Решить уравнение √2 cos2x = cosx+sinx...

Мигель7 @Мигель7

August 2022 1 12 Report

Решить уравнение
√2 cos2x = cosx+sinx

Answers & Comments

Dимасuk

Verified answer

√2cos2x = cosx+sinx
√2(cos²x - sin²x) - (cosx + sinx) = 0
√2(sinx + cosx)(cosx - sinx) - (cosx + sinx) = 0
(sinx + cosx)(√2cosx - √2sinx - 1) = 0
1) sinx + cosx = 0
sinx = -cosx
tgx = -1
x = -π/4 + πn, n ∈ Z
2) √2cosx - √2sinx - 1 = 0
√2cosx - √2sinx = 1
√2/2cosx - √2/2sinx = 1/2
cosx·cos(arccos(√2/2) - sinx·sin(arccos(√2/2)) = 1/2
cos(x + arccos(√2/2)) = 1/2
cosx(x + π/4) = 1/2
x + π/4 = ±π/3 + 2πk, k ∈ Z
x = ± π/3 - π/4 + 2πk, k ∈ Z
Ответ: x = -π/4 + πn, n ∈ Z; ± π/3 - π/4 + 2πk, k ∈ Z.

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