Cos^2x=sin^2 (x/2). 13 задание егэ математика профиль

Cos ² x=Sin ² (x/2)

(1 - 2Sin ²x/2) ² - Sin²x/2 = 0

1 - 4Sin²x/2 + 4Sin⁴x/2 - Sin²x/2 = 0

4Sin⁴x/2 - 5Sin²x/2 + 1 = 0

Sin²x/2 = t

4t² - 5t + 1 = 0

D = 25 - 16 = 9

t₁ = 1 t₂=1/4

Sin²x/2 = 1 Sin²x/2 = 1/4

Sinx/2 = + - 1 Sinx/2 = + - 1/2

а) Sinx/2 = 1

x/2 = π/2 + 2πk, k ∈Z

x = π + 4πk, k ∈Z

б) Sin x/2 = - 1

x/2 = - π/2 + 2πn, n∈Z

x = - π + 4πn, n∈Z

в) Sinx/2 = 1/2

x/2 = (-1) ^m * π/6 + mπ, m ∈Z

x = (-1) ^m*π/3 + 2πm, m∈Z

г) Sinx/2 = - 1/2

x/2 = (-1) ^ (c+1) * π/6 + cπ, c ∈ Z

x = (-1) ^ (c+1) * π/3 + 2cπ, c∈Z