Решение неравенств/ cos(t)>-1/4

cos(t)>-1/4 (неравенство)

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    Укажите решение неравенства: cos(t)>-1/4 (множество решений неравенства)

    Решение

    Вы ввели [src]
    cos(t) > -1/4
    cos⁡(t)>−14\cos{\left(t \right)} > - \frac{1}{4}
    Подробное решение
    Дано неравенство:
    cos⁡(t)>−14\cos{\left (t \right )} > - \frac{1}{4}
    Чтобы решить это нер-во - надо сначала решить соотвествующее ур-ние:
    cos⁡(t)=−14\cos{\left (t \right )} = - \frac{1}{4}
    Решаем:
    Дано уравнение
    cos⁡(t)=−14\cos{\left (t \right )} = - \frac{1}{4}
    преобразуем
    cos⁡(t)+14=0\cos{\left (t \right )} + \frac{1}{4} = 0
    cos⁡(t)+14=0\cos{\left (t \right )} + \frac{1}{4} = 0
    Сделаем замену
    w=cos⁡(t)w = \cos{\left (t \right )}
    Переносим свободные слагаемые (без w)
    из левой части в правую, получим:
    w=−14w = - \frac{1}{4}
    Получим ответ: w = -1/4
    делаем обратную замену
    cos⁡(t)=w\cos{\left (t \right )} = w
    подставляем w:
    x1=−23.3092646468x_{1} = -23.3092646468
    x2=89.7880708825x_{2} = 89.7880708825
    x3=23.3092646468x_{3} = 23.3092646468
    x4=45.8057737322x_{4} = 45.8057737322
    x5=−64.6553296537x_{5} = -64.6553296537
    x6=−45.8057737322x_{6} = -45.8057737322
    x7=73.5747471042x_{7} = 73.5747471042
    x8=−83.5048855753x_{8} = -83.5048855753
    x9=−70.9385149609x_{9} = -70.9385149609
    x10=−54.7251911827x_{10} = -54.7251911827
    x11=52.0889590394x_{11} = 52.0889590394
    x12=−35.8756352611x_{12} = -35.8756352611
    x13=−86.1411177186x_{13} = -86.1411177186
    x14=79.8579324114x_{14} = 79.8579324114
    x15=−89.7880708825x_{15} = -89.7880708825
    x16=64.6553296537x_{16} = 64.6553296537
    x17=54.7251911827x_{17} = 54.7251911827
    x18=67.291561797x_{18} = 67.291561797
    x19=42.1588205683x_{19} = 42.1588205683
    x20=8.10666188912x_{20} = 8.10666188912
    x21=33.2394031178x_{21} = 33.2394031178
    x22=−61.0083764899x_{22} = -61.0083764899
    x23=20.6730325035x_{23} = 20.6730325035
    x24=10.7428940324x_{24} = 10.7428940324
    x25=35.8756352611x_{25} = 35.8756352611
    x26=96.0712561896x_{26} = 96.0712561896
    x27=61.0083764899x_{27} = 61.0083764899
    x28=48.4420058755x_{28} = 48.4420058755
    x29=83.5048855753x_{29} = 83.5048855753
    x30=29.592449954x_{30} = 29.592449954
    x31=−98.7074883329x_{31} = -98.7074883329
    x32=−52.0889590394x_{32} = -52.0889590394
    x33=−42.1588205683x_{33} = -42.1588205683
    x34=86.1411177186x_{34} = 86.1411177186
    x35=−92.4243030258x_{35} = -92.4243030258
    x36=−67.291561797x_{36} = -67.291561797
    x37=−10.7428940324x_{37} = -10.7428940324
    x38=−77.2217002681x_{38} = -77.2217002681
    x39=14.3898471963x_{39} = 14.3898471963
    x40=70.9385149609x_{40} = 70.9385149609
    x41=98.7074883329x_{41} = 98.7074883329
    x42=−39.522588425x_{42} = -39.522588425
    x43=−48.4420058755x_{43} = -48.4420058755
    x44=−79.8579324114x_{44} = -79.8579324114
    x45=4.45970872524x_{45} = 4.45970872524
    x46=−1.82347658194x_{46} = -1.82347658194
    x47=−33.2394031178x_{47} = -33.2394031178
    x48=−58.3721443466x_{48} = -58.3721443466
    x49=−14.3898471963x_{49} = -14.3898471963
    x50=26.9562178107x_{50} = 26.9562178107
    x51=−29.592449954x_{51} = -29.592449954
    x52=−20.6730325035x_{52} = -20.6730325035
    x53=−73.5747471042x_{53} = -73.5747471042
    x54=−4.45970872524x_{54} = -4.45970872524
    x55=92.4243030258x_{55} = 92.4243030258
    x56=17.0260793396x_{56} = 17.0260793396
    x57=1.82347658194x_{57} = 1.82347658194
    x58=−8.10666188912x_{58} = -8.10666188912
    x59=−17.0260793396x_{59} = -17.0260793396
    x60=77.2217002681x_{60} = 77.2217002681
    x61=−26.9562178107x_{61} = -26.9562178107
    x62=58.3721443466x_{62} = 58.3721443466
    x63=39.522588425x_{63} = 39.522588425
    x64=−96.0712561896x_{64} = -96.0712561896
    x1=−23.3092646468x_{1} = -23.3092646468
    x2=89.7880708825x_{2} = 89.7880708825
    x3=23.3092646468x_{3} = 23.3092646468
    x4=45.8057737322x_{4} = 45.8057737322
    x5=−64.6553296537x_{5} = -64.6553296537
    x6=−45.8057737322x_{6} = -45.8057737322
    x7=73.5747471042x_{7} = 73.5747471042
    x8=−83.5048855753x_{8} = -83.5048855753
    x9=−70.9385149609x_{9} = -70.9385149609
    x10=−54.7251911827x_{10} = -54.7251911827
    x11=52.0889590394x_{11} = 52.0889590394
    x12=−35.8756352611x_{12} = -35.8756352611
    x13=−86.1411177186x_{13} = -86.1411177186
    x14=79.8579324114x_{14} = 79.8579324114
    x15=−89.7880708825x_{15} = -89.7880708825
    x16=64.6553296537x_{16} = 64.6553296537
    x17=54.7251911827x_{17} = 54.7251911827
    x18=67.291561797x_{18} = 67.291561797
    x19=42.1588205683x_{19} = 42.1588205683
    x20=8.10666188912x_{20} = 8.10666188912
    x21=33.2394031178x_{21} = 33.2394031178
    x22=−61.0083764899x_{22} = -61.0083764899
    x23=20.6730325035x_{23} = 20.6730325035
    x24=10.7428940324x_{24} = 10.7428940324
    x25=35.8756352611x_{25} = 35.8756352611
    x26=96.0712561896x_{26} = 96.0712561896
    x27=61.0083764899x_{27} = 61.0083764899
    x28=48.4420058755x_{28} = 48.4420058755
    x29=83.5048855753x_{29} = 83.5048855753
    x30=29.592449954x_{30} = 29.592449954
    x31=−98.7074883329x_{31} = -98.7074883329
    x32=−52.0889590394x_{32} = -52.0889590394
    x33=−42.1588205683x_{33} = -42.1588205683
    x34=86.1411177186x_{34} = 86.1411177186
    x35=−92.4243030258x_{35} = -92.4243030258
    x36=−67.291561797x_{36} = -67.291561797
    x37=−10.7428940324x_{37} = -10.7428940324
    x38=−77.2217002681x_{38} = -77.2217002681
    x39=14.3898471963x_{39} = 14.3898471963
    x40=70.9385149609x_{40} = 70.9385149609
    x41=98.7074883329x_{41} = 98.7074883329
    x42=−39.522588425x_{42} = -39.522588425
    x43=−48.4420058755x_{43} = -48.4420058755
    x44=−79.8579324114x_{44} = -79.8579324114
    x45=4.45970872524x_{45} = 4.45970872524
    x46=−1.82347658194x_{46} = -1.82347658194
    x47=−33.2394031178x_{47} = -33.2394031178
    x48=−58.3721443466x_{48} = -58.3721443466
    x49=−14.3898471963x_{49} = -14.3898471963
    x50=26.9562178107x_{50} = 26.9562178107
    x51=−29.592449954x_{51} = -29.592449954
    x52=−20.6730325035x_{52} = -20.6730325035
    x53=−73.5747471042x_{53} = -73.5747471042
    x54=−4.45970872524x_{54} = -4.45970872524
    x55=92.4243030258x_{55} = 92.4243030258
    x56=17.0260793396x_{56} = 17.0260793396
    x57=1.82347658194x_{57} = 1.82347658194
    x58=−8.10666188912x_{58} = -8.10666188912
    x59=−17.0260793396x_{59} = -17.0260793396
    x60=77.2217002681x_{60} = 77.2217002681
    x61=−26.9562178107x_{61} = -26.9562178107
    x62=58.3721443466x_{62} = 58.3721443466
    x63=39.522588425x_{63} = 39.522588425
    x64=−96.0712561896x_{64} = -96.0712561896
    Данные корни
    x31=−98.7074883329x_{31} = -98.7074883329
    x64=−96.0712561896x_{64} = -96.0712561896
    x35=−92.4243030258x_{35} = -92.4243030258
    x15=−89.7880708825x_{15} = -89.7880708825
    x13=−86.1411177186x_{13} = -86.1411177186
    x8=−83.5048855753x_{8} = -83.5048855753
    x44=−79.8579324114x_{44} = -79.8579324114
    x38=−77.2217002681x_{38} = -77.2217002681
    x53=−73.5747471042x_{53} = -73.5747471042
    x9=−70.9385149609x_{9} = -70.9385149609
    x36=−67.291561797x_{36} = -67.291561797
    x5=−64.6553296537x_{5} = -64.6553296537
    x22=−61.0083764899x_{22} = -61.0083764899
    x48=−58.3721443466x_{48} = -58.3721443466
    x10=−54.7251911827x_{10} = -54.7251911827
    x32=−52.0889590394x_{32} = -52.0889590394
    x43=−48.4420058755x_{43} = -48.4420058755
    x6=−45.8057737322x_{6} = -45.8057737322
    x33=−42.1588205683x_{33} = -42.1588205683
    x42=−39.522588425x_{42} = -39.522588425
    x12=−35.8756352611x_{12} = -35.8756352611
    x47=−33.2394031178x_{47} = -33.2394031178
    x51=−29.592449954x_{51} = -29.592449954
    x61=−26.9562178107x_{61} = -26.9562178107
    x1=−23.3092646468x_{1} = -23.3092646468
    x52=−20.6730325035x_{52} = -20.6730325035
    x59=−17.0260793396x_{59} = -17.0260793396
    x49=−14.3898471963x_{49} = -14.3898471963
    x37=−10.7428940324x_{37} = -10.7428940324
    x58=−8.10666188912x_{58} = -8.10666188912
    x54=−4.45970872524x_{54} = -4.45970872524
    x46=−1.82347658194x_{46} = -1.82347658194
    x57=1.82347658194x_{57} = 1.82347658194
    x45=4.45970872524x_{45} = 4.45970872524
    x20=8.10666188912x_{20} = 8.10666188912
    x24=10.7428940324x_{24} = 10.7428940324
    x39=14.3898471963x_{39} = 14.3898471963
    x56=17.0260793396x_{56} = 17.0260793396
    x23=20.6730325035x_{23} = 20.6730325035
    x3=23.3092646468x_{3} = 23.3092646468
    x50=26.9562178107x_{50} = 26.9562178107
    x30=29.592449954x_{30} = 29.592449954
    x21=33.2394031178x_{21} = 33.2394031178
    x25=35.8756352611x_{25} = 35.8756352611
    x63=39.522588425x_{63} = 39.522588425
    x19=42.1588205683x_{19} = 42.1588205683
    x4=45.8057737322x_{4} = 45.8057737322
    x28=48.4420058755x_{28} = 48.4420058755
    x11=52.0889590394x_{11} = 52.0889590394
    x17=54.7251911827x_{17} = 54.7251911827
    x62=58.3721443466x_{62} = 58.3721443466
    x27=61.0083764899x_{27} = 61.0083764899
    x16=64.6553296537x_{16} = 64.6553296537
    x18=67.291561797x_{18} = 67.291561797
    x40=70.9385149609x_{40} = 70.9385149609
    x7=73.5747471042x_{7} = 73.5747471042
    x60=77.2217002681x_{60} = 77.2217002681
    x14=79.8579324114x_{14} = 79.8579324114
    x29=83.5048855753x_{29} = 83.5048855753
    x34=86.1411177186x_{34} = 86.1411177186
    x2=89.7880708825x_{2} = 89.7880708825
    x55=92.4243030258x_{55} = 92.4243030258
    x26=96.0712561896x_{26} = 96.0712561896
    x41=98.7074883329x_{41} = 98.7074883329
    являются точками смены знака неравенства в решениях.
    Сначала определимся со знаком до крайней левой точки:
    x0<x31x_{0} < x_{31}
    Возьмём например точку
    x0=x31−110x_{0} = x_{31} - \frac{1}{10}
    =
    −98.8074883329-98.8074883329
    =
    −98.8074883329-98.8074883329
    подставляем в выражение
    cos⁡(t)>−14\cos{\left (t \right )} > - \frac{1}{4}
    cos⁡(t)>−14\cos{\left (t \right )} > - \frac{1}{4}
    cos(t) > -1/4

    Тогда
    x<−98.7074883329x < -98.7074883329
    не выполняется
    значит одно из решений нашего неравенства будет при:
    x>−98.7074883329∧x<−96.0712561896x > -98.7074883329 \wedge x < -96.0712561896
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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-------
       x31      x64      x35      x15      x13      x8      x44      x38      x53      x9      x36      x5      x22      x48      x10      x32      x43      x6      x33      x42      x12      x47      x51      x61      x1      x52      x59      x49      x37      x58      x54      x46      x57      x45      x20      x24      x39      x56      x23      x3      x50      x30      x21      x25      x63      x19      x4      x28      x11      x17      x62      x27      x16      x18      x40      x7      x60      x14      x29      x34      x2      x55      x26      x41

    Другие решения неравенства будем получать переходом на следующий полюс
    и т.д.
    Ответ:
    x>−98.7074883329∧x<−96.0712561896x > -98.7074883329 \wedge x < -96.0712561896
    x>−92.4243030258∧x<−89.7880708825x > -92.4243030258 \wedge x < -89.7880708825
    x>−86.1411177186∧x<−83.5048855753x > -86.1411177186 \wedge x < -83.5048855753
    x>−79.8579324114∧x<−77.2217002681x > -79.8579324114 \wedge x < -77.2217002681
    x>−73.5747471042∧x<−70.9385149609x > -73.5747471042 \wedge x < -70.9385149609
    x>−67.291561797∧x<−64.6553296537x > -67.291561797 \wedge x < -64.6553296537
    x>−61.0083764899∧x<−58.3721443466x > -61.0083764899 \wedge x < -58.3721443466
    x>−54.7251911827∧x<−52.0889590394x > -54.7251911827 \wedge x < -52.0889590394
    x>−48.4420058755∧x<−45.8057737322x > -48.4420058755 \wedge x < -45.8057737322
    x>−42.1588205683∧x<−39.522588425x > -42.1588205683 \wedge x < -39.522588425
    x>−35.8756352611∧x<−33.2394031178x > -35.8756352611 \wedge x < -33.2394031178
    x>−29.592449954∧x<−26.9562178107x > -29.592449954 \wedge x < -26.9562178107
    x>−23.3092646468∧x<−20.6730325035x > -23.3092646468 \wedge x < -20.6730325035
    x>−17.0260793396∧x<−14.3898471963x > -17.0260793396 \wedge x < -14.3898471963
    x>−10.7428940324∧x<−8.10666188912x > -10.7428940324 \wedge x < -8.10666188912
    x>−4.45970872524∧x<−1.82347658194x > -4.45970872524 \wedge x < -1.82347658194
    x>1.82347658194∧x<4.45970872524x > 1.82347658194 \wedge x < 4.45970872524
    x>8.10666188912∧x<10.7428940324x > 8.10666188912 \wedge x < 10.7428940324
    x>14.3898471963∧x<17.0260793396x > 14.3898471963 \wedge x < 17.0260793396
    x>20.6730325035∧x<23.3092646468x > 20.6730325035 \wedge x < 23.3092646468
    x>26.9562178107∧x<29.592449954x > 26.9562178107 \wedge x < 29.592449954
    x>33.2394031178∧x<35.8756352611x > 33.2394031178 \wedge x < 35.8756352611
    x>39.522588425∧x<42.1588205683x > 39.522588425 \wedge x < 42.1588205683
    x>45.8057737322∧x<48.4420058755x > 45.8057737322 \wedge x < 48.4420058755
    x>52.0889590394∧x<54.7251911827x > 52.0889590394 \wedge x < 54.7251911827
    x>58.3721443466∧x<61.0083764899x > 58.3721443466 \wedge x < 61.0083764899
    x>64.6553296537∧x<67.291561797x > 64.6553296537 \wedge x < 67.291561797
    x>70.9385149609∧x<73.5747471042x > 70.9385149609 \wedge x < 73.5747471042
    x>77.2217002681∧x<79.8579324114x > 77.2217002681 \wedge x < 79.8579324114
    x>83.5048855753∧x<86.1411177186x > 83.5048855753 \wedge x < 86.1411177186
    x>89.7880708825∧x<92.4243030258x > 89.7880708825 \wedge x < 92.4243030258
    x>96.0712561896∧x<98.7074883329x > 96.0712561896 \wedge x < 98.7074883329
    Быстрый ответ [src]
      /   /                     /  ____\\     /                   /  ____\    \\
Or\And\0 <= t, t < pi - atan\\/ 15 //, And\t < 2*pi, pi + atan\\/ 15 / < t//
    (0≤t∧t<π−atan⁡(15))∨(t<2π∧atan⁡(15)+π<t)\left(0 \leq t \wedge t < \pi - \operatorname{atan}{\left(\sqrt{15} \right)}\right) \vee \left(t < 2 \pi \wedge \operatorname{atan}{\left(\sqrt{15} \right)} + \pi < t\right)
    Быстрый ответ 2 [src]
                 /  ____\              /  ____\       
[0, pi - atan\\/ 15 /) U (pi + atan\\/ 15 /, 2*pi)
    x in [0,π−atan⁡(15))∪(atan⁡(15)+π,2π)x\ in\ \left[0, \pi - \operatorname{atan}{\left(\sqrt{15} \right)}\right) \cup \left(\operatorname{atan}{\left(\sqrt{15} \right)} + \pi, 2 \pi\right)