Resuelve las siguientes ecuaciones para 0 < x < 2Pi
3. a) 2cos x + 3 = 2
4. b) sen3x - 2 = -3sen3x
5. c) senx(2 - senx) = cos2x
Resuelve las siguientes ecuaciones para 0 < x < 2Pi
6. a) 2cos x + 3 = 2
7. b) sen3x - 2 = -3sen3x
8. c) senx(2 - senx) = cos2x
Respuestas
Respuesta dada por:
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Hola.
![1) \\ 2\cos \left(x\right)+3=2,\:0\ \textless \ x\ \textless \ 2\pi \\ \\ 2\cos \left(x\right)=2-3 \\ \\ 2\cos \left(x\right)=-1 \\ \\ \cos \left(x\right)=-\frac{1}{2} \\ \\ x=\frac{2\pi }{3}+2\pi n,\:x=\frac{4\pi }{3}+2\pi n \\ \\ \boxed{x=\frac{2\pi }{3},\:x=\frac{4\pi }{3}} 1) \\ 2\cos \left(x\right)+3=2,\:0\ \textless \ x\ \textless \ 2\pi \\ \\ 2\cos \left(x\right)=2-3 \\ \\ 2\cos \left(x\right)=-1 \\ \\ \cos \left(x\right)=-\frac{1}{2} \\ \\ x=\frac{2\pi }{3}+2\pi n,\:x=\frac{4\pi }{3}+2\pi n \\ \\ \boxed{x=\frac{2\pi }{3},\:x=\frac{4\pi }{3}}](https://tex.z-dn.net/?f=1%29++%5C%5C+2%5Ccos+%5Cleft%28x%5Cright%29%2B3%3D2%2C%5C%3A0%5C+%5Ctextless+%5C+x%5C+%5Ctextless+%5C+2%5Cpi++%5C%5C++%5C%5C+2%5Ccos+%5Cleft%28x%5Cright%29%3D2-3+%5C%5C++%5C%5C+2%5Ccos+%5Cleft%28x%5Cright%29%3D-1+%5C%5C++%5C%5C+%5Ccos+%5Cleft%28x%5Cright%29%3D-%5Cfrac%7B1%7D%7B2%7D+%5C%5C++%5C%5C+x%3D%5Cfrac%7B2%5Cpi+%7D%7B3%7D%2B2%5Cpi+n%2C%5C%3Ax%3D%5Cfrac%7B4%5Cpi+%7D%7B3%7D%2B2%5Cpi+n+%5C%5C++%5C%5C+%5Cboxed%7Bx%3D%5Cfrac%7B2%5Cpi+%7D%7B3%7D%2C%5C%3Ax%3D%5Cfrac%7B4%5Cpi+%7D%7B3%7D%7D)
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![2) \\ \\ \sin \left(3x\right)-2=-3\sin \left(3x\right),\:0\ \textless \ x\ \textless \ 2\pi \\ \\ Sustitucion: \sin \left(3x\right)=u \\ \\ u-2=-3u \\ \\ u=-3u+2 \\ \\ u+3u=2 \\ \\ 4u=2 \\ \\ \frac{4u}{4}=\frac{2}{4} \\ \\ u=\frac{1}{2} \\ \\ Volvemos \ a \ la \ original: \\ \\ \sin \left(3x\right)=\frac{1}{2} \\ \\ \sin \left(3x\right)=\frac{1}{2},\:0\ \textless \ x\ \textless \ 2\pi \\ \\ \boxed{x=\frac{\pi }{18},\:x=\frac{5\pi }{18},\:x=\frac{17\pi }{18},\:x=\frac{29\pi }{18},\:x=\frac{13\pi }{18},\:x=\frac{25\pi }{18}} 2) \\ \\ \sin \left(3x\right)-2=-3\sin \left(3x\right),\:0\ \textless \ x\ \textless \ 2\pi \\ \\ Sustitucion: \sin \left(3x\right)=u \\ \\ u-2=-3u \\ \\ u=-3u+2 \\ \\ u+3u=2 \\ \\ 4u=2 \\ \\ \frac{4u}{4}=\frac{2}{4} \\ \\ u=\frac{1}{2} \\ \\ Volvemos \ a \ la \ original: \\ \\ \sin \left(3x\right)=\frac{1}{2} \\ \\ \sin \left(3x\right)=\frac{1}{2},\:0\ \textless \ x\ \textless \ 2\pi \\ \\ \boxed{x=\frac{\pi }{18},\:x=\frac{5\pi }{18},\:x=\frac{17\pi }{18},\:x=\frac{29\pi }{18},\:x=\frac{13\pi }{18},\:x=\frac{25\pi }{18}}](https://tex.z-dn.net/?f=2%29++%5C%5C++%5C%5C+%5Csin+%5Cleft%283x%5Cright%29-2%3D-3%5Csin+%5Cleft%283x%5Cright%29%2C%5C%3A0%5C+%5Ctextless+%5C+x%5C+%5Ctextless+%5C+2%5Cpi++%5C%5C++%5C%5C+Sustitucion%3A+%5Csin+%5Cleft%283x%5Cright%29%3Du+%5C%5C++%5C%5C+u-2%3D-3u+%5C%5C++%5C%5C+u%3D-3u%2B2+%5C%5C++%5C%5C+u%2B3u%3D2+%5C%5C++%5C%5C+4u%3D2+%5C%5C++%5C%5C+%5Cfrac%7B4u%7D%7B4%7D%3D%5Cfrac%7B2%7D%7B4%7D+%5C%5C++%5C%5C+u%3D%5Cfrac%7B1%7D%7B2%7D+%5C%5C++%5C%5C+Volvemos+%5C+a+%5C+la+%5C+original%3A+%5C%5C++%5C%5C+%5Csin+%5Cleft%283x%5Cright%29%3D%5Cfrac%7B1%7D%7B2%7D+%5C%5C++%5C%5C+%5Csin+%5Cleft%283x%5Cright%29%3D%5Cfrac%7B1%7D%7B2%7D%2C%5C%3A0%5C+%5Ctextless+%5C+x%5C+%5Ctextless+%5C+2%5Cpi++%5C%5C++%5C%5C+%5Cboxed%7Bx%3D%5Cfrac%7B%5Cpi+%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B5%5Cpi+%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B17%5Cpi+%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B29%5Cpi+%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B13%5Cpi+%7D%7B18%7D%2C%5C%3Ax%3D%5Cfrac%7B25%5Cpi+%7D%7B18%7D%7D)
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![3) \\ \\ \sin \left(x\right)\left(2-\sin \left(x\right)\right)=\cos \left(2x\right),\:0\ \textless \ x\ \textless \ 2\pi \\ \\ \sin \left(x\right)\left(2-\sin \left(x\right)\right)-\cos \left(2x\right)=0 \\ \\ Usamos: \cos \left(2x\right)=1-2\sin ^2\left(x\right) \\ \\ -\left(1-2\sin ^2\left(x\right)\right)+\left(2-\sin \left(x\right)\right)\sin \left(x\right)=0 \\ \\ -1+\sin ^2\left(x\right)+2\sin \left(x\right)=0 \\ \\ Sustituimos: \sin \left(x\right)=u \\ \\ -1+u^2+2u=0 \\ \\ 3) \\ \\ \sin \left(x\right)\left(2-\sin \left(x\right)\right)=\cos \left(2x\right),\:0\ \textless \ x\ \textless \ 2\pi \\ \\ \sin \left(x\right)\left(2-\sin \left(x\right)\right)-\cos \left(2x\right)=0 \\ \\ Usamos: \cos \left(2x\right)=1-2\sin ^2\left(x\right) \\ \\ -\left(1-2\sin ^2\left(x\right)\right)+\left(2-\sin \left(x\right)\right)\sin \left(x\right)=0 \\ \\ -1+\sin ^2\left(x\right)+2\sin \left(x\right)=0 \\ \\ Sustituimos: \sin \left(x\right)=u \\ \\ -1+u^2+2u=0 \\ \\](https://tex.z-dn.net/?f=3%29++%5C%5C++%5C%5C+%5Csin+%5Cleft%28x%5Cright%29%5Cleft%282-%5Csin+%5Cleft%28x%5Cright%29%5Cright%29%3D%5Ccos+%5Cleft%282x%5Cright%29%2C%5C%3A0%5C+%5Ctextless+%5C+x%5C+%5Ctextless+%5C+2%5Cpi++%5C%5C++%5C%5C+%5Csin+%5Cleft%28x%5Cright%29%5Cleft%282-%5Csin+%5Cleft%28x%5Cright%29%5Cright%29-%5Ccos+%5Cleft%282x%5Cright%29%3D0+%5C%5C++%5C%5C+Usamos%3A+%5Ccos+%5Cleft%282x%5Cright%29%3D1-2%5Csin+%5E2%5Cleft%28x%5Cright%29+%5C%5C++%5C%5C+-%5Cleft%281-2%5Csin+%5E2%5Cleft%28x%5Cright%29%5Cright%29%2B%5Cleft%282-%5Csin+%5Cleft%28x%5Cright%29%5Cright%29%5Csin+%5Cleft%28x%5Cright%29%3D0+%5C%5C++%5C%5C+-1%2B%5Csin+%5E2%5Cleft%28x%5Cright%29%2B2%5Csin+%5Cleft%28x%5Cright%29%3D0+%5C%5C++%5C%5C+Sustituimos%3A+%5Csin+%5Cleft%28x%5Cright%29%3Du+%5C%5C++%5C%5C+-1%2Bu%5E2%2B2u%3D0+%5C%5C++%5C%5C+)
![Ecuacion \ general: \\ \\ \frac{-2+\sqrt{2^2-4\cdot \:1\cdot \left(-1\right)}}{2\cdot \:1} \\ \\ Soluciones: \\ \\ u=\sqrt{2}-1,\:u=-1-\sqrt{2} \\ \\ Cambiamos \ a \ la \ original: \\ \\ \sin \left(x\right)=\sqrt{2}-1,\:\sin \left(x\right)=-1-\sqrt{2} \\ \\ \boxed{x=\pi -\arcsin \left(\sqrt{2}-1\right),\:x=\arcsin \left(\sqrt{2}-1\right)} Ecuacion \ general: \\ \\ \frac{-2+\sqrt{2^2-4\cdot \:1\cdot \left(-1\right)}}{2\cdot \:1} \\ \\ Soluciones: \\ \\ u=\sqrt{2}-1,\:u=-1-\sqrt{2} \\ \\ Cambiamos \ a \ la \ original: \\ \\ \sin \left(x\right)=\sqrt{2}-1,\:\sin \left(x\right)=-1-\sqrt{2} \\ \\ \boxed{x=\pi -\arcsin \left(\sqrt{2}-1\right),\:x=\arcsin \left(\sqrt{2}-1\right)}](https://tex.z-dn.net/?f=Ecuacion+%5C+general%3A++%5C%5C++%5C%5C+%5Cfrac%7B-2%2B%5Csqrt%7B2%5E2-4%5Ccdot+%5C%3A1%5Ccdot+%5Cleft%28-1%5Cright%29%7D%7D%7B2%5Ccdot+%5C%3A1%7D+%5C%5C++%5C%5C+Soluciones%3A+%5C%5C++%5C%5C++u%3D%5Csqrt%7B2%7D-1%2C%5C%3Au%3D-1-%5Csqrt%7B2%7D+%5C%5C++%5C%5C+Cambiamos+%5C++a+%5C+la+%5C+original%3A+%5C%5C++%5C%5C+%5Csin+%5Cleft%28x%5Cright%29%3D%5Csqrt%7B2%7D-1%2C%5C%3A%5Csin+%5Cleft%28x%5Cright%29%3D-1-%5Csqrt%7B2%7D+%5C%5C++%5C%5C++%5Cboxed%7Bx%3D%5Cpi+-%5Carcsin+%5Cleft%28%5Csqrt%7B2%7D-1%5Cright%29%2C%5C%3Ax%3D%5Carcsin+%5Cleft%28%5Csqrt%7B2%7D-1%5Cright%29%7D)
¡Espero haberte ayudado, saludos... G.G.H!
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¡Espero haberte ayudado, saludos... G.G.H!
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