Resuelve las siguientes ecuaciones para 0 < x < 2 pi6?

Resuelve las siguientes ecuaciones para 0 < x < 2π

6.

A) 2cos x + 3 = 2

7.

B) sen3x - 2 = - 3sen3x

8.

C) senx(2 - senx) = cos2x

Resolvemos :

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<img src="https://tex.z-dn.net/?f=b%29%20%20%5C%5C%20%20%5C%5C%20sen%20%5Cleft%283x%5Cright%29-2%3D-3%20sen%20%5Cleft%283x%5Cright%29%5CRightarrow%5C%3A0%5C%20%5Ctextless%20%5C%20x%5C%20%5Ctextless%20%5C%202%5Cpi%20%20%5C%5C%20%20%5C%5C%20Hacemos%20%5C%20una%20%5C%20sustituci%5C%27on%3A%20%20%5C%5C%20%20%5C%5C%20u%3Dsen%283x%29%20%5C%5C%20%20%5C%5C%20Reescribimos%3A%20%5C%5C%20%20%5C%5C%20u-2%3D-3u%20%5C%5C%20%20%5C%5C%20Agrupamos%20%5C%20t%5C%27erminos%20%5C%20semejantes%3A%20%5C%5C%20%20%5C%5C%20u%2B3u%3D2%20%5C%5C%20%20%5C%5C%20Hacemos%20%5C%20las%20%5C%20operaciones%3A%20%5C%5C%20%20%5C%5C%204u%3D2%20%5C%5C%20%20%5C%5C%20u%3D%20%5Cfrac%7B4%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20u%3D2%20%5C%5C%20%20%5C%5C%20Regresamos%20%5C%20a%20%5C%20la%20%5C%20original%3A%20%5C%5C%20%20%5C%5C%20%20sen%20%5Cleft%283x%5Cright%29%3D%5Cfrac%7B1%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20sen%20%5Cleft%283x%5Cright%29%3D%5Cfrac%7B1%7D%7B2%7D%3A%5Cquad%203x%3D%5Cfrac%7B%5Cpi%20%7D%7B6%7D%2B2%5Cpi%20n%2C%5C%3A%5Cquad%203x%3D%5Cfrac%7B5%5Cpi%20%7D%7B6%7D%2B2%5Cpi%20n%20%5C%5C%20%20%5C%5C%20" />

<img src="https://tex.z-dn.net/?f=Resolvemos%3A%20%5C%5C%20%5C%5C%203x%3D%5Cfrac%7B%5Cpi%20%7D%7B6%7D%2B2%5Cpi%20n%20%5C%5C%20%5C%5C%20x%3D%5Cfrac%7B%5Cfrac%7B%5Cpi%20%7D%7B6%7D%7D%7B3%7D%2B%5Cfrac%7B2%5Cpi%20n%7D%7B3%7D%20%5C%5C%20%5C%5C%20x%3D%5Cfrac%7B%5Cfrac%7B%5Cpi%20%7D%7B6%7D%2B2%5Cpi%20n%7D%7B3%7D%20%5C%5C%20%5C%5C%20x%3D%5Cfrac%7B%5Cpi%20%2B12%5Cpi%20n%7D%7B3%5Ccdot%20%5C%3A6%7D%20%5C%5C%20%5C%5C%20x%3D%5Cfrac%7B%5Cpi%20%2B12%5Cpi%20n%7D%7B18%7D%20%5C%5C%20%5C%5C%20Ahora%20%5C%20resolvemos%20%5C%20este%20%5C%5C%20%5C%5C%203x%3D%5Cfrac%7B5%5Cpi%20%7D%7B6%7D%2B2%5Cpi%20n%20%5C%5C%20%5C%5C%20x%3D%5Cfrac%7B%5Cfrac%7B5%5Cpi%20%7D%7B6%7D%7D%7B3%7D%2B%5Cfrac%7B2%5Cpi%20n%7D%7B3%7D%20%5C%5C%20%5C%5C%20Mcm%3D%206%20%5C%5C%20%5C%5C%20x%3D%5Cfrac%7B5%5Cpi%20%2B2%5Ccdot%20%5C%3A6%5Cpi%20n%7D%7B6%7D%20%5C%5C%20%5C%5C%20x%3D%5Cfrac%7B%5Cfrac%7B5%5Cpi%20%2B12%5Cpi%20n%7D%7B6%7D%7D%7B3%7D%20%5C%5C%20%5C%5C%20x%3D%5Cfrac%7B5%5Cpi%20%2B12%5Cpi%20n%7D%7B18%7D%20%5C%5C%20%5C%5C%20Unimos%20%5C%5C%20%5C%5C%20" />

<img src="https://tex.z-dn.net/?f=x%3D%5Cfrac%7B%5Cpi%20%2B12%5Cpi%20n%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B5%5Cpi%20%2B12%5Cpi%20n%7D%7B18%7D%20%20%5C%5C%20%20%5C%5C%20Entonces%20%20%5C%5C%20%20%5C%5C%20%5Cboxed%7Bx%3D%5Cfrac%7B%5Cpi%20%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B5%5Cpi%20%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B13%5Cpi%20%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B17%5Cpi%20%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B25%5Cpi%20%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B29%5Cpi%20%7D%7B18%7D%7D%20%5C%20%5Csurd" /> - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

<img src="https://tex.z-dn.net/?f=c%29%20%5C%5C%20%20%5C%5C%20sen%20%5Cleft%28x%5Cright%29%5Cleft%282-sen%20%5Cleft%28x%5Cright%29%5Cright%29%3D%5Ccos%20%5Cleft%282x%5Cright%29%5CRightarrow%5C%3A0%5C%20%5Ctextless%20%5C%20x%5C%20%5Ctextless%20%5C%202%5Cpi%20%20%5C%5C%20%20%5C%5C%20sen%20%5Cleft%28x%5Cright%29%5Cleft%282-sen%20%5Cleft%28x%5Cright%29%5Cright%29-%5Ccos%20%5Cleft%282x%5Cright%29%3D0%20%20%5C%5C%20%20%5C%5C%20Usamos%20%5C%20est%5C%27a%20%5C%20identidad%3A%20%5C%5C%20%20%5C%5C%20%5Ccos%20%5Cleft%282x%5Cright%29%3D1-2sen%20%5E2%5Cleft%28x%5Cright%29%20%5C%5C%20%20%5C%5C%20Resolvemos%20%5C%5C%20%5C%5C%20-1%2B2sen%20%5E2%5Cleft%28x%5Cright%29%2B2sen%20%5Cleft%28x%5Cright%29-sen%20%5E2%5Cleft%28x%5Cright%29%20%5C%5C%20%20%5C%5C%20Sumamos%20%5C%20elementos%20%5C%20similares%3A%20%5C%5C%20%20%5C%5C%20sen%20%5E2%5Cleft%28x%5Cright%29%2B2sen%20%5Cleft%28x%5Cright%29-1%3D0%20%5C%5C%20%20%5C%5C%20Sustituci%5C%27on%3A%20%5C%5C%20%20%5C%5C%20u%3Dsen%28x%29%20%5C%5C%20" />

<img src="https://tex.z-dn.net/?f=Reescribimos%3A%20%5C%5C%20%5C%5C%20-1%2Bu%5E2%2B2u%3D0%20%5C%5C%20%20%5C%5C%20Utilizamos%20%5C%20la%20%5C%20formula%20%5C%20general%3A%20%20%5C%5C%20%20%5C%5C%20a%3D1%2C%20%5C%5C%20%5C%3Ab%3D2%2C%20%5C%5C%20%5C%3Ac%3D-1%20%5C%5C%20%20%5C%5C%20%5Cquad%20u%3D%5Cfrac%7B-2%5Cpm%20%5Csqrt%7B2%5E2-4%5Ccdot%20%5C%3A1%5Cleft%28-1%5Cright%29%7D%7D%7B2%5Ccdot%20%5C%3A1%7D%20%5C%5C%20%20%5C%5C%20Resolvemos%20%5C%20y%20%5C%20nos%20%5C%20queda%3A%20%5C%5C%20%20%5C%5C%20u%3D%5Csqrt%7B2%7D-1%2C%5C%3Au%3D-1-%5Csqrt%7B2%7D%20%5C%5C%20%20%5C%5C%20Volvemos%20%5C%20a%20%5C%20la%20%5C%20original%3A%20%5C%5C%20%20%5C%5C%20sen%20%5Cleft%28x%5Cright%29%3D%5Csqrt%7B2%7D-1%20%5C%5C%20%20%5C%5C%20Entonces%20%5C%5C%20%5C%5C%20sen%20%5Cleft%28x%5Cright%29%3Da%5Cquad%20%5CRightarrow%20%5Cquad%20%5C%3Ax%3Darcsen%20%5Cleft%28a%5Cright%29%2B2%5Cpi%20n%2C%5C%3A%5Cquad%20%5C%3Ax%3D%5Cpi%20-arcsen%20%5Cleft%28a%5Cright%29%2B2%5Cpi%20n%20%20" />

<img src="https://tex.z-dn.net/?f=Soluci%5C%27on%20%5C%5C%20%5C%5C%20%5Cboxed%7Bx%3D%5Cpi%20-arcsen%20%5Cleft%28%5Csqrt%7B2%7D-1%5Cright%29%2C%5C%3Ax%3Darcsen%20%5Cleft%28%5Csqrt%7B2%7D-1%5Cright%29%7D%20%5C%20%5Csurd" />

¡Espero haberte ayudado, saludos.

G. G.

H! .