|sin(x)|<1 desigualdades

Se da la desigualdad:
$$\left|{\sin{\left(x \right)}}\right| < 1$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)}}\right| = 1$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)}}\right| = 1$$
cambiamos
$$\left|{\sin{\left(x \right)}}\right| - 1 = 0$$
$$\left|{\sin{\left(x \right)}}\right| - 1 = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)}}\right|$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = 1$$
Obtenemos la respuesta: w = 1
hacemos cambio inverso
$$\left|{\sin{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = 51.8362789031518$$
$$x_{2} = -17.2787590920677$$
$$x_{3} = -36.1283154173375$$
$$x_{4} = -39.2699084145515$$
$$x_{5} = -64.4026497466373$$
$$x_{6} = -48.6946857788076$$
$$x_{7} = 76.9690204681432$$
$$x_{8} = 83.2522056907544$$
$$x_{9} = 1.57079657289894$$
$$x_{10} = 83.252204888767$$
$$x_{11} = 48.6946866365921$$
$$x_{12} = 23.5619444059921$$
$$x_{13} = 54.9778710948428$$
$$x_{14} = 1.57079582971902$$
$$x_{15} = 45.5530929823099$$
$$x_{16} = -76.9690203748894$$
$$x_{17} = -98.9601689530982$$
$$x_{18} = -32.9867224188086$$
$$x_{19} = -51.8362786893284$$
$$x_{20} = -10.9955738413568$$
$$x_{21} = 70.6858352127237$$
$$x_{22} = 32.9867225164981$$
$$x_{23} = 76.9690196732095$$
$$x_{24} = 20.4203521477756$$
$$x_{25} = -98.9601681513438$$
$$x_{26} = -4.71238942125338$$
$$x_{27} = -58.1194639976905$$
$$x_{28} = 36.128315618579$$
$$x_{29} = 4.7123894841958$$
$$x_{30} = -92.6769829355125$$
$$x_{31} = 67.54424230971$$
$$x_{32} = 32.9867233134552$$
$$x_{33} = 98.9601690454399$$
$$x_{34} = 10.9955739381756$$
$$x_{35} = -1.57079643188553$$
$$x_{36} = -89.535390750197$$
$$x_{37} = 61.2610563112167$$
$$x_{38} = 64.4026493072124$$
$$x_{39} = -23.5619450115115$$
$$x_{40} = 89.5353901350773$$
$$x_{41} = -32.9867232184024$$
$$x_{42} = 54.9778718908148$$
$$x_{43} = -4.71238862219396$$
$$x_{44} = 67.5442415586719$$
$$x_{45} = -48.6946865760795$$
$$x_{46} = -45.5530935911043$$
$$x_{47} = 4.71238874329685$$
$$x_{48} = 73.8274274830848$$
$$x_{49} = -54.9778709962906$$
$$x_{50} = 80.1106131368654$$
$$x_{51} = -83.2522055723275$$
$$x_{52} = -73.8274272798455$$
$$x_{53} = 45.553093730794$$
$$x_{54} = 58.1194643979608$$
$$x_{55} = 26.703537322248$$
$$x_{56} = 39.2699077336963$$
$$x_{57} = -95.8185758680502$$
$$x_{58} = -80.1106125781572$$
$$x_{59} = 23.5619451518571$$
$$x_{60} = 26.7035380604159$$
$$x_{61} = 10.9955747360645$$
$$x_{62} = -86.3937982346452$$
$$x_{63} = -70.6858351534454$$
$$x_{64} = -39.2699076683741$$
$$x_{65} = 48.6946859012172$$
$$x_{66} = 92.6769830592094$$
$$x_{67} = -76.9690195738024$$
$$x_{68} = -92.6769837307794$$
$$x_{69} = 70.6858344802043$$
$$x_{70} = -83.2522048211133$$
$$x_{71} = -61.2610569934486$$
$$x_{72} = 17.2787591562062$$
$$x_{73} = 29.8451303231501$$
$$x_{74} = -26.7035379986821$$
$$x_{75} = -14.1371668370864$$
$$x_{76} = -42.4115005850814$$
$$x_{77} = -29.8451300954883$$
$$x_{78} = 7.85398174307326$$
$$x_{79} = 95.8185760629547$$
$$x_{80} = -42.411501235316$$
$$x_{81} = -54.9778717966574$$
$$x_{82} = 14.1371671100222$$
$$x_{83} = -17.2787598356363$$
$$x_{84} = -7.85398149665124$$
$$x_{85} = -20.4203520060805$$
$$x_{86} = 92.6769837789098$$
$$x_{87} = 89.535390888605$$
$$x_{88} = -70.6858343571487$$
$$x_{89} = 42.4115007274741$$
$$x_{90} = -20.4203527082932$$
$$x_{91} = 61.2610571125526$$
$$x_{92} = 39.2699085343272$$
$$x_{93} = -10.9955746401247$$
$$x_{94} = -67.5442421706656$$
$$x_{95} = -64.4026491641039$$
$$x_{96} = -86.3937977431483$$
$$x_{97} = 17.2787599560783$$
$$x_{98} = -26.7035372004893$$
$$x_{99} = 98.9601682515978$$
$$x_{100} = 86.3937978869933$$
$$x_{101} = -61.2610562447228$$
$$x_{1} = 51.8362789031518$$
$$x_{2} = -17.2787590920677$$
$$x_{3} = -36.1283154173375$$
$$x_{4} = -39.2699084145515$$
$$x_{5} = -64.4026497466373$$
$$x_{6} = -48.6946857788076$$
$$x_{7} = 76.9690204681432$$
$$x_{8} = 83.2522056907544$$
$$x_{9} = 1.57079657289894$$
$$x_{10} = 83.252204888767$$
$$x_{11} = 48.6946866365921$$
$$x_{12} = 23.5619444059921$$
$$x_{13} = 54.9778710948428$$
$$x_{14} = 1.57079582971902$$
$$x_{15} = 45.5530929823099$$
$$x_{16} = -76.9690203748894$$
$$x_{17} = -98.9601689530982$$
$$x_{18} = -32.9867224188086$$
$$x_{19} = -51.8362786893284$$
$$x_{20} = -10.9955738413568$$
$$x_{21} = 70.6858352127237$$
$$x_{22} = 32.9867225164981$$
$$x_{23} = 76.9690196732095$$
$$x_{24} = 20.4203521477756$$
$$x_{25} = -98.9601681513438$$
$$x_{26} = -4.71238942125338$$
$$x_{27} = -58.1194639976905$$
$$x_{28} = 36.128315618579$$
$$x_{29} = 4.7123894841958$$
$$x_{30} = -92.6769829355125$$
$$x_{31} = 67.54424230971$$
$$x_{32} = 32.9867233134552$$
$$x_{33} = 98.9601690454399$$
$$x_{34} = 10.9955739381756$$
$$x_{35} = -1.57079643188553$$
$$x_{36} = -89.535390750197$$
$$x_{37} = 61.2610563112167$$
$$x_{38} = 64.4026493072124$$
$$x_{39} = -23.5619450115115$$
$$x_{40} = 89.5353901350773$$
$$x_{41} = -32.9867232184024$$
$$x_{42} = 54.9778718908148$$
$$x_{43} = -4.71238862219396$$
$$x_{44} = 67.5442415586719$$
$$x_{45} = -48.6946865760795$$
$$x_{46} = -45.5530935911043$$
$$x_{47} = 4.71238874329685$$
$$x_{48} = 73.8274274830848$$
$$x_{49} = -54.9778709962906$$
$$x_{50} = 80.1106131368654$$
$$x_{51} = -83.2522055723275$$
$$x_{52} = -73.8274272798455$$
$$x_{53} = 45.553093730794$$
$$x_{54} = 58.1194643979608$$
$$x_{55} = 26.703537322248$$
$$x_{56} = 39.2699077336963$$
$$x_{57} = -95.8185758680502$$
$$x_{58} = -80.1106125781572$$
$$x_{59} = 23.5619451518571$$
$$x_{60} = 26.7035380604159$$
$$x_{61} = 10.9955747360645$$
$$x_{62} = -86.3937982346452$$
$$x_{63} = -70.6858351534454$$
$$x_{64} = -39.2699076683741$$
$$x_{65} = 48.6946859012172$$
$$x_{66} = 92.6769830592094$$
$$x_{67} = -76.9690195738024$$
$$x_{68} = -92.6769837307794$$
$$x_{69} = 70.6858344802043$$
$$x_{70} = -83.2522048211133$$
$$x_{71} = -61.2610569934486$$
$$x_{72} = 17.2787591562062$$
$$x_{73} = 29.8451303231501$$
$$x_{74} = -26.7035379986821$$
$$x_{75} = -14.1371668370864$$
$$x_{76} = -42.4115005850814$$
$$x_{77} = -29.8451300954883$$
$$x_{78} = 7.85398174307326$$
$$x_{79} = 95.8185760629547$$
$$x_{80} = -42.411501235316$$
$$x_{81} = -54.9778717966574$$
$$x_{82} = 14.1371671100222$$
$$x_{83} = -17.2787598356363$$
$$x_{84} = -7.85398149665124$$
$$x_{85} = -20.4203520060805$$
$$x_{86} = 92.6769837789098$$
$$x_{87} = 89.535390888605$$
$$x_{88} = -70.6858343571487$$
$$x_{89} = 42.4115007274741$$
$$x_{90} = -20.4203527082932$$
$$x_{91} = 61.2610571125526$$
$$x_{92} = 39.2699085343272$$
$$x_{93} = -10.9955746401247$$
$$x_{94} = -67.5442421706656$$
$$x_{95} = -64.4026491641039$$
$$x_{96} = -86.3937977431483$$
$$x_{97} = 17.2787599560783$$
$$x_{98} = -26.7035372004893$$
$$x_{99} = 98.9601682515978$$
$$x_{100} = 86.3937978869933$$
$$x_{101} = -61.2610562447228$$
Las raíces dadas
$$x_{17} = -98.9601689530982$$
$$x_{25} = -98.9601681513438$$
$$x_{57} = -95.8185758680502$$
$$x_{68} = -92.6769837307794$$
$$x_{30} = -92.6769829355125$$
$$x_{36} = -89.535390750197$$
$$x_{62} = -86.3937982346452$$
$$x_{96} = -86.3937977431483$$
$$x_{51} = -83.2522055723275$$
$$x_{70} = -83.2522048211133$$
$$x_{58} = -80.1106125781572$$
$$x_{16} = -76.9690203748894$$
$$x_{67} = -76.9690195738024$$
$$x_{52} = -73.8274272798455$$
$$x_{63} = -70.6858351534454$$
$$x_{88} = -70.6858343571487$$
$$x_{94} = -67.5442421706656$$
$$x_{5} = -64.4026497466373$$
$$x_{95} = -64.4026491641039$$
$$x_{71} = -61.2610569934486$$
$$x_{101} = -61.2610562447228$$
$$x_{27} = -58.1194639976905$$
$$x_{81} = -54.9778717966574$$
$$x_{49} = -54.9778709962906$$
$$x_{19} = -51.8362786893284$$
$$x_{45} = -48.6946865760795$$
$$x_{6} = -48.6946857788076$$
$$x_{46} = -45.5530935911043$$
$$x_{80} = -42.411501235316$$
$$x_{76} = -42.4115005850814$$
$$x_{4} = -39.2699084145515$$
$$x_{64} = -39.2699076683741$$
$$x_{3} = -36.1283154173375$$
$$x_{41} = -32.9867232184024$$
$$x_{18} = -32.9867224188086$$
$$x_{77} = -29.8451300954883$$
$$x_{74} = -26.7035379986821$$
$$x_{98} = -26.7035372004893$$
$$x_{39} = -23.5619450115115$$
$$x_{90} = -20.4203527082932$$
$$x_{85} = -20.4203520060805$$
$$x_{83} = -17.2787598356363$$
$$x_{2} = -17.2787590920677$$
$$x_{75} = -14.1371668370864$$
$$x_{93} = -10.9955746401247$$
$$x_{20} = -10.9955738413568$$
$$x_{84} = -7.85398149665124$$
$$x_{26} = -4.71238942125338$$
$$x_{43} = -4.71238862219396$$
$$x_{35} = -1.57079643188553$$
$$x_{14} = 1.57079582971902$$
$$x_{9} = 1.57079657289894$$
$$x_{47} = 4.71238874329685$$
$$x_{29} = 4.7123894841958$$
$$x_{78} = 7.85398174307326$$
$$x_{34} = 10.9955739381756$$
$$x_{61} = 10.9955747360645$$
$$x_{82} = 14.1371671100222$$
$$x_{72} = 17.2787591562062$$
$$x_{97} = 17.2787599560783$$
$$x_{24} = 20.4203521477756$$
$$x_{12} = 23.5619444059921$$
$$x_{59} = 23.5619451518571$$
$$x_{55} = 26.703537322248$$
$$x_{60} = 26.7035380604159$$
$$x_{73} = 29.8451303231501$$
$$x_{22} = 32.9867225164981$$
$$x_{32} = 32.9867233134552$$
$$x_{28} = 36.128315618579$$
$$x_{56} = 39.2699077336963$$
$$x_{92} = 39.2699085343272$$
$$x_{89} = 42.4115007274741$$
$$x_{15} = 45.5530929823099$$
$$x_{53} = 45.553093730794$$
$$x_{65} = 48.6946859012172$$
$$x_{11} = 48.6946866365921$$
$$x_{1} = 51.8362789031518$$
$$x_{13} = 54.9778710948428$$
$$x_{42} = 54.9778718908148$$
$$x_{54} = 58.1194643979608$$
$$x_{37} = 61.2610563112167$$
$$x_{91} = 61.2610571125526$$
$$x_{38} = 64.4026493072124$$
$$x_{44} = 67.5442415586719$$
$$x_{31} = 67.54424230971$$
$$x_{69} = 70.6858344802043$$
$$x_{21} = 70.6858352127237$$
$$x_{48} = 73.8274274830848$$
$$x_{23} = 76.9690196732095$$
$$x_{7} = 76.9690204681432$$
$$x_{50} = 80.1106131368654$$
$$x_{10} = 83.252204888767$$
$$x_{8} = 83.2522056907544$$
$$x_{100} = 86.3937978869933$$
$$x_{40} = 89.5353901350773$$
$$x_{87} = 89.535390888605$$
$$x_{66} = 92.6769830592094$$
$$x_{86} = 92.6769837789098$$
$$x_{79} = 95.8185760629547$$
$$x_{99} = 98.9601682515978$$
$$x_{33} = 98.9601690454399$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{17}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{17} - \frac{1}{10}$$
=
$$-98.9601689530982 + - \frac{1}{10}$$
=
$$-99.0601689530982$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| < 1$$
$$\left|{\sin{\left(-99.0601689530982 \right)}}\right| < 1$$

0.995004128836795 < 1

significa que una de las soluciones de nuestra ecuación será con:
$$x < -98.9601689530982$$

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-------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-------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-------
       x17      x25      x57      x68      x30      x36      x62      x96      x51      x70      x58      x16      x67      x52      x63      x88      x94      x5      x95      x71      x101      x27      x81      x49      x19      x45      x6      x46      x80      x76      x4      x64      x3      x41      x18      x77      x74      x98      x39      x90      x85      x83      x2      x75      x93      x20      x84      x26      x43      x35      x14      x9      x47      x29      x78      x34      x61      x82      x72      x97      x24      x12      x59      x55      x60      x73      x22      x32      x28      x56      x92      x89      x15      x53      x65      x11      x1      x13      x42      x54      x37      x91      x38      x44      x31      x69      x21      x48      x23      x7      x50      x10      x8      x100      x40      x87      x66      x86      x79      x99      x33

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < -98.9601689530982$$
$$x > -98.9601681513438 \wedge x < -95.8185758680502$$
$$x > -92.6769837307794 \wedge x < -92.6769829355125$$
$$x > -89.535390750197 \wedge x < -86.3937982346452$$
$$x > -86.3937977431483 \wedge x < -83.2522055723275$$
$$x > -83.2522048211133 \wedge x < -80.1106125781572$$
$$x > -76.9690203748894 \wedge x < -76.9690195738024$$
$$x > -73.8274272798455 \wedge x < -70.6858351534454$$
$$x > -70.6858343571487 \wedge x < -67.5442421706656$$
$$x > -64.4026497466373 \wedge x < -64.4026491641039$$
$$x > -61.2610569934486 \wedge x < -61.2610562447228$$
$$x > -58.1194639976905 \wedge x < -54.9778717966574$$
$$x > -54.9778709962906 \wedge x < -51.8362786893284$$
$$x > -48.6946865760795 \wedge x < -48.6946857788076$$
$$x > -45.5530935911043 \wedge x < -42.411501235316$$
$$x > -42.4115005850814 \wedge x < -39.2699084145515$$
$$x > -39.2699076683741 \wedge x < -36.1283154173375$$
$$x > -32.9867232184024 \wedge x < -32.9867224188086$$
$$x > -29.8451300954883 \wedge x < -26.7035379986821$$
$$x > -26.7035372004893 \wedge x < -23.5619450115115$$
$$x > -20.4203527082932 \wedge x < -20.4203520060805$$
$$x > -17.2787598356363 \wedge x < -17.2787590920677$$
$$x > -14.1371668370864 \wedge x < -10.9955746401247$$
$$x > -10.9955738413568 \wedge x < -7.85398149665124$$
$$x > -4.71238942125338 \wedge x < -4.71238862219396$$
$$x > -1.57079643188553 \wedge x < 1.57079582971902$$
$$x > 1.57079657289894 \wedge x < 4.71238874329685$$
$$x > 4.7123894841958 \wedge x < 7.85398174307326$$
$$x > 10.9955739381756 \wedge x < 10.9955747360645$$
$$x > 14.1371671100222 \wedge x < 17.2787591562062$$
$$x > 17.2787599560783 \wedge x < 20.4203521477756$$
$$x > 23.5619444059921 \wedge x < 23.5619451518571$$
$$x > 26.703537322248 \wedge x < 26.7035380604159$$
$$x > 29.8451303231501 \wedge x < 32.9867225164981$$
$$x > 32.9867233134552 \wedge x < 36.128315618579$$
$$x > 39.2699077336963 \wedge x < 39.2699085343272$$
$$x > 42.4115007274741 \wedge x < 45.5530929823099$$
$$x > 45.553093730794 \wedge x < 48.6946859012172$$
$$x > 48.6946866365921 \wedge x < 51.8362789031518$$
$$x > 54.9778710948428 \wedge x < 54.9778718908148$$
$$x > 58.1194643979608 \wedge x < 61.2610563112167$$
$$x > 61.2610571125526 \wedge x < 64.4026493072124$$
$$x > 67.5442415586719 \wedge x < 67.54424230971$$
$$x > 70.6858344802043 \wedge x < 70.6858352127237$$
$$x > 73.8274274830848 \wedge x < 76.9690196732095$$
$$x > 76.9690204681432 \wedge x < 80.1106131368654$$
$$x > 83.252204888767 \wedge x < 83.2522056907544$$
$$x > 86.3937978869933 \wedge x < 89.5353901350773$$
$$x > 89.535390888605 \wedge x < 92.6769830592094$$
$$x > 92.6769837789098 \wedge x < 95.8185760629547$$
$$x > 98.9601682515978 \wedge x < 98.9601690454399$$