Question

Sinx+sin5x=корень5cos2x

Answer 1

Sinx + sin5x = sqrt(5) * cos2x
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Sinx + sin5x  = 2 * sin ((x+5x)/2) * сos ((5x-x)/2) = 2 * sin3x * cos 2x
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2 * sin3x * cos 2x - sqrt(5) * cos2x = 0
cos2x * (2 * sin3x - sqrt(5)) = 0
1) cos2x = 0
2x = pi/2 + pi*n (n ∈ z)
x =  pi/4 + (pi*n)/2 (n ∈ z)
2)2 * sin3x - sqrt(5) = 0
sin3x = sqrt(5)/2
3x = (-1)^n * arcsin sqrt(5)/2 + pi*n (n ∈ z)
x = (-1)^n * (arcsin sqrt(5)/2)/3 + (pi*n)/3 (n ∈ z)

P.s. Точно в условии корень из 5 стоит?

Answer 2

Sinx + sin5x = sqrt(5) * cos2x
Sinx + sin5x  = 2 * sin ((x+5x)/2) * сos ((5x-x)/2) = 2 * sin3x * cos 2x
2 * sin3x * cos 2x - sqrt(5) * cos2x = 0
cos2x * (2 * sin3x - sqrt(5)) = 0
1) cos2x = 0
2x = pi/2 + pi*n (n ∈ z)
x =  pi/4 + (pi*n)/2 (n ∈ z)
2)2 * sin3x - sqrt(5) = 0
sin3x = sqrt(5)/2