Mister Exam
|sin(x)|<=1 inequation
The solution
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|sin(x)| <= 1
$$\left|{\sin{\left(x \right)}}\right| \leq 1$$
Abs(sin(x)) <= 1
Detail solution
Given the inequality:
$$\left|{\sin{\left(x \right)}}\right| \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\sin{\left(x \right)}}\right| = 1$$
Solve:
Given the equation
$$\left|{\sin{\left(x \right)}}\right| = 1$$
transform
$$\left|{\sin{\left(x \right)}}\right| - 1 = 0$$
$$\left|{\sin{\left(x \right)}}\right| - 1 = 0$$
Do replacement
$$w = \left|{\sin{\left(x \right)}}\right|$$
Move free summands (without w)
from left part to right part, we given:
$$w = 1$$
We get the answer: w = 1
do backward replacement
$$\left|{\sin{\left(x \right)}}\right| = w$$
substitute w:
$$x_{1} = 39.2699077336963$$
$$x_{2} = -7.85398149665124$$
$$x_{3} = -64.4026491641039$$
$$x_{4} = -98.9601681513438$$
$$x_{5} = -42.4115005850814$$
$$x_{6} = -76.9690203748894$$
$$x_{7} = -92.6769837307794$$
$$x_{8} = -39.2699084145515$$
$$x_{9} = -32.9867224188086$$
$$x_{10} = 39.2699085343272$$
$$x_{11} = 42.4115007274741$$
$$x_{12} = 92.6769830592094$$
$$x_{13} = 70.6858344802043$$
$$x_{14} = -23.5619450115115$$
$$x_{15} = 14.1371671100222$$
$$x_{16} = 92.6769837789098$$
$$x_{17} = -20.4203527082932$$
$$x_{18} = -4.71238862219396$$
$$x_{19} = -61.2610562447228$$
$$x_{20} = -83.2522055723275$$
$$x_{21} = 67.54424230971$$
$$x_{22} = 76.9690196732095$$
$$x_{23} = 1.57079582971902$$
$$x_{24} = 95.8185760629547$$
$$x_{25} = 51.8362789031518$$
$$x_{26} = 76.9690204681432$$
$$x_{27} = 23.5619444059921$$
$$x_{28} = -20.4203520060805$$
$$x_{29} = 98.9601682515978$$
$$x_{30} = -45.5530935911043$$
$$x_{31} = 80.1106131368654$$
$$x_{32} = 61.2610563112167$$
$$x_{33} = -70.6858343571487$$
$$x_{34} = -17.2787590920677$$
$$x_{35} = -39.2699076683741$$
$$x_{36} = -48.6946865760795$$
$$x_{37} = -67.5442421706656$$
$$x_{38} = -54.9778709962906$$
$$x_{39} = 86.3937978869933$$
$$x_{40} = 83.2522056907544$$
$$x_{41} = -26.7035379986821$$
$$x_{42} = 32.9867233134552$$
$$x_{43} = 61.2610571125526$$
$$x_{44} = 70.6858352127237$$
$$x_{45} = 73.8274274830848$$
$$x_{46} = -17.2787598356363$$
$$x_{47} = 45.553093730794$$
$$x_{48} = -92.6769829355125$$
$$x_{49} = -14.1371668370864$$
$$x_{50} = 64.4026493072124$$
$$x_{51} = 17.2787591562062$$
$$x_{52} = 48.6946866365921$$
$$x_{53} = 4.7123894841958$$
$$x_{54} = 54.9778710948428$$
$$x_{55} = -10.9955746401247$$
$$x_{56} = 54.9778718908148$$
$$x_{57} = -48.6946857788076$$
$$x_{58} = -36.1283154173375$$
$$x_{59} = 26.7035380604159$$
$$x_{60} = -58.1194639976905$$
$$x_{61} = 98.9601690454399$$
$$x_{62} = 89.535390888605$$
$$x_{63} = -42.411501235316$$
$$x_{64} = 45.5530929823099$$
$$x_{65} = 4.71238874329685$$
$$x_{66} = -54.9778717966574$$
$$x_{67} = -32.9867232184024$$
$$x_{68} = -4.71238942125338$$
$$x_{69} = 10.9955747360645$$
$$x_{70} = 7.85398174307326$$
$$x_{71} = -76.9690195738024$$
$$x_{72} = 1.57079657289894$$
$$x_{73} = 36.128315618579$$
$$x_{74} = -29.8451300954883$$
$$x_{75} = 20.4203521477756$$
$$x_{76} = 17.2787599560783$$
$$x_{77} = -86.3937982346452$$
$$x_{78} = 89.5353901350773$$
$$x_{79} = 58.1194643979608$$
$$x_{80} = 67.5442415586719$$
$$x_{81} = 83.252204888767$$
$$x_{82} = -51.8362786893284$$
$$x_{83} = -10.9955738413568$$
$$x_{84} = -89.535390750197$$
$$x_{85} = 23.5619451518571$$
$$x_{86} = -95.8185758680502$$
$$x_{87} = 26.703537322248$$
$$x_{88} = 32.9867225164981$$
$$x_{89} = -64.4026497466373$$
$$x_{90} = -86.3937977431483$$
$$x_{91} = -80.1106125781572$$
$$x_{92} = -98.9601689530982$$
$$x_{93} = 48.6946859012172$$
$$x_{94} = -61.2610569934486$$
$$x_{95} = -1.57079643188553$$
$$x_{96} = -73.8274272798455$$
$$x_{97} = 10.9955739381756$$
$$x_{98} = -26.7035372004893$$
$$x_{99} = -83.2522048211133$$
$$x_{100} = -70.6858351534454$$
$$x_{101} = 29.8451303231501$$
$$x_{1} = 39.2699077336963$$
$$x_{2} = -7.85398149665124$$
$$x_{3} = -64.4026491641039$$
$$x_{4} = -98.9601681513438$$
$$x_{5} = -42.4115005850814$$
$$x_{6} = -76.9690203748894$$
$$x_{7} = -92.6769837307794$$
$$x_{8} = -39.2699084145515$$
$$x_{9} = -32.9867224188086$$
$$x_{10} = 39.2699085343272$$
$$x_{11} = 42.4115007274741$$
$$x_{12} = 92.6769830592094$$
$$x_{13} = 70.6858344802043$$
$$x_{14} = -23.5619450115115$$
$$x_{15} = 14.1371671100222$$
$$x_{16} = 92.6769837789098$$
$$x_{17} = -20.4203527082932$$
$$x_{18} = -4.71238862219396$$
$$x_{19} = -61.2610562447228$$
$$x_{20} = -83.2522055723275$$
$$x_{21} = 67.54424230971$$
$$x_{22} = 76.9690196732095$$
$$x_{23} = 1.57079582971902$$
$$x_{24} = 95.8185760629547$$
$$x_{25} = 51.8362789031518$$
$$x_{26} = 76.9690204681432$$
$$x_{27} = 23.5619444059921$$
$$x_{28} = -20.4203520060805$$
$$x_{29} = 98.9601682515978$$
$$x_{30} = -45.5530935911043$$
$$x_{31} = 80.1106131368654$$
$$x_{32} = 61.2610563112167$$
$$x_{33} = -70.6858343571487$$
$$x_{34} = -17.2787590920677$$
$$x_{35} = -39.2699076683741$$
$$x_{36} = -48.6946865760795$$
$$x_{37} = -67.5442421706656$$
$$x_{38} = -54.9778709962906$$
$$x_{39} = 86.3937978869933$$
$$x_{40} = 83.2522056907544$$
$$x_{41} = -26.7035379986821$$
$$x_{42} = 32.9867233134552$$
$$x_{43} = 61.2610571125526$$
$$x_{44} = 70.6858352127237$$
$$x_{45} = 73.8274274830848$$
$$x_{46} = -17.2787598356363$$
$$x_{47} = 45.553093730794$$
$$x_{48} = -92.6769829355125$$
$$x_{49} = -14.1371668370864$$
$$x_{50} = 64.4026493072124$$
$$x_{51} = 17.2787591562062$$
$$x_{52} = 48.6946866365921$$
$$x_{53} = 4.7123894841958$$
$$x_{54} = 54.9778710948428$$
$$x_{55} = -10.9955746401247$$
$$x_{56} = 54.9778718908148$$
$$x_{57} = -48.6946857788076$$
$$x_{58} = -36.1283154173375$$
$$x_{59} = 26.7035380604159$$
$$x_{60} = -58.1194639976905$$
$$x_{61} = 98.9601690454399$$
$$x_{62} = 89.535390888605$$
$$x_{63} = -42.411501235316$$
$$x_{64} = 45.5530929823099$$
$$x_{65} = 4.71238874329685$$
$$x_{66} = -54.9778717966574$$
$$x_{67} = -32.9867232184024$$
$$x_{68} = -4.71238942125338$$
$$x_{69} = 10.9955747360645$$
$$x_{70} = 7.85398174307326$$
$$x_{71} = -76.9690195738024$$
$$x_{72} = 1.57079657289894$$
$$x_{73} = 36.128315618579$$
$$x_{74} = -29.8451300954883$$
$$x_{75} = 20.4203521477756$$
$$x_{76} = 17.2787599560783$$
$$x_{77} = -86.3937982346452$$
$$x_{78} = 89.5353901350773$$
$$x_{79} = 58.1194643979608$$
$$x_{80} = 67.5442415586719$$
$$x_{81} = 83.252204888767$$
$$x_{82} = -51.8362786893284$$
$$x_{83} = -10.9955738413568$$
$$x_{84} = -89.535390750197$$
$$x_{85} = 23.5619451518571$$
$$x_{86} = -95.8185758680502$$
$$x_{87} = 26.703537322248$$
$$x_{88} = 32.9867225164981$$
$$x_{89} = -64.4026497466373$$
$$x_{90} = -86.3937977431483$$
$$x_{91} = -80.1106125781572$$
$$x_{92} = -98.9601689530982$$
$$x_{93} = 48.6946859012172$$
$$x_{94} = -61.2610569934486$$
$$x_{95} = -1.57079643188553$$
$$x_{96} = -73.8274272798455$$
$$x_{97} = 10.9955739381756$$
$$x_{98} = -26.7035372004893$$
$$x_{99} = -83.2522048211133$$
$$x_{100} = -70.6858351534454$$
$$x_{101} = 29.8451303231501$$
This roots
$$x_{92} = -98.9601689530982$$
$$x_{4} = -98.9601681513438$$
$$x_{86} = -95.8185758680502$$
$$x_{7} = -92.6769837307794$$
$$x_{48} = -92.6769829355125$$
$$x_{84} = -89.535390750197$$
$$x_{77} = -86.3937982346452$$
$$x_{90} = -86.3937977431483$$
$$x_{20} = -83.2522055723275$$
$$x_{99} = -83.2522048211133$$
$$x_{91} = -80.1106125781572$$
$$x_{6} = -76.9690203748894$$
$$x_{71} = -76.9690195738024$$
$$x_{96} = -73.8274272798455$$
$$x_{100} = -70.6858351534454$$
$$x_{33} = -70.6858343571487$$
$$x_{37} = -67.5442421706656$$
$$x_{89} = -64.4026497466373$$
$$x_{3} = -64.4026491641039$$
$$x_{94} = -61.2610569934486$$
$$x_{19} = -61.2610562447228$$
$$x_{60} = -58.1194639976905$$
$$x_{66} = -54.9778717966574$$
$$x_{38} = -54.9778709962906$$
$$x_{82} = -51.8362786893284$$
$$x_{36} = -48.6946865760795$$
$$x_{57} = -48.6946857788076$$
$$x_{30} = -45.5530935911043$$
$$x_{63} = -42.411501235316$$
$$x_{5} = -42.4115005850814$$
$$x_{8} = -39.2699084145515$$
$$x_{35} = -39.2699076683741$$
$$x_{58} = -36.1283154173375$$
$$x_{67} = -32.9867232184024$$
$$x_{9} = -32.9867224188086$$
$$x_{74} = -29.8451300954883$$
$$x_{41} = -26.7035379986821$$
$$x_{98} = -26.7035372004893$$
$$x_{14} = -23.5619450115115$$
$$x_{17} = -20.4203527082932$$
$$x_{28} = -20.4203520060805$$
$$x_{46} = -17.2787598356363$$
$$x_{34} = -17.2787590920677$$
$$x_{49} = -14.1371668370864$$
$$x_{55} = -10.9955746401247$$
$$x_{83} = -10.9955738413568$$
$$x_{2} = -7.85398149665124$$
$$x_{68} = -4.71238942125338$$
$$x_{18} = -4.71238862219396$$
$$x_{95} = -1.57079643188553$$
$$x_{23} = 1.57079582971902$$
$$x_{72} = 1.57079657289894$$
$$x_{65} = 4.71238874329685$$
$$x_{53} = 4.7123894841958$$
$$x_{70} = 7.85398174307326$$
$$x_{97} = 10.9955739381756$$
$$x_{69} = 10.9955747360645$$
$$x_{15} = 14.1371671100222$$
$$x_{51} = 17.2787591562062$$
$$x_{76} = 17.2787599560783$$
$$x_{75} = 20.4203521477756$$
$$x_{27} = 23.5619444059921$$
$$x_{85} = 23.5619451518571$$
$$x_{87} = 26.703537322248$$
$$x_{59} = 26.7035380604159$$
$$x_{101} = 29.8451303231501$$
$$x_{88} = 32.9867225164981$$
$$x_{42} = 32.9867233134552$$
$$x_{73} = 36.128315618579$$
$$x_{1} = 39.2699077336963$$
$$x_{10} = 39.2699085343272$$
$$x_{11} = 42.4115007274741$$
$$x_{64} = 45.5530929823099$$
$$x_{47} = 45.553093730794$$
$$x_{93} = 48.6946859012172$$
$$x_{52} = 48.6946866365921$$
$$x_{25} = 51.8362789031518$$
$$x_{54} = 54.9778710948428$$
$$x_{56} = 54.9778718908148$$
$$x_{79} = 58.1194643979608$$
$$x_{32} = 61.2610563112167$$
$$x_{43} = 61.2610571125526$$
$$x_{50} = 64.4026493072124$$
$$x_{80} = 67.5442415586719$$
$$x_{21} = 67.54424230971$$
$$x_{13} = 70.6858344802043$$
$$x_{44} = 70.6858352127237$$
$$x_{45} = 73.8274274830848$$
$$x_{22} = 76.9690196732095$$
$$x_{26} = 76.9690204681432$$
$$x_{31} = 80.1106131368654$$
$$x_{81} = 83.252204888767$$
$$x_{40} = 83.2522056907544$$
$$x_{39} = 86.3937978869933$$
$$x_{78} = 89.5353901350773$$
$$x_{62} = 89.535390888605$$
$$x_{12} = 92.6769830592094$$
$$x_{16} = 92.6769837789098$$
$$x_{24} = 95.8185760629547$$
$$x_{29} = 98.9601682515978$$
$$x_{61} = 98.9601690454399$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{92}$$
For example, let's take the point
$$x_{0} = x_{92} - \frac{1}{10}$$
=
$$-98.9601689530982 + - \frac{1}{10}$$
=
$$-99.0601689530982$$
substitute to the expression
$$\left|{\sin{\left(x \right)}}\right| \leq 1$$
$$\left|{\sin{\left(-99.0601689530982 \right)}}\right| \leq 1$$
one of the solutions of our inequality is:
$$x \leq -98.9601689530982$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -98.9601689530982$$
$$x \geq -98.9601681513438 \wedge x \leq -95.8185758680502$$
$$x \geq -92.6769837307794 \wedge x \leq -92.6769829355125$$
$$x \geq -89.535390750197 \wedge x \leq -86.3937982346452$$
$$x \geq -86.3937977431483 \wedge x \leq -83.2522055723275$$
$$x \geq -83.2522048211133 \wedge x \leq -80.1106125781572$$
$$x \geq -76.9690203748894 \wedge x \leq -76.9690195738024$$
$$x \geq -73.8274272798455 \wedge x \leq -70.6858351534454$$
$$x \geq -70.6858343571487 \wedge x \leq -67.5442421706656$$
$$x \geq -64.4026497466373 \wedge x \leq -64.4026491641039$$
$$x \geq -61.2610569934486 \wedge x \leq -61.2610562447228$$
$$x \geq -58.1194639976905 \wedge x \leq -54.9778717966574$$
$$x \geq -54.9778709962906 \wedge x \leq -51.8362786893284$$
$$x \geq -48.6946865760795 \wedge x \leq -48.6946857788076$$
$$x \geq -45.5530935911043 \wedge x \leq -42.411501235316$$
$$x \geq -42.4115005850814 \wedge x \leq -39.2699084145515$$
$$x \geq -39.2699076683741 \wedge x \leq -36.1283154173375$$
$$x \geq -32.9867232184024 \wedge x \leq -32.9867224188086$$
$$x \geq -29.8451300954883 \wedge x \leq -26.7035379986821$$
$$x \geq -26.7035372004893 \wedge x \leq -23.5619450115115$$
$$x \geq -20.4203527082932 \wedge x \leq -20.4203520060805$$
$$x \geq -17.2787598356363 \wedge x \leq -17.2787590920677$$
$$x \geq -14.1371668370864 \wedge x \leq -10.9955746401247$$
$$x \geq -10.9955738413568 \wedge x \leq -7.85398149665124$$
$$x \geq -4.71238942125338 \wedge x \leq -4.71238862219396$$
$$x \geq -1.57079643188553 \wedge x \leq 1.57079582971902$$
$$x \geq 1.57079657289894 \wedge x \leq 4.71238874329685$$
$$x \geq 4.7123894841958 \wedge x \leq 7.85398174307326$$
$$x \geq 10.9955739381756 \wedge x \leq 10.9955747360645$$
$$x \geq 14.1371671100222 \wedge x \leq 17.2787591562062$$
$$x \geq 17.2787599560783 \wedge x \leq 20.4203521477756$$
$$x \geq 23.5619444059921 \wedge x \leq 23.5619451518571$$
$$x \geq 26.703537322248 \wedge x \leq 26.7035380604159$$
$$x \geq 29.8451303231501 \wedge x \leq 32.9867225164981$$
$$x \geq 32.9867233134552 \wedge x \leq 36.128315618579$$
$$x \geq 39.2699077336963 \wedge x \leq 39.2699085343272$$
$$x \geq 42.4115007274741 \wedge x \leq 45.5530929823099$$
$$x \geq 45.553093730794 \wedge x \leq 48.6946859012172$$
$$x \geq 48.6946866365921 \wedge x \leq 51.8362789031518$$
$$x \geq 54.9778710948428 \wedge x \leq 54.9778718908148$$
$$x \geq 58.1194643979608 \wedge x \leq 61.2610563112167$$
$$x \geq 61.2610571125526 \wedge x \leq 64.4026493072124$$
$$x \geq 67.5442415586719 \wedge x \leq 67.54424230971$$
$$x \geq 70.6858344802043 \wedge x \leq 70.6858352127237$$
$$x \geq 73.8274274830848 \wedge x \leq 76.9690196732095$$
$$x \geq 76.9690204681432 \wedge x \leq 80.1106131368654$$
$$x \geq 83.252204888767 \wedge x \leq 83.2522056907544$$
$$x \geq 86.3937978869933 \wedge x \leq 89.5353901350773$$
$$x \geq 89.535390888605 \wedge x \leq 92.6769830592094$$
$$x \geq 92.6769837789098 \wedge x \leq 95.8185760629547$$
$$x \geq 98.9601682515978 \wedge x \leq 98.9601690454399$$
$$\left|{\sin{\left(x \right)}}\right| \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\sin{\left(x \right)}}\right| = 1$$
Solve:
Given the equation
$$\left|{\sin{\left(x \right)}}\right| = 1$$
transform
$$\left|{\sin{\left(x \right)}}\right| - 1 = 0$$
$$\left|{\sin{\left(x \right)}}\right| - 1 = 0$$
Do replacement
$$w = \left|{\sin{\left(x \right)}}\right|$$
Move free summands (without w)
from left part to right part, we given:
$$w = 1$$
We get the answer: w = 1
do backward replacement
$$\left|{\sin{\left(x \right)}}\right| = w$$
substitute w:
$$x_{1} = 39.2699077336963$$
$$x_{2} = -7.85398149665124$$
$$x_{3} = -64.4026491641039$$
$$x_{4} = -98.9601681513438$$
$$x_{5} = -42.4115005850814$$
$$x_{6} = -76.9690203748894$$
$$x_{7} = -92.6769837307794$$
$$x_{8} = -39.2699084145515$$
$$x_{9} = -32.9867224188086$$
$$x_{10} = 39.2699085343272$$
$$x_{11} = 42.4115007274741$$
$$x_{12} = 92.6769830592094$$
$$x_{13} = 70.6858344802043$$
$$x_{14} = -23.5619450115115$$
$$x_{15} = 14.1371671100222$$
$$x_{16} = 92.6769837789098$$
$$x_{17} = -20.4203527082932$$
$$x_{18} = -4.71238862219396$$
$$x_{19} = -61.2610562447228$$
$$x_{20} = -83.2522055723275$$
$$x_{21} = 67.54424230971$$
$$x_{22} = 76.9690196732095$$
$$x_{23} = 1.57079582971902$$
$$x_{24} = 95.8185760629547$$
$$x_{25} = 51.8362789031518$$
$$x_{26} = 76.9690204681432$$
$$x_{27} = 23.5619444059921$$
$$x_{28} = -20.4203520060805$$
$$x_{29} = 98.9601682515978$$
$$x_{30} = -45.5530935911043$$
$$x_{31} = 80.1106131368654$$
$$x_{32} = 61.2610563112167$$
$$x_{33} = -70.6858343571487$$
$$x_{34} = -17.2787590920677$$
$$x_{35} = -39.2699076683741$$
$$x_{36} = -48.6946865760795$$
$$x_{37} = -67.5442421706656$$
$$x_{38} = -54.9778709962906$$
$$x_{39} = 86.3937978869933$$
$$x_{40} = 83.2522056907544$$
$$x_{41} = -26.7035379986821$$
$$x_{42} = 32.9867233134552$$
$$x_{43} = 61.2610571125526$$
$$x_{44} = 70.6858352127237$$
$$x_{45} = 73.8274274830848$$
$$x_{46} = -17.2787598356363$$
$$x_{47} = 45.553093730794$$
$$x_{48} = -92.6769829355125$$
$$x_{49} = -14.1371668370864$$
$$x_{50} = 64.4026493072124$$
$$x_{51} = 17.2787591562062$$
$$x_{52} = 48.6946866365921$$
$$x_{53} = 4.7123894841958$$
$$x_{54} = 54.9778710948428$$
$$x_{55} = -10.9955746401247$$
$$x_{56} = 54.9778718908148$$
$$x_{57} = -48.6946857788076$$
$$x_{58} = -36.1283154173375$$
$$x_{59} = 26.7035380604159$$
$$x_{60} = -58.1194639976905$$
$$x_{61} = 98.9601690454399$$
$$x_{62} = 89.535390888605$$
$$x_{63} = -42.411501235316$$
$$x_{64} = 45.5530929823099$$
$$x_{65} = 4.71238874329685$$
$$x_{66} = -54.9778717966574$$
$$x_{67} = -32.9867232184024$$
$$x_{68} = -4.71238942125338$$
$$x_{69} = 10.9955747360645$$
$$x_{70} = 7.85398174307326$$
$$x_{71} = -76.9690195738024$$
$$x_{72} = 1.57079657289894$$
$$x_{73} = 36.128315618579$$
$$x_{74} = -29.8451300954883$$
$$x_{75} = 20.4203521477756$$
$$x_{76} = 17.2787599560783$$
$$x_{77} = -86.3937982346452$$
$$x_{78} = 89.5353901350773$$
$$x_{79} = 58.1194643979608$$
$$x_{80} = 67.5442415586719$$
$$x_{81} = 83.252204888767$$
$$x_{82} = -51.8362786893284$$
$$x_{83} = -10.9955738413568$$
$$x_{84} = -89.535390750197$$
$$x_{85} = 23.5619451518571$$
$$x_{86} = -95.8185758680502$$
$$x_{87} = 26.703537322248$$
$$x_{88} = 32.9867225164981$$
$$x_{89} = -64.4026497466373$$
$$x_{90} = -86.3937977431483$$
$$x_{91} = -80.1106125781572$$
$$x_{92} = -98.9601689530982$$
$$x_{93} = 48.6946859012172$$
$$x_{94} = -61.2610569934486$$
$$x_{95} = -1.57079643188553$$
$$x_{96} = -73.8274272798455$$
$$x_{97} = 10.9955739381756$$
$$x_{98} = -26.7035372004893$$
$$x_{99} = -83.2522048211133$$
$$x_{100} = -70.6858351534454$$
$$x_{101} = 29.8451303231501$$
$$x_{1} = 39.2699077336963$$
$$x_{2} = -7.85398149665124$$
$$x_{3} = -64.4026491641039$$
$$x_{4} = -98.9601681513438$$
$$x_{5} = -42.4115005850814$$
$$x_{6} = -76.9690203748894$$
$$x_{7} = -92.6769837307794$$
$$x_{8} = -39.2699084145515$$
$$x_{9} = -32.9867224188086$$
$$x_{10} = 39.2699085343272$$
$$x_{11} = 42.4115007274741$$
$$x_{12} = 92.6769830592094$$
$$x_{13} = 70.6858344802043$$
$$x_{14} = -23.5619450115115$$
$$x_{15} = 14.1371671100222$$
$$x_{16} = 92.6769837789098$$
$$x_{17} = -20.4203527082932$$
$$x_{18} = -4.71238862219396$$
$$x_{19} = -61.2610562447228$$
$$x_{20} = -83.2522055723275$$
$$x_{21} = 67.54424230971$$
$$x_{22} = 76.9690196732095$$
$$x_{23} = 1.57079582971902$$
$$x_{24} = 95.8185760629547$$
$$x_{25} = 51.8362789031518$$
$$x_{26} = 76.9690204681432$$
$$x_{27} = 23.5619444059921$$
$$x_{28} = -20.4203520060805$$
$$x_{29} = 98.9601682515978$$
$$x_{30} = -45.5530935911043$$
$$x_{31} = 80.1106131368654$$
$$x_{32} = 61.2610563112167$$
$$x_{33} = -70.6858343571487$$
$$x_{34} = -17.2787590920677$$
$$x_{35} = -39.2699076683741$$
$$x_{36} = -48.6946865760795$$
$$x_{37} = -67.5442421706656$$
$$x_{38} = -54.9778709962906$$
$$x_{39} = 86.3937978869933$$
$$x_{40} = 83.2522056907544$$
$$x_{41} = -26.7035379986821$$
$$x_{42} = 32.9867233134552$$
$$x_{43} = 61.2610571125526$$
$$x_{44} = 70.6858352127237$$
$$x_{45} = 73.8274274830848$$
$$x_{46} = -17.2787598356363$$
$$x_{47} = 45.553093730794$$
$$x_{48} = -92.6769829355125$$
$$x_{49} = -14.1371668370864$$
$$x_{50} = 64.4026493072124$$
$$x_{51} = 17.2787591562062$$
$$x_{52} = 48.6946866365921$$
$$x_{53} = 4.7123894841958$$
$$x_{54} = 54.9778710948428$$
$$x_{55} = -10.9955746401247$$
$$x_{56} = 54.9778718908148$$
$$x_{57} = -48.6946857788076$$
$$x_{58} = -36.1283154173375$$
$$x_{59} = 26.7035380604159$$
$$x_{60} = -58.1194639976905$$
$$x_{61} = 98.9601690454399$$
$$x_{62} = 89.535390888605$$
$$x_{63} = -42.411501235316$$
$$x_{64} = 45.5530929823099$$
$$x_{65} = 4.71238874329685$$
$$x_{66} = -54.9778717966574$$
$$x_{67} = -32.9867232184024$$
$$x_{68} = -4.71238942125338$$
$$x_{69} = 10.9955747360645$$
$$x_{70} = 7.85398174307326$$
$$x_{71} = -76.9690195738024$$
$$x_{72} = 1.57079657289894$$
$$x_{73} = 36.128315618579$$
$$x_{74} = -29.8451300954883$$
$$x_{75} = 20.4203521477756$$
$$x_{76} = 17.2787599560783$$
$$x_{77} = -86.3937982346452$$
$$x_{78} = 89.5353901350773$$
$$x_{79} = 58.1194643979608$$
$$x_{80} = 67.5442415586719$$
$$x_{81} = 83.252204888767$$
$$x_{82} = -51.8362786893284$$
$$x_{83} = -10.9955738413568$$
$$x_{84} = -89.535390750197$$
$$x_{85} = 23.5619451518571$$
$$x_{86} = -95.8185758680502$$
$$x_{87} = 26.703537322248$$
$$x_{88} = 32.9867225164981$$
$$x_{89} = -64.4026497466373$$
$$x_{90} = -86.3937977431483$$
$$x_{91} = -80.1106125781572$$
$$x_{92} = -98.9601689530982$$
$$x_{93} = 48.6946859012172$$
$$x_{94} = -61.2610569934486$$
$$x_{95} = -1.57079643188553$$
$$x_{96} = -73.8274272798455$$
$$x_{97} = 10.9955739381756$$
$$x_{98} = -26.7035372004893$$
$$x_{99} = -83.2522048211133$$
$$x_{100} = -70.6858351534454$$
$$x_{101} = 29.8451303231501$$
This roots
$$x_{92} = -98.9601689530982$$
$$x_{4} = -98.9601681513438$$
$$x_{86} = -95.8185758680502$$
$$x_{7} = -92.6769837307794$$
$$x_{48} = -92.6769829355125$$
$$x_{84} = -89.535390750197$$
$$x_{77} = -86.3937982346452$$
$$x_{90} = -86.3937977431483$$
$$x_{20} = -83.2522055723275$$
$$x_{99} = -83.2522048211133$$
$$x_{91} = -80.1106125781572$$
$$x_{6} = -76.9690203748894$$
$$x_{71} = -76.9690195738024$$
$$x_{96} = -73.8274272798455$$
$$x_{100} = -70.6858351534454$$
$$x_{33} = -70.6858343571487$$
$$x_{37} = -67.5442421706656$$
$$x_{89} = -64.4026497466373$$
$$x_{3} = -64.4026491641039$$
$$x_{94} = -61.2610569934486$$
$$x_{19} = -61.2610562447228$$
$$x_{60} = -58.1194639976905$$
$$x_{66} = -54.9778717966574$$
$$x_{38} = -54.9778709962906$$
$$x_{82} = -51.8362786893284$$
$$x_{36} = -48.6946865760795$$
$$x_{57} = -48.6946857788076$$
$$x_{30} = -45.5530935911043$$
$$x_{63} = -42.411501235316$$
$$x_{5} = -42.4115005850814$$
$$x_{8} = -39.2699084145515$$
$$x_{35} = -39.2699076683741$$
$$x_{58} = -36.1283154173375$$
$$x_{67} = -32.9867232184024$$
$$x_{9} = -32.9867224188086$$
$$x_{74} = -29.8451300954883$$
$$x_{41} = -26.7035379986821$$
$$x_{98} = -26.7035372004893$$
$$x_{14} = -23.5619450115115$$
$$x_{17} = -20.4203527082932$$
$$x_{28} = -20.4203520060805$$
$$x_{46} = -17.2787598356363$$
$$x_{34} = -17.2787590920677$$
$$x_{49} = -14.1371668370864$$
$$x_{55} = -10.9955746401247$$
$$x_{83} = -10.9955738413568$$
$$x_{2} = -7.85398149665124$$
$$x_{68} = -4.71238942125338$$
$$x_{18} = -4.71238862219396$$
$$x_{95} = -1.57079643188553$$
$$x_{23} = 1.57079582971902$$
$$x_{72} = 1.57079657289894$$
$$x_{65} = 4.71238874329685$$
$$x_{53} = 4.7123894841958$$
$$x_{70} = 7.85398174307326$$
$$x_{97} = 10.9955739381756$$
$$x_{69} = 10.9955747360645$$
$$x_{15} = 14.1371671100222$$
$$x_{51} = 17.2787591562062$$
$$x_{76} = 17.2787599560783$$
$$x_{75} = 20.4203521477756$$
$$x_{27} = 23.5619444059921$$
$$x_{85} = 23.5619451518571$$
$$x_{87} = 26.703537322248$$
$$x_{59} = 26.7035380604159$$
$$x_{101} = 29.8451303231501$$
$$x_{88} = 32.9867225164981$$
$$x_{42} = 32.9867233134552$$
$$x_{73} = 36.128315618579$$
$$x_{1} = 39.2699077336963$$
$$x_{10} = 39.2699085343272$$
$$x_{11} = 42.4115007274741$$
$$x_{64} = 45.5530929823099$$
$$x_{47} = 45.553093730794$$
$$x_{93} = 48.6946859012172$$
$$x_{52} = 48.6946866365921$$
$$x_{25} = 51.8362789031518$$
$$x_{54} = 54.9778710948428$$
$$x_{56} = 54.9778718908148$$
$$x_{79} = 58.1194643979608$$
$$x_{32} = 61.2610563112167$$
$$x_{43} = 61.2610571125526$$
$$x_{50} = 64.4026493072124$$
$$x_{80} = 67.5442415586719$$
$$x_{21} = 67.54424230971$$
$$x_{13} = 70.6858344802043$$
$$x_{44} = 70.6858352127237$$
$$x_{45} = 73.8274274830848$$
$$x_{22} = 76.9690196732095$$
$$x_{26} = 76.9690204681432$$
$$x_{31} = 80.1106131368654$$
$$x_{81} = 83.252204888767$$
$$x_{40} = 83.2522056907544$$
$$x_{39} = 86.3937978869933$$
$$x_{78} = 89.5353901350773$$
$$x_{62} = 89.535390888605$$
$$x_{12} = 92.6769830592094$$
$$x_{16} = 92.6769837789098$$
$$x_{24} = 95.8185760629547$$
$$x_{29} = 98.9601682515978$$
$$x_{61} = 98.9601690454399$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{92}$$
For example, let's take the point
$$x_{0} = x_{92} - \frac{1}{10}$$
=
$$-98.9601689530982 + - \frac{1}{10}$$
=
$$-99.0601689530982$$
substitute to the expression
$$\left|{\sin{\left(x \right)}}\right| \leq 1$$
$$\left|{\sin{\left(-99.0601689530982 \right)}}\right| \leq 1$$
0.995004128836795 <= 1
one of the solutions of our inequality is:
$$x \leq -98.9601689530982$$
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x92 x4 x86 x7 x48 x84 x77 x90 x20 x99 x91 x6 x71 x96 x100 x33 x37 x89 x3 x94 x19 x60 x66 x38 x82 x36 x57 x30 x63 x5 x8 x35 x58 x67 x9 x74 x41 x98 x14 x17 x28 x46 x34 x49 x55 x83 x2 x68 x18 x95 x23 x72 x65 x53 x70 x97 x69 x15 x51 x76 x75 x27 x85 x87 x59 x101 x88 x42 x73 x1 x10 x11 x64 x47 x93 x52 x25 x54 x56 x79 x32 x43 x50 x80 x21 x13 x44 x45 x22 x26 x31 x81 x40 x39 x78 x62 x12 x16 x24 x29 x61Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -98.9601689530982$$
$$x \geq -98.9601681513438 \wedge x \leq -95.8185758680502$$
$$x \geq -92.6769837307794 \wedge x \leq -92.6769829355125$$
$$x \geq -89.535390750197 \wedge x \leq -86.3937982346452$$
$$x \geq -86.3937977431483 \wedge x \leq -83.2522055723275$$
$$x \geq -83.2522048211133 \wedge x \leq -80.1106125781572$$
$$x \geq -76.9690203748894 \wedge x \leq -76.9690195738024$$
$$x \geq -73.8274272798455 \wedge x \leq -70.6858351534454$$
$$x \geq -70.6858343571487 \wedge x \leq -67.5442421706656$$
$$x \geq -64.4026497466373 \wedge x \leq -64.4026491641039$$
$$x \geq -61.2610569934486 \wedge x \leq -61.2610562447228$$
$$x \geq -58.1194639976905 \wedge x \leq -54.9778717966574$$
$$x \geq -54.9778709962906 \wedge x \leq -51.8362786893284$$
$$x \geq -48.6946865760795 \wedge x \leq -48.6946857788076$$
$$x \geq -45.5530935911043 \wedge x \leq -42.411501235316$$
$$x \geq -42.4115005850814 \wedge x \leq -39.2699084145515$$
$$x \geq -39.2699076683741 \wedge x \leq -36.1283154173375$$
$$x \geq -32.9867232184024 \wedge x \leq -32.9867224188086$$
$$x \geq -29.8451300954883 \wedge x \leq -26.7035379986821$$
$$x \geq -26.7035372004893 \wedge x \leq -23.5619450115115$$
$$x \geq -20.4203527082932 \wedge x \leq -20.4203520060805$$
$$x \geq -17.2787598356363 \wedge x \leq -17.2787590920677$$
$$x \geq -14.1371668370864 \wedge x \leq -10.9955746401247$$
$$x \geq -10.9955738413568 \wedge x \leq -7.85398149665124$$
$$x \geq -4.71238942125338 \wedge x \leq -4.71238862219396$$
$$x \geq -1.57079643188553 \wedge x \leq 1.57079582971902$$
$$x \geq 1.57079657289894 \wedge x \leq 4.71238874329685$$
$$x \geq 4.7123894841958 \wedge x \leq 7.85398174307326$$
$$x \geq 10.9955739381756 \wedge x \leq 10.9955747360645$$
$$x \geq 14.1371671100222 \wedge x \leq 17.2787591562062$$
$$x \geq 17.2787599560783 \wedge x \leq 20.4203521477756$$
$$x \geq 23.5619444059921 \wedge x \leq 23.5619451518571$$
$$x \geq 26.703537322248 \wedge x \leq 26.7035380604159$$
$$x \geq 29.8451303231501 \wedge x \leq 32.9867225164981$$
$$x \geq 32.9867233134552 \wedge x \leq 36.128315618579$$
$$x \geq 39.2699077336963 \wedge x \leq 39.2699085343272$$
$$x \geq 42.4115007274741 \wedge x \leq 45.5530929823099$$
$$x \geq 45.553093730794 \wedge x \leq 48.6946859012172$$
$$x \geq 48.6946866365921 \wedge x \leq 51.8362789031518$$
$$x \geq 54.9778710948428 \wedge x \leq 54.9778718908148$$
$$x \geq 58.1194643979608 \wedge x \leq 61.2610563112167$$
$$x \geq 61.2610571125526 \wedge x \leq 64.4026493072124$$
$$x \geq 67.5442415586719 \wedge x \leq 67.54424230971$$
$$x \geq 70.6858344802043 \wedge x \leq 70.6858352127237$$
$$x \geq 73.8274274830848 \wedge x \leq 76.9690196732095$$
$$x \geq 76.9690204681432 \wedge x \leq 80.1106131368654$$
$$x \geq 83.252204888767 \wedge x \leq 83.2522056907544$$
$$x \geq 86.3937978869933 \wedge x \leq 89.5353901350773$$
$$x \geq 89.535390888605 \wedge x \leq 92.6769830592094$$
$$x \geq 92.6769837789098 \wedge x \leq 95.8185760629547$$
$$x \geq 98.9601682515978 \wedge x \leq 98.9601690454399$$
Solving inequality on a graph