Sr Examen
Otras calculadoras
- Sistemas de desigualdades paso a paso
- Ecuaciones básicas paso a paso
- Sistemas de ecuaciones paso a paso
- Solución de desigualdades trigonométricas en línea
- Resolución de desigualdades exponenciales en línea
- Resolución de desigualdades con un módulo en línea
-
¿Cómo usar?
- Desigualdades:
-
x^2>1
-
-x^2+3x-2<0
-
(x-2)/(x-4)>0
-
x+1>0
- Integral de d{x}:
- |sin(x)|
-
Expresiones idénticas
- |sin(x)|<=sqrt(tres)/ dos
- módulo de seno de (x)| menos o igual a raíz cuadrada de (3) dividir por 2
- módulo de seno de (x)| menos o igual a raíz cuadrada de (tres) dividir por dos
- |sin(x)|<=√(3)/2
- |sinx|<=sqrt3/2
- |sin(x)|<=sqrt(3) dividir por 2
-
Expresiones semejantes
- |sinx|<=sqrt(3)/2
-
Expresiones con funciones
- Seno sin
- sin(x)^2<=1/2
- sin(x)>√2
- sin(x^2)<=1/2
- sin(-x/2)<=-1/2
- sin(x/2)>0
- Raíz cuadrada sqrt
- sqrt(x+5)+sqrt(5-x)>4
- sqrt(x+3)-sqrt(x-1)<sqrt(x-2)
- sqrt(x+5)/(x-1)<1
- sqrt(x^3-2*x^2+4*x-2)>=x
- sqrt(x-5)>-2
- Módulo |
- |x^2+10x+16|>=|x^2-16|
- |x-12|<=5
- |x-1|+|x+1|<4
- |x+1|+|x-4|>7
- |x+1|<=5
|sin(x)|<=sqrt(3)/2 desigualdades
Solución
Ha introducido
[src]
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|sin(x)| <= -----
2
$$\left|{\sin{\left(x \right)}}\right| \leq \frac{\sqrt{3}}{2}$$
Abs(sin(x)) <= sqrt(3)/2
Solución detallada
Se da la desigualdad:
$$\left|{\sin{\left(x \right)}}\right| \leq \frac{\sqrt{3}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{3}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{3}}{2}$$
cambiamos
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w - \frac{\sqrt{3}}{2} = 1$$
Dividamos ambos miembros de la ecuación en (w - sqrt(3)/2)/w
Obtenemos la respuesta: w = 1 + sqrt(3)/2
hacemos cambio inverso
$$\left|{\sin{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = -99.4837673636768$$
$$x_{2} = -60.7374579694027$$
$$x_{3} = -5.23598775598299$$
$$x_{4} = 2.0943951023932$$
$$x_{5} = 90.0589894029074$$
$$x_{6} = -93.2005820564972$$
$$x_{7} = -17.8023583703422$$
$$x_{8} = 19.8967534727354$$
$$x_{9} = -39.7935069454707$$
$$x_{10} = 61.7846555205993$$
$$x_{11} = -48.1710873550435$$
$$x_{12} = -10.471975511966$$
$$x_{13} = -70.162235930172$$
$$x_{14} = -79.5870138909414$$
$$x_{15} = -32.4631240870945$$
$$x_{16} = -71.2094334813686$$
$$x_{17} = -86.9173967493176$$
$$x_{18} = 83.7758040957278$$
$$x_{19} = -41.8879020478639$$
$$x_{20} = 30.3687289847013$$
$$x_{21} = -92.1533845053006$$
$$x_{22} = 89.0117918517108$$
$$x_{23} = 99.4837673636768$$
$$x_{24} = 86.9173967493176$$
$$x_{25} = 76.4454212373516$$
$$x_{26} = -19.8967534727354$$
$$x_{27} = 4.18879020478639$$
$$x_{28} = 85.870199198121$$
$$x_{29} = 8.37758040957278$$
$$x_{30} = -77.4926187885482$$
$$x_{31} = 60.7374579694027$$
$$x_{32} = -13.6135681655558$$
$$x_{33} = -49.2182849062401$$
$$x_{34} = 5.23598775598299$$
$$x_{35} = 68.0678408277789$$
$$x_{36} = 11.5191730631626$$
$$x_{37} = 77.4926187885482$$
$$x_{38} = -90.0589894029074$$
$$x_{39} = -85.870199198121$$
$$x_{40} = -57.5958653158129$$
$$x_{41} = -63.8790506229925$$
$$x_{42} = 33.5103216382911$$
$$x_{43} = -24.0855436775217$$
$$x_{44} = 1.0471975511966$$
$$x_{45} = 45.0294947014537$$
$$x_{46} = -16.7551608191456$$
$$x_{47} = 74.3510261349584$$
$$x_{48} = -35.6047167406843$$
$$x_{49} = -83.7758040957278$$
$$x_{50} = 96.342174710087$$
$$x_{51} = -98.4365698124802$$
$$x_{52} = -42.9350995990605$$
$$x_{53} = 71.2094334813686$$
$$x_{54} = -33.5103216382911$$
$$x_{55} = -61.7846555205993$$
$$x_{56} = -26.1799387799149$$
$$x_{57} = -36.6519142918809$$
$$x_{58} = 26.1799387799149$$
$$x_{59} = -80.634211442138$$
$$x_{60} = -38.7463093942741$$
$$x_{61} = 67.0206432765823$$
$$x_{62} = -76.4454212373516$$
$$x_{63} = 70.162235930172$$
$$x_{64} = 10.471975511966$$
$$x_{65} = 16.7551608191456$$
$$x_{66} = 55.5014702134197$$
$$x_{67} = 17.8023583703422$$
$$x_{68} = 48.1710873550435$$
$$x_{69} = 98.4365698124802$$
$$x_{70} = 82.7286065445312$$
$$x_{71} = -11.5191730631626$$
$$x_{72} = 92.1533845053006$$
$$x_{73} = -82.7286065445312$$
$$x_{74} = 52.3598775598299$$
$$x_{75} = 38.7463093942741$$
$$x_{76} = 27.2271363311115$$
$$x_{77} = 41.8879020478639$$
$$x_{78} = -4.18879020478639$$
$$x_{79} = 32.4631240870945$$
$$x_{80} = -55.5014702134197$$
$$x_{81} = -20.943951023932$$
$$x_{82} = -54.4542726622231$$
$$x_{83} = 39.7935069454707$$
$$x_{84} = 23.0383461263252$$
$$x_{85} = -2.0943951023932$$
$$x_{86} = -68.0678408277789$$
$$x_{87} = -64.9262481741891$$
$$x_{88} = 93.2005820564972$$
$$x_{89} = 24.0855436775217$$
$$x_{90} = 46.0766922526503$$
$$x_{91} = 49.2182849062401$$
$$x_{92} = 54.4542726622231$$
$$x_{93} = 63.8790506229925$$
$$x_{94} = -46.0766922526503$$
$$x_{95} = -27.2271363311115$$
$$x_{1} = -99.4837673636768$$
$$x_{2} = -60.7374579694027$$
$$x_{3} = -5.23598775598299$$
$$x_{4} = 2.0943951023932$$
$$x_{5} = 90.0589894029074$$
$$x_{6} = -93.2005820564972$$
$$x_{7} = -17.8023583703422$$
$$x_{8} = 19.8967534727354$$
$$x_{9} = -39.7935069454707$$
$$x_{10} = 61.7846555205993$$
$$x_{11} = -48.1710873550435$$
$$x_{12} = -10.471975511966$$
$$x_{13} = -70.162235930172$$
$$x_{14} = -79.5870138909414$$
$$x_{15} = -32.4631240870945$$
$$x_{16} = -71.2094334813686$$
$$x_{17} = -86.9173967493176$$
$$x_{18} = 83.7758040957278$$
$$x_{19} = -41.8879020478639$$
$$x_{20} = 30.3687289847013$$
$$x_{21} = -92.1533845053006$$
$$x_{22} = 89.0117918517108$$
$$x_{23} = 99.4837673636768$$
$$x_{24} = 86.9173967493176$$
$$x_{25} = 76.4454212373516$$
$$x_{26} = -19.8967534727354$$
$$x_{27} = 4.18879020478639$$
$$x_{28} = 85.870199198121$$
$$x_{29} = 8.37758040957278$$
$$x_{30} = -77.4926187885482$$
$$x_{31} = 60.7374579694027$$
$$x_{32} = -13.6135681655558$$
$$x_{33} = -49.2182849062401$$
$$x_{34} = 5.23598775598299$$
$$x_{35} = 68.0678408277789$$
$$x_{36} = 11.5191730631626$$
$$x_{37} = 77.4926187885482$$
$$x_{38} = -90.0589894029074$$
$$x_{39} = -85.870199198121$$
$$x_{40} = -57.5958653158129$$
$$x_{41} = -63.8790506229925$$
$$x_{42} = 33.5103216382911$$
$$x_{43} = -24.0855436775217$$
$$x_{44} = 1.0471975511966$$
$$x_{45} = 45.0294947014537$$
$$x_{46} = -16.7551608191456$$
$$x_{47} = 74.3510261349584$$
$$x_{48} = -35.6047167406843$$
$$x_{49} = -83.7758040957278$$
$$x_{50} = 96.342174710087$$
$$x_{51} = -98.4365698124802$$
$$x_{52} = -42.9350995990605$$
$$x_{53} = 71.2094334813686$$
$$x_{54} = -33.5103216382911$$
$$x_{55} = -61.7846555205993$$
$$x_{56} = -26.1799387799149$$
$$x_{57} = -36.6519142918809$$
$$x_{58} = 26.1799387799149$$
$$x_{59} = -80.634211442138$$
$$x_{60} = -38.7463093942741$$
$$x_{61} = 67.0206432765823$$
$$x_{62} = -76.4454212373516$$
$$x_{63} = 70.162235930172$$
$$x_{64} = 10.471975511966$$
$$x_{65} = 16.7551608191456$$
$$x_{66} = 55.5014702134197$$
$$x_{67} = 17.8023583703422$$
$$x_{68} = 48.1710873550435$$
$$x_{69} = 98.4365698124802$$
$$x_{70} = 82.7286065445312$$
$$x_{71} = -11.5191730631626$$
$$x_{72} = 92.1533845053006$$
$$x_{73} = -82.7286065445312$$
$$x_{74} = 52.3598775598299$$
$$x_{75} = 38.7463093942741$$
$$x_{76} = 27.2271363311115$$
$$x_{77} = 41.8879020478639$$
$$x_{78} = -4.18879020478639$$
$$x_{79} = 32.4631240870945$$
$$x_{80} = -55.5014702134197$$
$$x_{81} = -20.943951023932$$
$$x_{82} = -54.4542726622231$$
$$x_{83} = 39.7935069454707$$
$$x_{84} = 23.0383461263252$$
$$x_{85} = -2.0943951023932$$
$$x_{86} = -68.0678408277789$$
$$x_{87} = -64.9262481741891$$
$$x_{88} = 93.2005820564972$$
$$x_{89} = 24.0855436775217$$
$$x_{90} = 46.0766922526503$$
$$x_{91} = 49.2182849062401$$
$$x_{92} = 54.4542726622231$$
$$x_{93} = 63.8790506229925$$
$$x_{94} = -46.0766922526503$$
$$x_{95} = -27.2271363311115$$
Las raíces dadas
$$x_{1} = -99.4837673636768$$
$$x_{51} = -98.4365698124802$$
$$x_{6} = -93.2005820564972$$
$$x_{21} = -92.1533845053006$$
$$x_{38} = -90.0589894029074$$
$$x_{17} = -86.9173967493176$$
$$x_{39} = -85.870199198121$$
$$x_{49} = -83.7758040957278$$
$$x_{73} = -82.7286065445312$$
$$x_{59} = -80.634211442138$$
$$x_{14} = -79.5870138909414$$
$$x_{30} = -77.4926187885482$$
$$x_{62} = -76.4454212373516$$
$$x_{16} = -71.2094334813686$$
$$x_{13} = -70.162235930172$$
$$x_{86} = -68.0678408277789$$
$$x_{87} = -64.9262481741891$$
$$x_{41} = -63.8790506229925$$
$$x_{55} = -61.7846555205993$$
$$x_{2} = -60.7374579694027$$
$$x_{40} = -57.5958653158129$$
$$x_{80} = -55.5014702134197$$
$$x_{82} = -54.4542726622231$$
$$x_{33} = -49.2182849062401$$
$$x_{11} = -48.1710873550435$$
$$x_{94} = -46.0766922526503$$
$$x_{52} = -42.9350995990605$$
$$x_{19} = -41.8879020478639$$
$$x_{9} = -39.7935069454707$$
$$x_{60} = -38.7463093942741$$
$$x_{57} = -36.6519142918809$$
$$x_{48} = -35.6047167406843$$
$$x_{54} = -33.5103216382911$$
$$x_{15} = -32.4631240870945$$
$$x_{95} = -27.2271363311115$$
$$x_{56} = -26.1799387799149$$
$$x_{43} = -24.0855436775217$$
$$x_{81} = -20.943951023932$$
$$x_{26} = -19.8967534727354$$
$$x_{7} = -17.8023583703422$$
$$x_{46} = -16.7551608191456$$
$$x_{32} = -13.6135681655558$$
$$x_{71} = -11.5191730631626$$
$$x_{12} = -10.471975511966$$
$$x_{3} = -5.23598775598299$$
$$x_{78} = -4.18879020478639$$
$$x_{85} = -2.0943951023932$$
$$x_{44} = 1.0471975511966$$
$$x_{4} = 2.0943951023932$$
$$x_{27} = 4.18879020478639$$
$$x_{34} = 5.23598775598299$$
$$x_{29} = 8.37758040957278$$
$$x_{64} = 10.471975511966$$
$$x_{36} = 11.5191730631626$$
$$x_{65} = 16.7551608191456$$
$$x_{67} = 17.8023583703422$$
$$x_{8} = 19.8967534727354$$
$$x_{84} = 23.0383461263252$$
$$x_{89} = 24.0855436775217$$
$$x_{58} = 26.1799387799149$$
$$x_{76} = 27.2271363311115$$
$$x_{20} = 30.3687289847013$$
$$x_{79} = 32.4631240870945$$
$$x_{42} = 33.5103216382911$$
$$x_{75} = 38.7463093942741$$
$$x_{83} = 39.7935069454707$$
$$x_{77} = 41.8879020478639$$
$$x_{45} = 45.0294947014537$$
$$x_{90} = 46.0766922526503$$
$$x_{68} = 48.1710873550435$$
$$x_{91} = 49.2182849062401$$
$$x_{74} = 52.3598775598299$$
$$x_{92} = 54.4542726622231$$
$$x_{66} = 55.5014702134197$$
$$x_{31} = 60.7374579694027$$
$$x_{10} = 61.7846555205993$$
$$x_{93} = 63.8790506229925$$
$$x_{61} = 67.0206432765823$$
$$x_{35} = 68.0678408277789$$
$$x_{63} = 70.162235930172$$
$$x_{53} = 71.2094334813686$$
$$x_{47} = 74.3510261349584$$
$$x_{25} = 76.4454212373516$$
$$x_{37} = 77.4926187885482$$
$$x_{70} = 82.7286065445312$$
$$x_{18} = 83.7758040957278$$
$$x_{28} = 85.870199198121$$
$$x_{24} = 86.9173967493176$$
$$x_{22} = 89.0117918517108$$
$$x_{5} = 90.0589894029074$$
$$x_{72} = 92.1533845053006$$
$$x_{88} = 93.2005820564972$$
$$x_{50} = 96.342174710087$$
$$x_{69} = 98.4365698124802$$
$$x_{23} = 99.4837673636768$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-99.4837673636768 + - \frac{1}{10}$$
=
$$-99.5837673636768$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| \leq \frac{\sqrt{3}}{2}$$
$$\left|{\sin{\left(-99.5837673636768 \right)}}\right| \leq \frac{\sqrt{3}}{2}$$
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -99.4837673636768$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -99.4837673636768$$
$$x \geq -98.4365698124802 \wedge x \leq -93.2005820564972$$
$$x \geq -92.1533845053006 \wedge x \leq -90.0589894029074$$
$$x \geq -86.9173967493176 \wedge x \leq -85.870199198121$$
$$x \geq -83.7758040957278 \wedge x \leq -82.7286065445312$$
$$x \geq -80.634211442138 \wedge x \leq -79.5870138909414$$
$$x \geq -77.4926187885482 \wedge x \leq -76.4454212373516$$
$$x \geq -71.2094334813686 \wedge x \leq -70.162235930172$$
$$x \geq -68.0678408277789 \wedge x \leq -64.9262481741891$$
$$x \geq -63.8790506229925 \wedge x \leq -61.7846555205993$$
$$x \geq -60.7374579694027 \wedge x \leq -57.5958653158129$$
$$x \geq -55.5014702134197 \wedge x \leq -54.4542726622231$$
$$x \geq -49.2182849062401 \wedge x \leq -48.1710873550435$$
$$x \geq -46.0766922526503 \wedge x \leq -42.9350995990605$$
$$x \geq -41.8879020478639 \wedge x \leq -39.7935069454707$$
$$x \geq -38.7463093942741 \wedge x \leq -36.6519142918809$$
$$x \geq -35.6047167406843 \wedge x \leq -33.5103216382911$$
$$x \geq -32.4631240870945 \wedge x \leq -27.2271363311115$$
$$x \geq -26.1799387799149 \wedge x \leq -24.0855436775217$$
$$x \geq -20.943951023932 \wedge x \leq -19.8967534727354$$
$$x \geq -17.8023583703422 \wedge x \leq -16.7551608191456$$
$$x \geq -13.6135681655558 \wedge x \leq -11.5191730631626$$
$$x \geq -10.471975511966 \wedge x \leq -5.23598775598299$$
$$x \geq -4.18879020478639 \wedge x \leq -2.0943951023932$$
$$x \geq 1.0471975511966 \wedge x \leq 2.0943951023932$$
$$x \geq 4.18879020478639 \wedge x \leq 5.23598775598299$$
$$x \geq 8.37758040957278 \wedge x \leq 10.471975511966$$
$$x \geq 11.5191730631626 \wedge x \leq 16.7551608191456$$
$$x \geq 17.8023583703422 \wedge x \leq 19.8967534727354$$
$$x \geq 23.0383461263252 \wedge x \leq 24.0855436775217$$
$$x \geq 26.1799387799149 \wedge x \leq 27.2271363311115$$
$$x \geq 30.3687289847013 \wedge x \leq 32.4631240870945$$
$$x \geq 33.5103216382911 \wedge x \leq 38.7463093942741$$
$$x \geq 39.7935069454707 \wedge x \leq 41.8879020478639$$
$$x \geq 45.0294947014537 \wedge x \leq 46.0766922526503$$
$$x \geq 48.1710873550435 \wedge x \leq 49.2182849062401$$
$$x \geq 52.3598775598299 \wedge x \leq 54.4542726622231$$
$$x \geq 55.5014702134197 \wedge x \leq 60.7374579694027$$
$$x \geq 61.7846555205993 \wedge x \leq 63.8790506229925$$
$$x \geq 67.0206432765823 \wedge x \leq 68.0678408277789$$
$$x \geq 70.162235930172 \wedge x \leq 71.2094334813686$$
$$x \geq 74.3510261349584 \wedge x \leq 76.4454212373516$$
$$x \geq 77.4926187885482 \wedge x \leq 82.7286065445312$$
$$x \geq 83.7758040957278 \wedge x \leq 85.870199198121$$
$$x \geq 86.9173967493176 \wedge x \leq 89.0117918517108$$
$$x \geq 90.0589894029074 \wedge x \leq 92.1533845053006$$
$$x \geq 93.2005820564972 \wedge x \leq 96.342174710087$$
$$x \geq 98.4365698124802 \wedge x \leq 99.4837673636768$$
$$\left|{\sin{\left(x \right)}}\right| \leq \frac{\sqrt{3}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{3}}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)}}\right| = \frac{\sqrt{3}}{2}$$
cambiamos
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - 1 - \frac{\sqrt{3}}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)}}\right|$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 + w - sqrt3/2 = 0
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w - \frac{\sqrt{3}}{2} = 1$$
Dividamos ambos miembros de la ecuación en (w - sqrt(3)/2)/w
w = 1 / ((w - sqrt(3)/2)/w)
Obtenemos la respuesta: w = 1 + sqrt(3)/2
hacemos cambio inverso
$$\left|{\sin{\left(x \right)}}\right| = w$$
sustituimos w:
$$x_{1} = -99.4837673636768$$
$$x_{2} = -60.7374579694027$$
$$x_{3} = -5.23598775598299$$
$$x_{4} = 2.0943951023932$$
$$x_{5} = 90.0589894029074$$
$$x_{6} = -93.2005820564972$$
$$x_{7} = -17.8023583703422$$
$$x_{8} = 19.8967534727354$$
$$x_{9} = -39.7935069454707$$
$$x_{10} = 61.7846555205993$$
$$x_{11} = -48.1710873550435$$
$$x_{12} = -10.471975511966$$
$$x_{13} = -70.162235930172$$
$$x_{14} = -79.5870138909414$$
$$x_{15} = -32.4631240870945$$
$$x_{16} = -71.2094334813686$$
$$x_{17} = -86.9173967493176$$
$$x_{18} = 83.7758040957278$$
$$x_{19} = -41.8879020478639$$
$$x_{20} = 30.3687289847013$$
$$x_{21} = -92.1533845053006$$
$$x_{22} = 89.0117918517108$$
$$x_{23} = 99.4837673636768$$
$$x_{24} = 86.9173967493176$$
$$x_{25} = 76.4454212373516$$
$$x_{26} = -19.8967534727354$$
$$x_{27} = 4.18879020478639$$
$$x_{28} = 85.870199198121$$
$$x_{29} = 8.37758040957278$$
$$x_{30} = -77.4926187885482$$
$$x_{31} = 60.7374579694027$$
$$x_{32} = -13.6135681655558$$
$$x_{33} = -49.2182849062401$$
$$x_{34} = 5.23598775598299$$
$$x_{35} = 68.0678408277789$$
$$x_{36} = 11.5191730631626$$
$$x_{37} = 77.4926187885482$$
$$x_{38} = -90.0589894029074$$
$$x_{39} = -85.870199198121$$
$$x_{40} = -57.5958653158129$$
$$x_{41} = -63.8790506229925$$
$$x_{42} = 33.5103216382911$$
$$x_{43} = -24.0855436775217$$
$$x_{44} = 1.0471975511966$$
$$x_{45} = 45.0294947014537$$
$$x_{46} = -16.7551608191456$$
$$x_{47} = 74.3510261349584$$
$$x_{48} = -35.6047167406843$$
$$x_{49} = -83.7758040957278$$
$$x_{50} = 96.342174710087$$
$$x_{51} = -98.4365698124802$$
$$x_{52} = -42.9350995990605$$
$$x_{53} = 71.2094334813686$$
$$x_{54} = -33.5103216382911$$
$$x_{55} = -61.7846555205993$$
$$x_{56} = -26.1799387799149$$
$$x_{57} = -36.6519142918809$$
$$x_{58} = 26.1799387799149$$
$$x_{59} = -80.634211442138$$
$$x_{60} = -38.7463093942741$$
$$x_{61} = 67.0206432765823$$
$$x_{62} = -76.4454212373516$$
$$x_{63} = 70.162235930172$$
$$x_{64} = 10.471975511966$$
$$x_{65} = 16.7551608191456$$
$$x_{66} = 55.5014702134197$$
$$x_{67} = 17.8023583703422$$
$$x_{68} = 48.1710873550435$$
$$x_{69} = 98.4365698124802$$
$$x_{70} = 82.7286065445312$$
$$x_{71} = -11.5191730631626$$
$$x_{72} = 92.1533845053006$$
$$x_{73} = -82.7286065445312$$
$$x_{74} = 52.3598775598299$$
$$x_{75} = 38.7463093942741$$
$$x_{76} = 27.2271363311115$$
$$x_{77} = 41.8879020478639$$
$$x_{78} = -4.18879020478639$$
$$x_{79} = 32.4631240870945$$
$$x_{80} = -55.5014702134197$$
$$x_{81} = -20.943951023932$$
$$x_{82} = -54.4542726622231$$
$$x_{83} = 39.7935069454707$$
$$x_{84} = 23.0383461263252$$
$$x_{85} = -2.0943951023932$$
$$x_{86} = -68.0678408277789$$
$$x_{87} = -64.9262481741891$$
$$x_{88} = 93.2005820564972$$
$$x_{89} = 24.0855436775217$$
$$x_{90} = 46.0766922526503$$
$$x_{91} = 49.2182849062401$$
$$x_{92} = 54.4542726622231$$
$$x_{93} = 63.8790506229925$$
$$x_{94} = -46.0766922526503$$
$$x_{95} = -27.2271363311115$$
$$x_{1} = -99.4837673636768$$
$$x_{2} = -60.7374579694027$$
$$x_{3} = -5.23598775598299$$
$$x_{4} = 2.0943951023932$$
$$x_{5} = 90.0589894029074$$
$$x_{6} = -93.2005820564972$$
$$x_{7} = -17.8023583703422$$
$$x_{8} = 19.8967534727354$$
$$x_{9} = -39.7935069454707$$
$$x_{10} = 61.7846555205993$$
$$x_{11} = -48.1710873550435$$
$$x_{12} = -10.471975511966$$
$$x_{13} = -70.162235930172$$
$$x_{14} = -79.5870138909414$$
$$x_{15} = -32.4631240870945$$
$$x_{16} = -71.2094334813686$$
$$x_{17} = -86.9173967493176$$
$$x_{18} = 83.7758040957278$$
$$x_{19} = -41.8879020478639$$
$$x_{20} = 30.3687289847013$$
$$x_{21} = -92.1533845053006$$
$$x_{22} = 89.0117918517108$$
$$x_{23} = 99.4837673636768$$
$$x_{24} = 86.9173967493176$$
$$x_{25} = 76.4454212373516$$
$$x_{26} = -19.8967534727354$$
$$x_{27} = 4.18879020478639$$
$$x_{28} = 85.870199198121$$
$$x_{29} = 8.37758040957278$$
$$x_{30} = -77.4926187885482$$
$$x_{31} = 60.7374579694027$$
$$x_{32} = -13.6135681655558$$
$$x_{33} = -49.2182849062401$$
$$x_{34} = 5.23598775598299$$
$$x_{35} = 68.0678408277789$$
$$x_{36} = 11.5191730631626$$
$$x_{37} = 77.4926187885482$$
$$x_{38} = -90.0589894029074$$
$$x_{39} = -85.870199198121$$
$$x_{40} = -57.5958653158129$$
$$x_{41} = -63.8790506229925$$
$$x_{42} = 33.5103216382911$$
$$x_{43} = -24.0855436775217$$
$$x_{44} = 1.0471975511966$$
$$x_{45} = 45.0294947014537$$
$$x_{46} = -16.7551608191456$$
$$x_{47} = 74.3510261349584$$
$$x_{48} = -35.6047167406843$$
$$x_{49} = -83.7758040957278$$
$$x_{50} = 96.342174710087$$
$$x_{51} = -98.4365698124802$$
$$x_{52} = -42.9350995990605$$
$$x_{53} = 71.2094334813686$$
$$x_{54} = -33.5103216382911$$
$$x_{55} = -61.7846555205993$$
$$x_{56} = -26.1799387799149$$
$$x_{57} = -36.6519142918809$$
$$x_{58} = 26.1799387799149$$
$$x_{59} = -80.634211442138$$
$$x_{60} = -38.7463093942741$$
$$x_{61} = 67.0206432765823$$
$$x_{62} = -76.4454212373516$$
$$x_{63} = 70.162235930172$$
$$x_{64} = 10.471975511966$$
$$x_{65} = 16.7551608191456$$
$$x_{66} = 55.5014702134197$$
$$x_{67} = 17.8023583703422$$
$$x_{68} = 48.1710873550435$$
$$x_{69} = 98.4365698124802$$
$$x_{70} = 82.7286065445312$$
$$x_{71} = -11.5191730631626$$
$$x_{72} = 92.1533845053006$$
$$x_{73} = -82.7286065445312$$
$$x_{74} = 52.3598775598299$$
$$x_{75} = 38.7463093942741$$
$$x_{76} = 27.2271363311115$$
$$x_{77} = 41.8879020478639$$
$$x_{78} = -4.18879020478639$$
$$x_{79} = 32.4631240870945$$
$$x_{80} = -55.5014702134197$$
$$x_{81} = -20.943951023932$$
$$x_{82} = -54.4542726622231$$
$$x_{83} = 39.7935069454707$$
$$x_{84} = 23.0383461263252$$
$$x_{85} = -2.0943951023932$$
$$x_{86} = -68.0678408277789$$
$$x_{87} = -64.9262481741891$$
$$x_{88} = 93.2005820564972$$
$$x_{89} = 24.0855436775217$$
$$x_{90} = 46.0766922526503$$
$$x_{91} = 49.2182849062401$$
$$x_{92} = 54.4542726622231$$
$$x_{93} = 63.8790506229925$$
$$x_{94} = -46.0766922526503$$
$$x_{95} = -27.2271363311115$$
Las raíces dadas
$$x_{1} = -99.4837673636768$$
$$x_{51} = -98.4365698124802$$
$$x_{6} = -93.2005820564972$$
$$x_{21} = -92.1533845053006$$
$$x_{38} = -90.0589894029074$$
$$x_{17} = -86.9173967493176$$
$$x_{39} = -85.870199198121$$
$$x_{49} = -83.7758040957278$$
$$x_{73} = -82.7286065445312$$
$$x_{59} = -80.634211442138$$
$$x_{14} = -79.5870138909414$$
$$x_{30} = -77.4926187885482$$
$$x_{62} = -76.4454212373516$$
$$x_{16} = -71.2094334813686$$
$$x_{13} = -70.162235930172$$
$$x_{86} = -68.0678408277789$$
$$x_{87} = -64.9262481741891$$
$$x_{41} = -63.8790506229925$$
$$x_{55} = -61.7846555205993$$
$$x_{2} = -60.7374579694027$$
$$x_{40} = -57.5958653158129$$
$$x_{80} = -55.5014702134197$$
$$x_{82} = -54.4542726622231$$
$$x_{33} = -49.2182849062401$$
$$x_{11} = -48.1710873550435$$
$$x_{94} = -46.0766922526503$$
$$x_{52} = -42.9350995990605$$
$$x_{19} = -41.8879020478639$$
$$x_{9} = -39.7935069454707$$
$$x_{60} = -38.7463093942741$$
$$x_{57} = -36.6519142918809$$
$$x_{48} = -35.6047167406843$$
$$x_{54} = -33.5103216382911$$
$$x_{15} = -32.4631240870945$$
$$x_{95} = -27.2271363311115$$
$$x_{56} = -26.1799387799149$$
$$x_{43} = -24.0855436775217$$
$$x_{81} = -20.943951023932$$
$$x_{26} = -19.8967534727354$$
$$x_{7} = -17.8023583703422$$
$$x_{46} = -16.7551608191456$$
$$x_{32} = -13.6135681655558$$
$$x_{71} = -11.5191730631626$$
$$x_{12} = -10.471975511966$$
$$x_{3} = -5.23598775598299$$
$$x_{78} = -4.18879020478639$$
$$x_{85} = -2.0943951023932$$
$$x_{44} = 1.0471975511966$$
$$x_{4} = 2.0943951023932$$
$$x_{27} = 4.18879020478639$$
$$x_{34} = 5.23598775598299$$
$$x_{29} = 8.37758040957278$$
$$x_{64} = 10.471975511966$$
$$x_{36} = 11.5191730631626$$
$$x_{65} = 16.7551608191456$$
$$x_{67} = 17.8023583703422$$
$$x_{8} = 19.8967534727354$$
$$x_{84} = 23.0383461263252$$
$$x_{89} = 24.0855436775217$$
$$x_{58} = 26.1799387799149$$
$$x_{76} = 27.2271363311115$$
$$x_{20} = 30.3687289847013$$
$$x_{79} = 32.4631240870945$$
$$x_{42} = 33.5103216382911$$
$$x_{75} = 38.7463093942741$$
$$x_{83} = 39.7935069454707$$
$$x_{77} = 41.8879020478639$$
$$x_{45} = 45.0294947014537$$
$$x_{90} = 46.0766922526503$$
$$x_{68} = 48.1710873550435$$
$$x_{91} = 49.2182849062401$$
$$x_{74} = 52.3598775598299$$
$$x_{92} = 54.4542726622231$$
$$x_{66} = 55.5014702134197$$
$$x_{31} = 60.7374579694027$$
$$x_{10} = 61.7846555205993$$
$$x_{93} = 63.8790506229925$$
$$x_{61} = 67.0206432765823$$
$$x_{35} = 68.0678408277789$$
$$x_{63} = 70.162235930172$$
$$x_{53} = 71.2094334813686$$
$$x_{47} = 74.3510261349584$$
$$x_{25} = 76.4454212373516$$
$$x_{37} = 77.4926187885482$$
$$x_{70} = 82.7286065445312$$
$$x_{18} = 83.7758040957278$$
$$x_{28} = 85.870199198121$$
$$x_{24} = 86.9173967493176$$
$$x_{22} = 89.0117918517108$$
$$x_{5} = 90.0589894029074$$
$$x_{72} = 92.1533845053006$$
$$x_{88} = 93.2005820564972$$
$$x_{50} = 96.342174710087$$
$$x_{69} = 98.4365698124802$$
$$x_{23} = 99.4837673636768$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-99.4837673636768 + - \frac{1}{10}$$
=
$$-99.5837673636768$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)}}\right| \leq \frac{\sqrt{3}}{2}$$
$$\left|{\sin{\left(-99.5837673636768 \right)}}\right| \leq \frac{\sqrt{3}}{2}$$
___
\/ 3
0.811782175678687 <= -----
2
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -99.4837673636768$$
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x1 x51 x6 x21 x38 x17 x39 x49 x73 x59 x14 x30 x62 x16 x13 x86 x87 x41 x55 x2 x40 x80 x82 x33 x11 x94 x52 x19 x9 x60 x57 x48 x54 x15 x95 x56 x43 x81 x26 x7 x46 x32 x71 x12 x3 x78 x85 x44 x4 x27 x34 x29 x64 x36 x65 x67 x8 x84 x89 x58 x76 x20 x79 x42 x75 x83 x77 x45 x90 x68 x91 x74 x92 x66 x31 x10 x93 x61 x35 x63 x53 x47 x25 x37 x70 x18 x28 x24 x22 x5 x72 x88 x50 x69 x23Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -99.4837673636768$$
$$x \geq -98.4365698124802 \wedge x \leq -93.2005820564972$$
$$x \geq -92.1533845053006 \wedge x \leq -90.0589894029074$$
$$x \geq -86.9173967493176 \wedge x \leq -85.870199198121$$
$$x \geq -83.7758040957278 \wedge x \leq -82.7286065445312$$
$$x \geq -80.634211442138 \wedge x \leq -79.5870138909414$$
$$x \geq -77.4926187885482 \wedge x \leq -76.4454212373516$$
$$x \geq -71.2094334813686 \wedge x \leq -70.162235930172$$
$$x \geq -68.0678408277789 \wedge x \leq -64.9262481741891$$
$$x \geq -63.8790506229925 \wedge x \leq -61.7846555205993$$
$$x \geq -60.7374579694027 \wedge x \leq -57.5958653158129$$
$$x \geq -55.5014702134197 \wedge x \leq -54.4542726622231$$
$$x \geq -49.2182849062401 \wedge x \leq -48.1710873550435$$
$$x \geq -46.0766922526503 \wedge x \leq -42.9350995990605$$
$$x \geq -41.8879020478639 \wedge x \leq -39.7935069454707$$
$$x \geq -38.7463093942741 \wedge x \leq -36.6519142918809$$
$$x \geq -35.6047167406843 \wedge x \leq -33.5103216382911$$
$$x \geq -32.4631240870945 \wedge x \leq -27.2271363311115$$
$$x \geq -26.1799387799149 \wedge x \leq -24.0855436775217$$
$$x \geq -20.943951023932 \wedge x \leq -19.8967534727354$$
$$x \geq -17.8023583703422 \wedge x \leq -16.7551608191456$$
$$x \geq -13.6135681655558 \wedge x \leq -11.5191730631626$$
$$x \geq -10.471975511966 \wedge x \leq -5.23598775598299$$
$$x \geq -4.18879020478639 \wedge x \leq -2.0943951023932$$
$$x \geq 1.0471975511966 \wedge x \leq 2.0943951023932$$
$$x \geq 4.18879020478639 \wedge x \leq 5.23598775598299$$
$$x \geq 8.37758040957278 \wedge x \leq 10.471975511966$$
$$x \geq 11.5191730631626 \wedge x \leq 16.7551608191456$$
$$x \geq 17.8023583703422 \wedge x \leq 19.8967534727354$$
$$x \geq 23.0383461263252 \wedge x \leq 24.0855436775217$$
$$x \geq 26.1799387799149 \wedge x \leq 27.2271363311115$$
$$x \geq 30.3687289847013 \wedge x \leq 32.4631240870945$$
$$x \geq 33.5103216382911 \wedge x \leq 38.7463093942741$$
$$x \geq 39.7935069454707 \wedge x \leq 41.8879020478639$$
$$x \geq 45.0294947014537 \wedge x \leq 46.0766922526503$$
$$x \geq 48.1710873550435 \wedge x \leq 49.2182849062401$$
$$x \geq 52.3598775598299 \wedge x \leq 54.4542726622231$$
$$x \geq 55.5014702134197 \wedge x \leq 60.7374579694027$$
$$x \geq 61.7846555205993 \wedge x \leq 63.8790506229925$$
$$x \geq 67.0206432765823 \wedge x \leq 68.0678408277789$$
$$x \geq 70.162235930172 \wedge x \leq 71.2094334813686$$
$$x \geq 74.3510261349584 \wedge x \leq 76.4454212373516$$
$$x \geq 77.4926187885482 \wedge x \leq 82.7286065445312$$
$$x \geq 83.7758040957278 \wedge x \leq 85.870199198121$$
$$x \geq 86.9173967493176 \wedge x \leq 89.0117918517108$$
$$x \geq 90.0589894029074 \wedge x \leq 92.1533845053006$$
$$x \geq 93.2005820564972 \wedge x \leq 96.342174710087$$
$$x \geq 98.4365698124802 \wedge x \leq 99.4837673636768$$
Solución de la desigualdad en el gráfico