Question

sin^2(2x) + sin^2(4x) = 1

Answer 1

Ответ:

x={π/4+kπ/2; ±π/12+kπ/2}, k∈Z

Объяснение:

sin²2x + sin²4x = 1

Формула: sin²x=(1-cos2x)/2; sin²x+cos²x=1

sin²2x=(1-cos4x)/2; sin²4x=1-cos²4x

(1-cos4x)/2+(1-cos²4x)=1

1-cos4x+2-2cos²4x=2

2cos²4x+cos4x-1=0

cos4x=t⇒|t|≤1

2t²+t-1=0

D=1-4·2·(-1)=9=3²

t₁=(-1-3)/4=-1

t₂=(-1+3)/4=0,5

1) cos4x=-1

4x=π+2kπ

x=π/4+kπ/2, k∈Z

2) cos4x=0,5

4x=±π/3+2kπ

x=±π/12+kπ/2, k∈Z