Se da la desigualdad:
((−2sin(x)+3sin(2x))−3cos(x))+4>0Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
((−2sin(x)+3sin(2x))−3cos(x))+4=0Resolvemos:
x1=−57.2326136979463x2=187.811613282058x3=68.8001327840634x4=−1873.07316747285x5=−88.6485402338442x6=68.4310924456455x7=−38.0140174379896x8=−94.5626852026059x9=24.4487952953884x10=−19.5335018548687x11=−94.9317255410238x12=5.96827971226751x13=74.7142777528251x14=100.216059319961x15=−69.7989843123054x16=87.6496887056021x17=30.731980602568x18=93.5638336743638x19=−44.6662430835871x20=−163.677723581581x21=5.59923937384961x22=−50.9494283907667x23=80.9974630600046x24=43.667391555345x25=−6.59809090209166x26=37.3842062481654x27=−50.5803880523488x28=18.1656099882088x29=−75.7131292810671x30=−81.9963145882467x31=62.1479071384659x32=81.3665033984226x33=62.5169474768838x34=−31.73083213081x35=49.9505768625246x36=43.2983512169271x37=872.678811764633x38=−151.111352967222x39=87.2806483671842x40=11.8824246810292x41=−220.226391346198x42=−1703.427164179x43=93.9328740127817x44=12.2514650194471x45=−25.8166871620483x46=−6.96713124050957x47=−107.498096155383x48=−12.8812762092712x49=−13.2503165476892x50=56.2337621697042x51=−0.683945933329981x52=37.0151659097475x53=−63.5157990051258x54=−88.2794998954263x55=−201.376835424659x56=−44.2972027451692x57=18.5346503266267x58=−0.314905594912077x59=−578.367953855434x1=−57.2326136979463x2=187.811613282058x3=68.8001327840634x4=−1873.07316747285x5=−88.6485402338442x6=68.4310924456455x7=−38.0140174379896x8=−94.5626852026059x9=24.4487952953884x10=−19.5335018548687x11=−94.9317255410238x12=5.96827971226751x13=74.7142777528251x14=100.216059319961x15=−69.7989843123054x16=87.6496887056021x17=30.731980602568x18=93.5638336743638x19=−44.6662430835871x20=−163.677723581581x21=5.59923937384961x22=−50.9494283907667x23=80.9974630600046x24=43.667391555345x25=−6.59809090209166x26=37.3842062481654x27=−50.5803880523488x28=18.1656099882088x29=−75.7131292810671x30=−81.9963145882467x31=62.1479071384659x32=81.3665033984226x33=62.5169474768838x34=−31.73083213081x35=49.9505768625246x36=43.2983512169271x37=872.678811764633x38=−151.111352967222x39=87.2806483671842x40=11.8824246810292x41=−220.226391346198x42=−1703.427164179x43=93.9328740127817x44=12.2514650194471x45=−25.8166871620483x46=−6.96713124050957x47=−107.498096155383x48=−12.8812762092712x49=−13.2503165476892x50=56.2337621697042x51=−0.683945933329981x52=37.0151659097475x53=−63.5157990051258x54=−88.2794998954263x55=−201.376835424659x56=−44.2972027451692x57=18.5346503266267x58=−0.314905594912077x59=−578.367953855434Las raíces dadas
x4=−1873.07316747285x42=−1703.427164179x59=−578.367953855434x41=−220.226391346198x55=−201.376835424659x20=−163.677723581581x38=−151.111352967222x47=−107.498096155383x11=−94.9317255410238x8=−94.5626852026059x5=−88.6485402338442x54=−88.2794998954263x30=−81.9963145882467x29=−75.7131292810671x15=−69.7989843123054x53=−63.5157990051258x1=−57.2326136979463x22=−50.9494283907667x27=−50.5803880523488x19=−44.6662430835871x56=−44.2972027451692x7=−38.0140174379896x34=−31.73083213081x45=−25.8166871620483x10=−19.5335018548687x49=−13.2503165476892x48=−12.8812762092712x46=−6.96713124050957x25=−6.59809090209166x51=−0.683945933329981x58=−0.314905594912077x21=5.59923937384961x12=5.96827971226751x40=11.8824246810292x44=12.2514650194471x28=18.1656099882088x57=18.5346503266267x9=24.4487952953884x17=30.731980602568x52=37.0151659097475x26=37.3842062481654x36=43.2983512169271x24=43.667391555345x35=49.9505768625246x50=56.2337621697042x31=62.1479071384659x33=62.5169474768838x6=68.4310924456455x3=68.8001327840634x13=74.7142777528251x23=80.9974630600046x32=81.3665033984226x39=87.2806483671842x16=87.6496887056021x18=93.5638336743638x43=93.9328740127817x14=100.216059319961x2=187.811613282058x37=872.678811764633son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x4Consideremos, por ejemplo, el punto
x0=x4−101=
−1873.07316747285+−101=
−1873.17316747285lo sustituimos en la expresión
((−2sin(x)+3sin(2x))−3cos(x))+4>0(−3cos(−1873.17316747285)+(3sin((−1873.17316747285)2)−2sin(−1873.17316747285)))+4>00.287772211433395 > 0
significa que una de las soluciones de nuestra ecuación será con:
x<−1873.07316747285 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x4 x42 x59 x41 x55 x20 x38 x47 x11 x8 x5 x54 x30 x29 x15 x53 x1 x22 x27 x19 x56 x7 x34 x45 x10 x49 x48 x46 x25 x51 x58 x21 x12 x40 x44 x28 x57 x9 x17 x52 x26 x36 x24 x35 x50 x31 x33 x6 x3 x13 x23 x32 x39 x16 x18 x43 x14 x2 x37Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x<−1873.07316747285x>−1703.427164179∧x<−578.367953855434x>−220.226391346198∧x<−201.376835424659x>−163.677723581581∧x<−151.111352967222x>−107.498096155383∧x<−94.9317255410238x>−94.5626852026059∧x<−88.6485402338442x>−88.2794998954263∧x<−81.9963145882467x>−75.7131292810671∧x<−69.7989843123054x>−63.5157990051258∧x<−57.2326136979463x>−50.9494283907667∧x<−50.5803880523488x>−44.6662430835871∧x<−44.2972027451692x>−38.0140174379896∧x<−31.73083213081x>−25.8166871620483∧x<−19.5335018548687x>−13.2503165476892∧x<−12.8812762092712x>−6.96713124050957∧x<−6.59809090209166x>−0.683945933329981∧x<−0.314905594912077x>5.59923937384961∧x<5.96827971226751x>11.8824246810292∧x<12.2514650194471x>18.1656099882088∧x<18.5346503266267x>24.4487952953884∧x<30.731980602568x>37.0151659097475∧x<37.3842062481654x>43.2983512169271∧x<43.667391555345x>49.9505768625246∧x<56.2337621697042x>62.1479071384659∧x<62.5169474768838x>68.4310924456455∧x<68.8001327840634x>74.7142777528251∧x<80.9974630600046x>81.3665033984226∧x<87.2806483671842x>87.6496887056021∧x<93.5638336743638x>93.9328740127817∧x<100.216059319961x>187.811613282058∧x<872.678811764633
(x-6)*(x-14)>0
4x^2-4x+1<0
2^x>=2
x/(x-3)>1