Question

a expresão sen (3x)+2sen(2x)+senx é equivalente a 

Answer 1

(sen3x)/(senx) - (cos3x)/(cosx)

1º) sen(3x) e cos(3x):

sen(3x) = sen(2x+x) = sen(2x) cos(x) + cos(2x) sen(x) = 2 sen(x) cos(x) cos(x) + cos^2(x) sen(x) – sen^2(x) sen(x) = 3 cos^2(x) sen(x) – sen^3(x)

cos(3x) = cos(2x+x) = cos(2x) cos(x) – sen(2x) sen(x) = (cos^2(x) – sen^2(x))cos(x) – 2 sen(x) cos(x) sen(x) = cos^3(x)-3cos(x) sen^2(x)

Sustendo:

3cos^2(x) sin(x)-sin^3(x)…. cos^3(x)-3cos(x)sin^2(x)
------------------------------- - -------------------------------
....sin(x) ………………………………cos(x)

3 cos^2x – sin^2(x) – cos^2(x) + 3 sin^2(x) =

2 cos^2(x) + 2 sin^2(x) = 2( sen^2(x) + cos^2(x) = 2