Se da la desigualdad:
∣sin(x)∣+∣cos(x)∣>1Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣sin(x)∣+∣cos(x)∣=1Resolvemos:
x1=−59.6902604182061x2=87.9645943005142x3=73.8274273593601x4=−37.6991118430775x5=95.8185759344887x6=−81.6814089933346x7=−21.9911485751286x8=−58.1194640914112x9=−23.5619449019235x10=−15.707963267949x11=−89.5353906273091x12=26.7035375555132x13=−17.2787595947439x14=−87.9645943005142x15=4393.51732604533x16=72.2566310325652x17=42.4115008234622x18=−7.85398163397448x19=−14.1371669411541x20=−94.2477796076938x21=6.28318530717959x22=−1.5707963267949x23=1.5707963267949x24=29.845130209103x25=−51.8362787842316x26=−29.845130209103x27=65.9734457253857x28=0x29=−50.2654824574367x30=−18.8495559215388x31=−48.6946861306418x32=−73.8274273593601x33=−43.9822971502571x34=20.4203522483337x35=51.8362787842316x36=43.9822971502571x37=−67.5442420521806x38=−65.9734457253857x39=−45.553093477052x40=−28.2743338823081x41=−95.8185759344887x42=−32.9867228626928x43=−72.2566310325652x44=28.2743338823081x45=−36.1283155162826x46=21.9911485751286x47=7.85398163397448x48=50.2654824574367x49=−80.1106126665397x50=94.2477796076938x51=86.3937979737193x52=64.4026493985908x1=−59.6902604182061x2=87.9645943005142x3=73.8274273593601x4=−37.6991118430775x5=95.8185759344887x6=−81.6814089933346x7=−21.9911485751286x8=−58.1194640914112x9=−23.5619449019235x10=−15.707963267949x11=−89.5353906273091x12=26.7035375555132x13=−17.2787595947439x14=−87.9645943005142x15=4393.51732604533x16=72.2566310325652x17=42.4115008234622x18=−7.85398163397448x19=−14.1371669411541x20=−94.2477796076938x21=6.28318530717959x22=−1.5707963267949x23=1.5707963267949x24=29.845130209103x25=−51.8362787842316x26=−29.845130209103x27=65.9734457253857x28=0x29=−50.2654824574367x30=−18.8495559215388x31=−48.6946861306418x32=−73.8274273593601x33=−43.9822971502571x34=20.4203522483337x35=51.8362787842316x36=43.9822971502571x37=−67.5442420521806x38=−65.9734457253857x39=−45.553093477052x40=−28.2743338823081x41=−95.8185759344887x42=−32.9867228626928x43=−72.2566310325652x44=28.2743338823081x45=−36.1283155162826x46=21.9911485751286x47=7.85398163397448x48=50.2654824574367x49=−80.1106126665397x50=94.2477796076938x51=86.3937979737193x52=64.4026493985908Las raíces dadas
x41=−95.8185759344887x20=−94.2477796076938x11=−89.5353906273091x14=−87.9645943005142x6=−81.6814089933346x49=−80.1106126665397x32=−73.8274273593601x43=−72.2566310325652x37=−67.5442420521806x38=−65.9734457253857x1=−59.6902604182061x8=−58.1194640914112x25=−51.8362787842316x29=−50.2654824574367x31=−48.6946861306418x39=−45.553093477052x33=−43.9822971502571x4=−37.6991118430775x45=−36.1283155162826x42=−32.9867228626928x26=−29.845130209103x40=−28.2743338823081x9=−23.5619449019235x7=−21.9911485751286x30=−18.8495559215388x13=−17.2787595947439x10=−15.707963267949x19=−14.1371669411541x18=−7.85398163397448x22=−1.5707963267949x28=0x23=1.5707963267949x21=6.28318530717959x47=7.85398163397448x34=20.4203522483337x46=21.9911485751286x12=26.7035375555132x44=28.2743338823081x24=29.845130209103x17=42.4115008234622x36=43.9822971502571x48=50.2654824574367x35=51.8362787842316x52=64.4026493985908x27=65.9734457253857x16=72.2566310325652x3=73.8274273593601x51=86.3937979737193x2=87.9645943005142x50=94.2477796076938x5=95.8185759344887x15=4393.51732604533son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x41Consideremos, por ejemplo, el punto
x0=x41−101=
−95.8185759344887+−101=
−95.9185759344887lo sustituimos en la expresión
∣sin(x)∣+∣cos(x)∣>1∣cos(−95.9185759344887)∣+∣sin(−95.9185759344887)∣>11.09483758192485 > 1
significa que una de las soluciones de nuestra ecuación será con:
x<−95.8185759344887 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------
x41 x20 x11 x14 x6 x49 x32 x43 x37 x38 x1 x8 x25 x29 x31 x39 x33 x4 x45 x42 x26 x40 x9 x7 x30 x13 x10 x19 x18 x22 x28 x23 x21 x47 x34 x46 x12 x44 x24 x17 x36 x48 x35 x52 x27 x16 x3 x51 x2 x50 x5 x15Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x<−95.8185759344887x>−94.2477796076938∧x<−89.5353906273091x>−87.9645943005142∧x<−81.6814089933346x>−80.1106126665397∧x<−73.8274273593601x>−72.2566310325652∧x<−67.5442420521806x>−65.9734457253857∧x<−59.6902604182061x>−58.1194640914112∧x<−51.8362787842316x>−50.2654824574367∧x<−48.6946861306418x>−45.553093477052∧x<−43.9822971502571x>−37.6991118430775∧x<−36.1283155162826x>−32.9867228626928∧x<−29.845130209103x>−28.2743338823081∧x<−23.5619449019235x>−21.9911485751286∧x<−18.8495559215388x>−17.2787595947439∧x<−15.707963267949x>−14.1371669411541∧x<−7.85398163397448x>−1.5707963267949∧x<0x>1.5707963267949∧x<6.28318530717959x>7.85398163397448∧x<20.4203522483337x>21.9911485751286∧x<26.7035375555132x>28.2743338823081∧x<29.845130209103x>42.4115008234622∧x<43.9822971502571x>50.2654824574367∧x<51.8362787842316x>64.4026493985908∧x<65.9734457253857x>72.2566310325652∧x<73.8274273593601x>86.3937979737193∧x<87.9645943005142x>94.2477796076938∧x<95.8185759344887x>4393.51732604533
-x^2-2x+3>0
x^2>9
x^2-10x<0
(x^2-x-1)*(x^2-x-7)<-5