Verified answer

znanija.com/task/34452584

* * * * * * * * * * * * * * * * * * * * * * *

Решить уравнение√2cos(8x)cos(x+π/4)= 2cos(π /4)

Ответ:   x = - π/4+2πk , k∈ ℤ .

Объяснение:

√2cos(8x)cos(x+π/4)= 2cos(π /4) ⇔√2cos(8x)cos(x+π/4)= 2*√2/2⇔

cos(8x)cos(x+π/4)= 1 .

a) { cos(8x)= -1 ; cos(x+π/4) = -1 .

{ 8x =π+2πn  ; x+π/4 =π+2πk.⇔{ x =π/8+(π/4)*n ; x=3π/4+2πk. (n , k ∈ℤ)

⇒ x ∈ ∅ .

* * *π/8+(π/4)*n=3π/4+2πk ⇔1+2n =6+8k⇔ n - 4k =2,5 невозможно * * *

b) { cos(8x)= 1 ; cos(x+π/4) =1 .

{ 8x =2πn ; x+π/4 =2πk.    (n , k ∈ℤ)

{ x =(π/4)*n  ; x= - π/4+2πk. ⇒ x = - π/4+2πk  k∈ ℤ .