Question

si senx + cosx = 1/2 hallar k=cos3x - sen3x

Answer 1

$K=\cos 3x-\sin 3x\\ \\ K=(4\cos^3x-3\cos x)-(3\sin x-4\sin^3x)\\ \\ K=4(\cos^3x+\sin^3x)-3(\sin x+\cos x)\\ \\ K=4(\cos x+\sin x)(\cos^2x-\sin x\cos x+\sin^2 x)-\dfrac{3}{2}\\ \\ K=2(1-\sin x\cos x)-\dfrac{3}{2}$

Por otra parte

$\sin x+\cos x = 1/2\\ \\ (\sin x+\cos x)^2=1/4\\ \\ 1+2\sin x\cos x=1/4\\ \\ -\sin x\cos x = 3/8$

entonces

$K=2(1+3/8)-3/2\\ \\ \boxed{K=\dfrac{5}{4}}$

Answer 2

senx + cosx = 1/2
k = cos3x - sen3x

k = cons3x - sen3x

k = cos(2x + x) - sen(2x + x)
k = cos2x cosx - sen2x senx - sen2x cosx - cos2x senx
k = cos2x cosx - cos2x senx - sen2x senx - sen2x cosx
k = cos2x (cosx - senx) - sen2x (senx + cosx)
k = (cos^2 x - sen^2 x)(cosx - senx) - 2senx cosx (senx + cosx)
k = (cosx + senx)(cosx - senx) (cosx - senx) - 2 senx cosx (senx + cosx)
k = (cosx + senx) [(cosx - senx)(cosx - senx) - 2 cosxsenx]
k = (cosx + senx) (cos^2 x - 2cosx + sen^2 - 2 cosx senx)
k = (cosx + senx) (1 - 4 cosx senx)
k = (1/2) (1 - 4 cosx senx)
k = (1/2) - 2 cosx senx

senx + cosx = 1/2            Elevas al cuadrado ambos lados para no alterar
(senx + cosx)^2 = (1/2)^2
sen^2 x + 2senx cosx + cos^2 x = 1/4
1 + 2 senx cosx = 1/4
2 senx cosx = (1/4) - 1
2 senx cosx = -3/4

k = (1/2) - (-3/4)
k = (1/2) + (3/4)
k =   5/4