = -1/2

t₂ = (-3 - 1) / 4 = -4/4 = -1

cosx = -1/2 или cosx = -1

x = ±arccos(-1/2) + 2πk, k ∈ Z или x = arccos(-1) + 2πk, k ∈ Z

x = ±2π/3 + 2πk, k ∈ Z или x = π + 2πk, k ∈ Z

4sin²x - 8cosx + 1 = 0

4(1 - cos²x) - 8cosx + 1 = 0

4 - 4cos²x - 8cosx + 1 = 0

-4cos²x - 8cosx + 5 = 0

4cos²x + 8cosx - 5 = 0

Пусть cosx = t, тогда 4t² + 8t - 5 = 0

D = 8² - 4 * 4 * (-5) = 64 + 80 = 144

t₁ = (-8 + 12) / 8 = 4/8 = 1/2

t₂ = (-8 - 12) / 8 = -20/8 = -5/2 (не подходит, так как |cosx| ≤ 1)

cosx = 1/2

x = ±arccos(1/2) + 2πk, k ∈ Z

x = ±π/3 + 2πk, k ∈ Z

sin²x + 2sinxcosx - 3cos²x = 0

Разделим обе части уравнения на cos²x (если cosx ≠ 0):

tg²x + 2tgx - 3 = 0

Пусть tgx = t, тогда t² + 2t - 3 = 0

D = 2² - 4 * 1 * (-3) = 4 + 12 = 16

t₁ = (-2 + 4) / 2 = 2 / 2 = 1

t₂ = (-2 - 4) / 2 = -6 / 2 = -3

tgx = 1 или tgx = -3

x = arctg(1) + πk, k ∈ Z или x = arctg(-3) + πk, k ∈ Z

x = π/4 + πk, k ∈ Z или x = -arctg(3) + πk, k ∈ Z

Если cosx = 0, то sin²x = 1, и уравнение принимает вид: 1 + 0 - 0 = 0, что неверно. Следовательно, cosx ≠ 0.

7cosx - 4sin2x = 0

7cosx - 4 * 2sinxcosx = 0

7cosx - 8sinxcosx = 0

cosx(7 - 8sinx) = 0

cosx = 0 или 7 - 8sinx = 0

x = π/2 + πk, k ∈ Z или sinx = 7/8

x = π/2 + πk, k ∈ Z или x = arcsin(7/8) + 2πk, k ∈ Z или x = π - arcsin(7/8) + 2πk, k ∈ Z