Sr Examen

sint<0,4 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
sin(t) < 2/5
$$\sin{\left(t \right)} < \frac{2}{5}$$
sin(t) < 2/5
Solución detallada
Se da la desigualdad:
$$\sin{\left(t \right)} < \frac{2}{5}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sin{\left(t \right)} = \frac{2}{5}$$
Resolvemos:
Tenemos la ecuación
$$\sin{\left(t \right)} = \frac{2}{5}$$
cambiamos
$$\sin{\left(t \right)} - \frac{2}{5} = 0$$
$$\sin{\left(t \right)} - \frac{2}{5} = 0$$
Sustituimos
$$w = \sin{\left(t \right)}$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = \frac{2}{5}$$
Obtenemos la respuesta: w = 2/5
hacemos cambio inverso
$$\sin{\left(t \right)} = w$$
sustituimos w:
$$x_{1} = -87.5530774544467$$
$$x_{2} = 40.4291876505998$$
$$x_{3} = 82.0929258394021$$
$$x_{4} = -22.402665421196$$
$$x_{5} = -41.2522213427348$$
$$x_{6} = 6.69470215324707$$
$$x_{7} = -16.1194801140165$$
$$x_{8} = 12.9778874604267$$
$$x_{9} = 46.7123729577794$$
$$x_{10} = 75.8097405322225$$
$$x_{11} = -72.6681478786327$$
$$x_{12} = -68.703521532908$$
$$x_{13} = 63.2433699178634$$
$$x_{14} = -31.0044096898304$$
$$x_{15} = -62.4203362257284$$
$$x_{16} = -66.3849625714531$$
$$x_{17} = -93.8362627616263$$
$$x_{18} = 15.2964464218815$$
$$x_{19} = -43.5707803041896$$
$$x_{20} = -3.55310949965728$$
$$x_{21} = -81.2698921472671$$
$$x_{22} = 71.8451141864978$$
$$x_{23} = -28.6858507283756$$
$$x_{24} = -60.1017772642736$$
$$x_{25} = 84.4114848008569$$
$$x_{26} = -37.28759499701$$
$$x_{27} = 94.6592964537613$$
$$x_{28} = -91.5177038001715$$
$$x_{29} = 44.3938139963246$$
$$x_{30} = 34.1460023434202$$
$$x_{31} = 100.942481760941$$
$$x_{32} = -47.5354066499144$$
$$x_{33} = -74.9867068400875$$
$$x_{34} = 59.2787435721386$$
$$x_{35} = -34.9690360355552$$
$$x_{36} = 56.9601846106838$$
$$x_{37} = -85.2345184929919$$
$$x_{38} = 27.8628170362407$$
$$x_{39} = 65.5619288793182$$
$$x_{40} = -78.9513331858123$$
$$x_{41} = -18.4380390754713$$
$$x_{42} = -100.119448068806$$
$$x_{43} = 2.73007580752231$$
$$x_{44} = -49.8539656113692$$
$$x_{45} = -9.83629480683687$$
$$x_{46} = 0.411516846067488$$
$$x_{47} = 78.1282994936773$$
$$x_{48} = 697.845085943002$$
$$x_{49} = 52.995558264959$$
$$x_{50} = 21.5796317290611$$
$$x_{51} = -12.1548537682917$$
$$x_{52} = 19.2610727676062$$
$$x_{53} = 38.110628689145$$
$$x_{54} = 50.6769993035042$$
$$x_{55} = 88.3761111465817$$
$$x_{56} = 90.6946701080365$$
$$x_{57} = 96.9778554152161$$
$$x_{58} = -97.8008891073511$$
$$x_{59} = 25.5442580747858$$
$$x_{60} = -56.1371509185488$$
$$x_{61} = 9.01326111470189$$
$$x_{62} = -53.818591957094$$
$$x_{63} = -24.7212243826509$$
$$x_{64} = -5.8716684611121$$
$$x_{65} = 69.5265552250429$$
$$x_{66} = 31.8274433819654$$
$$x_{1} = -87.5530774544467$$
$$x_{2} = 40.4291876505998$$
$$x_{3} = 82.0929258394021$$
$$x_{4} = -22.402665421196$$
$$x_{5} = -41.2522213427348$$
$$x_{6} = 6.69470215324707$$
$$x_{7} = -16.1194801140165$$
$$x_{8} = 12.9778874604267$$
$$x_{9} = 46.7123729577794$$
$$x_{10} = 75.8097405322225$$
$$x_{11} = -72.6681478786327$$
$$x_{12} = -68.703521532908$$
$$x_{13} = 63.2433699178634$$
$$x_{14} = -31.0044096898304$$
$$x_{15} = -62.4203362257284$$
$$x_{16} = -66.3849625714531$$
$$x_{17} = -93.8362627616263$$
$$x_{18} = 15.2964464218815$$
$$x_{19} = -43.5707803041896$$
$$x_{20} = -3.55310949965728$$
$$x_{21} = -81.2698921472671$$
$$x_{22} = 71.8451141864978$$
$$x_{23} = -28.6858507283756$$
$$x_{24} = -60.1017772642736$$
$$x_{25} = 84.4114848008569$$
$$x_{26} = -37.28759499701$$
$$x_{27} = 94.6592964537613$$
$$x_{28} = -91.5177038001715$$
$$x_{29} = 44.3938139963246$$
$$x_{30} = 34.1460023434202$$
$$x_{31} = 100.942481760941$$
$$x_{32} = -47.5354066499144$$
$$x_{33} = -74.9867068400875$$
$$x_{34} = 59.2787435721386$$
$$x_{35} = -34.9690360355552$$
$$x_{36} = 56.9601846106838$$
$$x_{37} = -85.2345184929919$$
$$x_{38} = 27.8628170362407$$
$$x_{39} = 65.5619288793182$$
$$x_{40} = -78.9513331858123$$
$$x_{41} = -18.4380390754713$$
$$x_{42} = -100.119448068806$$
$$x_{43} = 2.73007580752231$$
$$x_{44} = -49.8539656113692$$
$$x_{45} = -9.83629480683687$$
$$x_{46} = 0.411516846067488$$
$$x_{47} = 78.1282994936773$$
$$x_{48} = 697.845085943002$$
$$x_{49} = 52.995558264959$$
$$x_{50} = 21.5796317290611$$
$$x_{51} = -12.1548537682917$$
$$x_{52} = 19.2610727676062$$
$$x_{53} = 38.110628689145$$
$$x_{54} = 50.6769993035042$$
$$x_{55} = 88.3761111465817$$
$$x_{56} = 90.6946701080365$$
$$x_{57} = 96.9778554152161$$
$$x_{58} = -97.8008891073511$$
$$x_{59} = 25.5442580747858$$
$$x_{60} = -56.1371509185488$$
$$x_{61} = 9.01326111470189$$
$$x_{62} = -53.818591957094$$
$$x_{63} = -24.7212243826509$$
$$x_{64} = -5.8716684611121$$
$$x_{65} = 69.5265552250429$$
$$x_{66} = 31.8274433819654$$
Las raíces dadas
$$x_{42} = -100.119448068806$$
$$x_{58} = -97.8008891073511$$
$$x_{17} = -93.8362627616263$$
$$x_{28} = -91.5177038001715$$
$$x_{1} = -87.5530774544467$$
$$x_{37} = -85.2345184929919$$
$$x_{21} = -81.2698921472671$$
$$x_{40} = -78.9513331858123$$
$$x_{33} = -74.9867068400875$$
$$x_{11} = -72.6681478786327$$
$$x_{12} = -68.703521532908$$
$$x_{16} = -66.3849625714531$$
$$x_{15} = -62.4203362257284$$
$$x_{24} = -60.1017772642736$$
$$x_{60} = -56.1371509185488$$
$$x_{62} = -53.818591957094$$
$$x_{44} = -49.8539656113692$$
$$x_{32} = -47.5354066499144$$
$$x_{19} = -43.5707803041896$$
$$x_{5} = -41.2522213427348$$
$$x_{26} = -37.28759499701$$
$$x_{35} = -34.9690360355552$$
$$x_{14} = -31.0044096898304$$
$$x_{23} = -28.6858507283756$$
$$x_{63} = -24.7212243826509$$
$$x_{4} = -22.402665421196$$
$$x_{41} = -18.4380390754713$$
$$x_{7} = -16.1194801140165$$
$$x_{51} = -12.1548537682917$$
$$x_{45} = -9.83629480683687$$
$$x_{64} = -5.8716684611121$$
$$x_{20} = -3.55310949965728$$
$$x_{46} = 0.411516846067488$$
$$x_{43} = 2.73007580752231$$
$$x_{6} = 6.69470215324707$$
$$x_{61} = 9.01326111470189$$
$$x_{8} = 12.9778874604267$$
$$x_{18} = 15.2964464218815$$
$$x_{52} = 19.2610727676062$$
$$x_{50} = 21.5796317290611$$
$$x_{59} = 25.5442580747858$$
$$x_{38} = 27.8628170362407$$
$$x_{66} = 31.8274433819654$$
$$x_{30} = 34.1460023434202$$
$$x_{53} = 38.110628689145$$
$$x_{2} = 40.4291876505998$$
$$x_{29} = 44.3938139963246$$
$$x_{9} = 46.7123729577794$$
$$x_{54} = 50.6769993035042$$
$$x_{49} = 52.995558264959$$
$$x_{36} = 56.9601846106838$$
$$x_{34} = 59.2787435721386$$
$$x_{13} = 63.2433699178634$$
$$x_{39} = 65.5619288793182$$
$$x_{65} = 69.5265552250429$$
$$x_{22} = 71.8451141864978$$
$$x_{10} = 75.8097405322225$$
$$x_{47} = 78.1282994936773$$
$$x_{3} = 82.0929258394021$$
$$x_{25} = 84.4114848008569$$
$$x_{55} = 88.3761111465817$$
$$x_{56} = 90.6946701080365$$
$$x_{27} = 94.6592964537613$$
$$x_{57} = 96.9778554152161$$
$$x_{31} = 100.942481760941$$
$$x_{48} = 697.845085943002$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{42}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{42} - \frac{1}{10}$$
=
$$-100.119448068806 + - \frac{1}{10}$$
=
$$-100.219448068806$$
lo sustituimos en la expresión
$$\sin{\left(t \right)} < \frac{2}{5}$$
$$\sin{\left(t \right)} < \frac{2}{5}$$
sin(t) < 2/5

Entonces
$$x < -100.119448068806$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -100.119448068806 \wedge x < -97.8008891073511$$
         _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____  
        /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \  
-------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-------
       x42      x58      x17      x28      x1      x37      x21      x40      x33      x11      x12      x16      x15      x24      x60      x62      x44      x32      x19      x5      x26      x35      x14      x23      x63      x4      x41      x7      x51      x45      x64      x20      x46      x43      x6      x61      x8      x18      x52      x50      x59      x38      x66      x30      x53      x2      x29      x9      x54      x49      x36      x34      x13      x39      x65      x22      x10      x47      x3      x25      x55      x56      x27      x57      x31      x48

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -100.119448068806 \wedge x < -97.8008891073511$$
$$x > -93.8362627616263 \wedge x < -91.5177038001715$$
$$x > -87.5530774544467 \wedge x < -85.2345184929919$$
$$x > -81.2698921472671 \wedge x < -78.9513331858123$$
$$x > -74.9867068400875 \wedge x < -72.6681478786327$$
$$x > -68.703521532908 \wedge x < -66.3849625714531$$
$$x > -62.4203362257284 \wedge x < -60.1017772642736$$
$$x > -56.1371509185488 \wedge x < -53.818591957094$$
$$x > -49.8539656113692 \wedge x < -47.5354066499144$$
$$x > -43.5707803041896 \wedge x < -41.2522213427348$$
$$x > -37.28759499701 \wedge x < -34.9690360355552$$
$$x > -31.0044096898304 \wedge x < -28.6858507283756$$
$$x > -24.7212243826509 \wedge x < -22.402665421196$$
$$x > -18.4380390754713 \wedge x < -16.1194801140165$$
$$x > -12.1548537682917 \wedge x < -9.83629480683687$$
$$x > -5.8716684611121 \wedge x < -3.55310949965728$$
$$x > 0.411516846067488 \wedge x < 2.73007580752231$$
$$x > 6.69470215324707 \wedge x < 9.01326111470189$$
$$x > 12.9778874604267 \wedge x < 15.2964464218815$$
$$x > 19.2610727676062 \wedge x < 21.5796317290611$$
$$x > 25.5442580747858 \wedge x < 27.8628170362407$$
$$x > 31.8274433819654 \wedge x < 34.1460023434202$$
$$x > 38.110628689145 \wedge x < 40.4291876505998$$
$$x > 44.3938139963246 \wedge x < 46.7123729577794$$
$$x > 50.6769993035042 \wedge x < 52.995558264959$$
$$x > 56.9601846106838 \wedge x < 59.2787435721386$$
$$x > 63.2433699178634 \wedge x < 65.5619288793182$$
$$x > 69.5265552250429 \wedge x < 71.8451141864978$$
$$x > 75.8097405322225 \wedge x < 78.1282994936773$$
$$x > 82.0929258394021 \wedge x < 84.4114848008569$$
$$x > 88.3761111465817 \wedge x < 90.6946701080365$$
$$x > 94.6592964537613 \wedge x < 96.9778554152161$$
$$x > 100.942481760941 \wedge x < 697.845085943002$$
Respuesta rápida [src]
  /   /                /    ____\\     /                    /    ____\    \\
  |   |                |2*\/ 21 ||     |                    |2*\/ 21 |    ||
Or|And|0 <= t, t < atan|--------||, And|t <= 2*pi, pi - atan|--------| < t||
  \   \                \   21   //     \                    \   21   /    //
$$\left(0 \leq t \wedge t < \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)}\right) \vee \left(t \leq 2 \pi \wedge \pi - \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)} < t\right)$$
((0 <= t)∧(t < atan(2*sqrt(21)/21)))∨((t <= 2*pi)∧(pi - atan(2*sqrt(21)/21) < t))
Respuesta rápida 2 [src]
        /    ____\              /    ____\       
        |2*\/ 21 |              |2*\/ 21 |       
[0, atan|--------|) U (pi - atan|--------|, 2*pi]
        \   21   /              \   21   /       
$$x\ in\ \left[0, \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)}\right) \cup \left(\pi - \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)}, 2 \pi\right]$$
x in Union(Interval.Ropen(0, atan(2*sqrt(21)/21)), Interval.Lopen(pi - atan(2*sqrt(21)/21), 2*pi))