|sin(x)|cos(x)>1/4 inequation

Given the inequality:
$$\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| > \frac{1}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| = \frac{1}{4}$$
Solve:
$$x_{1} = -13.8753675533549$$
$$x_{2} = 56.2868683768171$$
$$x_{3} = -50.0036830696375$$
$$x_{4} = 93.9859802198946$$
$$x_{5} = 80.3724120543389$$
$$x_{6} = 36.3901149040818$$
$$x_{7} = 57.857664703612$$
$$x_{8} = -3707.34113062376$$
$$x_{9} = 7.59218224617533$$
$$x_{10} = -55.2396708256205$$
$$x_{11} = -82.9904059323304$$
$$x_{12} = -30.1069295969022$$
$$x_{13} = -95.5567765466895$$
$$x_{14} = -74.0892267471593$$
$$x_{15} = 12.30457122656$$
$$x_{16} = -43.720497762458$$
$$x_{17} = 45.2912940892529$$
$$x_{18} = 6.54498469497874$$
$$x_{19} = -12.8281700021583$$
$$x_{20} = -51.5744793964324$$
$$x_{21} = -231.168859426649$$
$$x_{22} = -70.4240353179712$$
$$x_{23} = -17.540558982543$$
$$x_{24} = -61.5228561328001$$
$$x_{25} = 100.269165527074$$
$$x_{26} = -0.261799387799149$$
$$x_{27} = -12.30457122656$$
$$x_{28} = -6.02138591938044$$
$$x_{29} = 51.5744793964324$$
$$x_{30} = -19.1113553093379$$
$$x_{31} = 95.5567765466895$$
$$x_{32} = -20.1585528605345$$
$$x_{33} = -80.3724120543389$$
$$x_{34} = 20.1585528605345$$
$$x_{35} = 50.0036830696375$$
$$x_{36} = 70.4240353179712$$
$$x_{37} = 67.8060414399797$$
$$x_{38} = 107.075949609852$$
$$x_{39} = -36.3901149040818$$
$$x_{40} = -68.8532389911763$$
$$x_{41} = -50.5272818452358$$
$$x_{42} = -1368.42540002615$$
$$x_{43} = 0.261799387799149$$
$$x_{44} = -42.6733002112614$$
$$x_{45} = -75.6600230739542$$
$$x_{46} = -86.6555973615185$$
$$x_{47} = 86.6555973615185$$
$$x_{48} = -87.7027949127151$$
$$x_{49} = -37.9609112308767$$
$$x_{50} = 64.1408500107916$$
$$x_{51} = -67.8060414399797$$
$$x_{52} = -1.30899693899575$$
$$x_{53} = -100.792764302673$$
$$x_{54} = 76.7072206251508$$
$$x_{55} = 26.4417381677141$$
$$x_{56} = -37.4373124552784$$
$$x_{57} = 23.8237442897226$$
$$x_{58} = -93.9859802198946$$
$$x_{59} = -31.6777259236971$$
$$x_{60} = 30.1069295969022$$
$$x_{61} = 81.9432083811338$$
$$x_{62} = 6.02138591938044$$
$$x_{63} = -7.59218224617533$$
$$x_{64} = -81.4196096055355$$
$$x_{65} = -25.3945406165175$$
$$x_{66} = -81.9432083811338$$
$$x_{67} = 55.2396708256205$$
$$x_{68} = 74.0892267471593$$
$$x_{69} = 37.9609112308767$$
$$x_{70} = 48.9564855184409$$
$$x_{71} = 88.2263936883134$$
$$x_{72} = -89.27359123951$$
$$x_{73} = 44.2440965380563$$
$$x_{74} = 114.406332468228$$
$$x_{75} = -64.1408500107916$$
$$x_{76} = -23.8237442897226$$
$$x_{77} = 13.8753675533549$$
$$x_{78} = -56.8104671524154$$
$$x_{79} = -45.2912940892529$$
$$x_{80} = -57.857664703612$$
$$x_{81} = 31.1541271480988$$
$$x_{82} = 42.6733002112614$$
$$x_{1} = -13.8753675533549$$
$$x_{2} = 56.2868683768171$$
$$x_{3} = -50.0036830696375$$
$$x_{4} = 93.9859802198946$$
$$x_{5} = 80.3724120543389$$
$$x_{6} = 36.3901149040818$$
$$x_{7} = 57.857664703612$$
$$x_{8} = -3707.34113062376$$
$$x_{9} = 7.59218224617533$$
$$x_{10} = -55.2396708256205$$
$$x_{11} = -82.9904059323304$$
$$x_{12} = -30.1069295969022$$
$$x_{13} = -95.5567765466895$$
$$x_{14} = -74.0892267471593$$
$$x_{15} = 12.30457122656$$
$$x_{16} = -43.720497762458$$
$$x_{17} = 45.2912940892529$$
$$x_{18} = 6.54498469497874$$
$$x_{19} = -12.8281700021583$$
$$x_{20} = -51.5744793964324$$
$$x_{21} = -231.168859426649$$
$$x_{22} = -70.4240353179712$$
$$x_{23} = -17.540558982543$$
$$x_{24} = -61.5228561328001$$
$$x_{25} = 100.269165527074$$
$$x_{26} = -0.261799387799149$$
$$x_{27} = -12.30457122656$$
$$x_{28} = -6.02138591938044$$
$$x_{29} = 51.5744793964324$$
$$x_{30} = -19.1113553093379$$
$$x_{31} = 95.5567765466895$$
$$x_{32} = -20.1585528605345$$
$$x_{33} = -80.3724120543389$$
$$x_{34} = 20.1585528605345$$
$$x_{35} = 50.0036830696375$$
$$x_{36} = 70.4240353179712$$
$$x_{37} = 67.8060414399797$$
$$x_{38} = 107.075949609852$$
$$x_{39} = -36.3901149040818$$
$$x_{40} = -68.8532389911763$$
$$x_{41} = -50.5272818452358$$
$$x_{42} = -1368.42540002615$$
$$x_{43} = 0.261799387799149$$
$$x_{44} = -42.6733002112614$$
$$x_{45} = -75.6600230739542$$
$$x_{46} = -86.6555973615185$$
$$x_{47} = 86.6555973615185$$
$$x_{48} = -87.7027949127151$$
$$x_{49} = -37.9609112308767$$
$$x_{50} = 64.1408500107916$$
$$x_{51} = -67.8060414399797$$
$$x_{52} = -1.30899693899575$$
$$x_{53} = -100.792764302673$$
$$x_{54} = 76.7072206251508$$
$$x_{55} = 26.4417381677141$$
$$x_{56} = -37.4373124552784$$
$$x_{57} = 23.8237442897226$$
$$x_{58} = -93.9859802198946$$
$$x_{59} = -31.6777259236971$$
$$x_{60} = 30.1069295969022$$
$$x_{61} = 81.9432083811338$$
$$x_{62} = 6.02138591938044$$
$$x_{63} = -7.59218224617533$$
$$x_{64} = -81.4196096055355$$
$$x_{65} = -25.3945406165175$$
$$x_{66} = -81.9432083811338$$
$$x_{67} = 55.2396708256205$$
$$x_{68} = 74.0892267471593$$
$$x_{69} = 37.9609112308767$$
$$x_{70} = 48.9564855184409$$
$$x_{71} = 88.2263936883134$$
$$x_{72} = -89.27359123951$$
$$x_{73} = 44.2440965380563$$
$$x_{74} = 114.406332468228$$
$$x_{75} = -64.1408500107916$$
$$x_{76} = -23.8237442897226$$
$$x_{77} = 13.8753675533549$$
$$x_{78} = -56.8104671524154$$
$$x_{79} = -45.2912940892529$$
$$x_{80} = -57.857664703612$$
$$x_{81} = 31.1541271480988$$
$$x_{82} = 42.6733002112614$$
This roots
$$x_{8} = -3707.34113062376$$
$$x_{42} = -1368.42540002615$$
$$x_{21} = -231.168859426649$$
$$x_{53} = -100.792764302673$$
$$x_{13} = -95.5567765466895$$
$$x_{58} = -93.9859802198946$$
$$x_{72} = -89.27359123951$$
$$x_{48} = -87.7027949127151$$
$$x_{46} = -86.6555973615185$$
$$x_{11} = -82.9904059323304$$
$$x_{66} = -81.9432083811338$$
$$x_{64} = -81.4196096055355$$
$$x_{33} = -80.3724120543389$$
$$x_{45} = -75.6600230739542$$
$$x_{14} = -74.0892267471593$$
$$x_{22} = -70.4240353179712$$
$$x_{40} = -68.8532389911763$$
$$x_{51} = -67.8060414399797$$
$$x_{75} = -64.1408500107916$$
$$x_{24} = -61.5228561328001$$
$$x_{80} = -57.857664703612$$
$$x_{78} = -56.8104671524154$$
$$x_{10} = -55.2396708256205$$
$$x_{20} = -51.5744793964324$$
$$x_{41} = -50.5272818452358$$
$$x_{3} = -50.0036830696375$$
$$x_{79} = -45.2912940892529$$
$$x_{16} = -43.720497762458$$
$$x_{44} = -42.6733002112614$$
$$x_{49} = -37.9609112308767$$
$$x_{56} = -37.4373124552784$$
$$x_{39} = -36.3901149040818$$
$$x_{59} = -31.6777259236971$$
$$x_{12} = -30.1069295969022$$
$$x_{65} = -25.3945406165175$$
$$x_{76} = -23.8237442897226$$
$$x_{32} = -20.1585528605345$$
$$x_{30} = -19.1113553093379$$
$$x_{23} = -17.540558982543$$
$$x_{1} = -13.8753675533549$$
$$x_{19} = -12.8281700021583$$
$$x_{27} = -12.30457122656$$
$$x_{63} = -7.59218224617533$$
$$x_{28} = -6.02138591938044$$
$$x_{52} = -1.30899693899575$$
$$x_{26} = -0.261799387799149$$
$$x_{43} = 0.261799387799149$$
$$x_{62} = 6.02138591938044$$
$$x_{18} = 6.54498469497874$$
$$x_{9} = 7.59218224617533$$
$$x_{15} = 12.30457122656$$
$$x_{77} = 13.8753675533549$$
$$x_{34} = 20.1585528605345$$
$$x_{57} = 23.8237442897226$$
$$x_{55} = 26.4417381677141$$
$$x_{60} = 30.1069295969022$$
$$x_{81} = 31.1541271480988$$
$$x_{6} = 36.3901149040818$$
$$x_{69} = 37.9609112308767$$
$$x_{82} = 42.6733002112614$$
$$x_{73} = 44.2440965380563$$
$$x_{17} = 45.2912940892529$$
$$x_{70} = 48.9564855184409$$
$$x_{35} = 50.0036830696375$$
$$x_{29} = 51.5744793964324$$
$$x_{67} = 55.2396708256205$$
$$x_{2} = 56.2868683768171$$
$$x_{7} = 57.857664703612$$
$$x_{50} = 64.1408500107916$$
$$x_{37} = 67.8060414399797$$
$$x_{36} = 70.4240353179712$$
$$x_{68} = 74.0892267471593$$
$$x_{54} = 76.7072206251508$$
$$x_{5} = 80.3724120543389$$
$$x_{61} = 81.9432083811338$$
$$x_{47} = 86.6555973615185$$
$$x_{71} = 88.2263936883134$$
$$x_{4} = 93.9859802198946$$
$$x_{31} = 95.5567765466895$$
$$x_{25} = 100.269165527074$$
$$x_{38} = 107.075949609852$$
$$x_{74} = 114.406332468228$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{8}$$
For example, let's take the point
$$x_{0} = x_{8} - \frac{1}{10}$$
=
$$-3707.34113062376 + - \frac{1}{10}$$
=
$$-3707.44113062376$$
substitute to the expression
$$\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right| > \frac{1}{4}$$
$$\cos{\left(-3707.44113062376 \right)} \left|{\sin{\left(-3707.44113062376 \right)}}\right| > \frac{1}{4}$$

0.331042988171074 > 1/4

one of the solutions of our inequality is:
$$x < -3707.34113062376$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -3707.34113062376$$
$$x > -1368.42540002615 \wedge x < -231.168859426649$$
$$x > -100.792764302673 \wedge x < -95.5567765466895$$
$$x > -93.9859802198946 \wedge x < -89.27359123951$$
$$x > -87.7027949127151 \wedge x < -86.6555973615185$$
$$x > -82.9904059323304 \wedge x < -81.9432083811338$$
$$x > -81.4196096055355 \wedge x < -80.3724120543389$$
$$x > -75.6600230739542 \wedge x < -74.0892267471593$$
$$x > -70.4240353179712 \wedge x < -68.8532389911763$$
$$x > -67.8060414399797 \wedge x < -64.1408500107916$$
$$x > -61.5228561328001 \wedge x < -57.857664703612$$
$$x > -56.8104671524154 \wedge x < -55.2396708256205$$
$$x > -51.5744793964324 \wedge x < -50.5272818452358$$
$$x > -50.0036830696375 \wedge x < -45.2912940892529$$
$$x > -43.720497762458 \wedge x < -42.6733002112614$$
$$x > -37.9609112308767 \wedge x < -37.4373124552784$$
$$x > -36.3901149040818 \wedge x < -31.6777259236971$$
$$x > -30.1069295969022 \wedge x < -25.3945406165175$$
$$x > -23.8237442897226 \wedge x < -20.1585528605345$$
$$x > -19.1113553093379 \wedge x < -17.540558982543$$
$$x > -13.8753675533549 \wedge x < -12.8281700021583$$
$$x > -12.30457122656 \wedge x < -7.59218224617533$$
$$x > -6.02138591938044 \wedge x < -1.30899693899575$$
$$x > -0.261799387799149 \wedge x < 0.261799387799149$$
$$x > 6.02138591938044 \wedge x < 6.54498469497874$$
$$x > 7.59218224617533 \wedge x < 12.30457122656$$
$$x > 13.8753675533549 \wedge x < 20.1585528605345$$
$$x > 23.8237442897226 \wedge x < 26.4417381677141$$
$$x > 30.1069295969022 \wedge x < 31.1541271480988$$
$$x > 36.3901149040818 \wedge x < 37.9609112308767$$
$$x > 42.6733002112614 \wedge x < 44.2440965380563$$
$$x > 45.2912940892529 \wedge x < 48.9564855184409$$
$$x > 50.0036830696375 \wedge x < 51.5744793964324$$
$$x > 55.2396708256205 \wedge x < 56.2868683768171$$
$$x > 57.857664703612 \wedge x < 64.1408500107916$$
$$x > 67.8060414399797 \wedge x < 70.4240353179712$$
$$x > 74.0892267471593 \wedge x < 76.7072206251508$$
$$x > 80.3724120543389 \wedge x < 81.9432083811338$$
$$x > 86.6555973615185 \wedge x < 88.2263936883134$$
$$x > 93.9859802198946 \wedge x < 95.5567765466895$$
$$x > 100.269165527074 \wedge x < 107.075949609852$$
$$x > 114.406332468228$$