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Solving inequalities/ cost≤1/2

cost≤1/2 inequation

A inequation with variable

The solution

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cos(t) <= 1/2
cos⁡(t)≤12\cos{\left(t \right)} \leq \frac{1}{2} 
cos(t) <= 1/2
Detail solution
Given the inequality:
cos⁡(t)≤12\cos{\left(t \right)} \leq \frac{1}{2}
To solve this inequality, we must first solve the corresponding equation:
cos⁡(t)=12\cos{\left(t \right)} = \frac{1}{2}
Solve:
Given the equation
cos⁡(t)=12\cos{\left(t \right)} = \frac{1}{2}
transform
cos⁡(t)−12=0\cos{\left(t \right)} - \frac{1}{2} = 0
cos⁡(t)−12=0\cos{\left(t \right)} - \frac{1}{2} = 0
Do replacement
w=cos⁡(t)w = \cos{\left(t \right)}
Move free summands (without w)
from left part to right part, we given:
w=12w = \frac{1}{2}
We get the answer: w = 1/2
do backward replacement
cos⁡(t)=w\cos{\left(t \right)} = w
substitute w:
x1=38.7463093942741x_{1} = 38.7463093942741
x2=13.6135681655558x_{2} = 13.6135681655558
x3=11.5191730631626x_{3} = 11.5191730631626
x4=−55.5014702134197x_{4} = -55.5014702134197
x5=−30.3687289847013x_{5} = -30.3687289847013
x6=−61.7846555205993x_{6} = -61.7846555205993
x7=−86.9173967493176x_{7} = -86.9173967493176
x8=89.0117918517108x_{8} = 89.0117918517108
x9=−51.3126800086333x_{9} = -51.3126800086333
x10=1.0471975511966x_{10} = 1.0471975511966
x11=−68.0678408277789x_{11} = -68.0678408277789
x12=36.6519142918809x_{12} = 36.6519142918809
x13=−70.162235930172x_{13} = -70.162235930172
x14=93.2005820564972x_{14} = 93.2005820564972
x15=−95.2949771588904x_{15} = -95.2949771588904
x16=49.2182849062401x_{16} = 49.2182849062401
x17=30.3687289847013x_{17} = 30.3687289847013
x18=17.8023583703422x_{18} = 17.8023583703422
x19=−32.4631240870945x_{19} = -32.4631240870945
x20=−99.4837673636768x_{20} = -99.4837673636768
x21=63.8790506229925x_{21} = 63.8790506229925
x22=42.9350995990605x_{22} = 42.9350995990605
x23=−225.147473507269x_{23} = -225.147473507269
x24=−93.2005820564972x_{24} = -93.2005820564972
x25=−17.8023583703422x_{25} = -17.8023583703422
x26=7.33038285837618x_{26} = 7.33038285837618
x27=−11.5191730631626x_{27} = -11.5191730631626
x28=57.5958653158129x_{28} = 57.5958653158129
x29=−42.9350995990605x_{29} = -42.9350995990605
x30=−24.0855436775217x_{30} = -24.0855436775217
x31=70.162235930172x_{31} = 70.162235930172
x32=−36.6519142918809x_{32} = -36.6519142918809
x33=−26.1799387799149x_{33} = -26.1799387799149
x34=19.8967534727354x_{34} = 19.8967534727354
x35=−76.4454212373516x_{35} = -76.4454212373516
x36=76.4454212373516x_{36} = 76.4454212373516
x37=−1.0471975511966x_{37} = -1.0471975511966
x38=1651.43053823704x_{38} = 1651.43053823704
x39=55.5014702134197x_{39} = 55.5014702134197
x40=−5.23598775598299x_{40} = -5.23598775598299
x41=51.3126800086333x_{41} = 51.3126800086333
x42=45.0294947014537x_{42} = 45.0294947014537
x43=68.0678408277789x_{43} = 68.0678408277789
x44=−45.0294947014537x_{44} = -45.0294947014537
x45=−82.7286065445312x_{45} = -82.7286065445312
x46=−19.8967534727354x_{46} = -19.8967534727354
x47=−57.5958653158129x_{47} = -57.5958653158129
x48=74.3510261349584x_{48} = 74.3510261349584
x49=−38.7463093942741x_{49} = -38.7463093942741
x50=80.634211442138x_{50} = 80.634211442138
x51=99.4837673636768x_{51} = 99.4837673636768
x52=−359.188760060433x_{52} = -359.188760060433
x53=26.1799387799149x_{53} = 26.1799387799149
x54=−63.8790506229925x_{54} = -63.8790506229925
x55=5.23598775598299x_{55} = 5.23598775598299
x56=−89.0117918517108x_{56} = -89.0117918517108
x57=82.7286065445312x_{57} = 82.7286065445312
x58=−13.6135681655558x_{58} = -13.6135681655558
x59=−80.634211442138x_{59} = -80.634211442138
x60=−74.3510261349584x_{60} = -74.3510261349584
x61=86.9173967493176x_{61} = 86.9173967493176
x62=24.0855436775217x_{62} = 24.0855436775217
x63=−49.2182849062401x_{63} = -49.2182849062401
x64=61.7846555205993x_{64} = 61.7846555205993
x65=95.2949771588904x_{65} = 95.2949771588904
x66=32.4631240870945x_{66} = 32.4631240870945
x67=−7.33038285837618x_{67} = -7.33038285837618
x1=38.7463093942741x_{1} = 38.7463093942741
x2=13.6135681655558x_{2} = 13.6135681655558
x3=11.5191730631626x_{3} = 11.5191730631626
x4=−55.5014702134197x_{4} = -55.5014702134197
x5=−30.3687289847013x_{5} = -30.3687289847013
x6=−61.7846555205993x_{6} = -61.7846555205993
x7=−86.9173967493176x_{7} = -86.9173967493176
x8=89.0117918517108x_{8} = 89.0117918517108
x9=−51.3126800086333x_{9} = -51.3126800086333
x10=1.0471975511966x_{10} = 1.0471975511966
x11=−68.0678408277789x_{11} = -68.0678408277789
x12=36.6519142918809x_{12} = 36.6519142918809
x13=−70.162235930172x_{13} = -70.162235930172
x14=93.2005820564972x_{14} = 93.2005820564972
x15=−95.2949771588904x_{15} = -95.2949771588904
x16=49.2182849062401x_{16} = 49.2182849062401
x17=30.3687289847013x_{17} = 30.3687289847013
x18=17.8023583703422x_{18} = 17.8023583703422
x19=−32.4631240870945x_{19} = -32.4631240870945
x20=−99.4837673636768x_{20} = -99.4837673636768
x21=63.8790506229925x_{21} = 63.8790506229925
x22=42.9350995990605x_{22} = 42.9350995990605
x23=−225.147473507269x_{23} = -225.147473507269
x24=−93.2005820564972x_{24} = -93.2005820564972
x25=−17.8023583703422x_{25} = -17.8023583703422
x26=7.33038285837618x_{26} = 7.33038285837618
x27=−11.5191730631626x_{27} = -11.5191730631626
x28=57.5958653158129x_{28} = 57.5958653158129
x29=−42.9350995990605x_{29} = -42.9350995990605
x30=−24.0855436775217x_{30} = -24.0855436775217
x31=70.162235930172x_{31} = 70.162235930172
x32=−36.6519142918809x_{32} = -36.6519142918809
x33=−26.1799387799149x_{33} = -26.1799387799149
x34=19.8967534727354x_{34} = 19.8967534727354
x35=−76.4454212373516x_{35} = -76.4454212373516
x36=76.4454212373516x_{36} = 76.4454212373516
x37=−1.0471975511966x_{37} = -1.0471975511966
x38=1651.43053823704x_{38} = 1651.43053823704
x39=55.5014702134197x_{39} = 55.5014702134197
x40=−5.23598775598299x_{40} = -5.23598775598299
x41=51.3126800086333x_{41} = 51.3126800086333
x42=45.0294947014537x_{42} = 45.0294947014537
x43=68.0678408277789x_{43} = 68.0678408277789
x44=−45.0294947014537x_{44} = -45.0294947014537
x45=−82.7286065445312x_{45} = -82.7286065445312
x46=−19.8967534727354x_{46} = -19.8967534727354
x47=−57.5958653158129x_{47} = -57.5958653158129
x48=74.3510261349584x_{48} = 74.3510261349584
x49=−38.7463093942741x_{49} = -38.7463093942741
x50=80.634211442138x_{50} = 80.634211442138
x51=99.4837673636768x_{51} = 99.4837673636768
x52=−359.188760060433x_{52} = -359.188760060433
x53=26.1799387799149x_{53} = 26.1799387799149
x54=−63.8790506229925x_{54} = -63.8790506229925
x55=5.23598775598299x_{55} = 5.23598775598299
x56=−89.0117918517108x_{56} = -89.0117918517108
x57=82.7286065445312x_{57} = 82.7286065445312
x58=−13.6135681655558x_{58} = -13.6135681655558
x59=−80.634211442138x_{59} = -80.634211442138
x60=−74.3510261349584x_{60} = -74.3510261349584
x61=86.9173967493176x_{61} = 86.9173967493176
x62=24.0855436775217x_{62} = 24.0855436775217
x63=−49.2182849062401x_{63} = -49.2182849062401
x64=61.7846555205993x_{64} = 61.7846555205993
x65=95.2949771588904x_{65} = 95.2949771588904
x66=32.4631240870945x_{66} = 32.4631240870945
x67=−7.33038285837618x_{67} = -7.33038285837618
This roots
x52=−359.188760060433x_{52} = -359.188760060433
x23=−225.147473507269x_{23} = -225.147473507269
x20=−99.4837673636768x_{20} = -99.4837673636768
x15=−95.2949771588904x_{15} = -95.2949771588904
x24=−93.2005820564972x_{24} = -93.2005820564972
x56=−89.0117918517108x_{56} = -89.0117918517108
x7=−86.9173967493176x_{7} = -86.9173967493176
x45=−82.7286065445312x_{45} = -82.7286065445312
x59=−80.634211442138x_{59} = -80.634211442138
x35=−76.4454212373516x_{35} = -76.4454212373516
x60=−74.3510261349584x_{60} = -74.3510261349584
x13=−70.162235930172x_{13} = -70.162235930172
x11=−68.0678408277789x_{11} = -68.0678408277789
x54=−63.8790506229925x_{54} = -63.8790506229925
x6=−61.7846555205993x_{6} = -61.7846555205993
x47=−57.5958653158129x_{47} = -57.5958653158129
x4=−55.5014702134197x_{4} = -55.5014702134197
x9=−51.3126800086333x_{9} = -51.3126800086333
x63=−49.2182849062401x_{63} = -49.2182849062401
x44=−45.0294947014537x_{44} = -45.0294947014537
x29=−42.9350995990605x_{29} = -42.9350995990605
x49=−38.7463093942741x_{49} = -38.7463093942741
x32=−36.6519142918809x_{32} = -36.6519142918809
x19=−32.4631240870945x_{19} = -32.4631240870945
x5=−30.3687289847013x_{5} = -30.3687289847013
x33=−26.1799387799149x_{33} = -26.1799387799149
x30=−24.0855436775217x_{30} = -24.0855436775217
x46=−19.8967534727354x_{46} = -19.8967534727354
x25=−17.8023583703422x_{25} = -17.8023583703422
x58=−13.6135681655558x_{58} = -13.6135681655558
x27=−11.5191730631626x_{27} = -11.5191730631626
x67=−7.33038285837618x_{67} = -7.33038285837618
x40=−5.23598775598299x_{40} = -5.23598775598299
x37=−1.0471975511966x_{37} = -1.0471975511966
x10=1.0471975511966x_{10} = 1.0471975511966
x55=5.23598775598299x_{55} = 5.23598775598299
x26=7.33038285837618x_{26} = 7.33038285837618
x3=11.5191730631626x_{3} = 11.5191730631626
x2=13.6135681655558x_{2} = 13.6135681655558
x18=17.8023583703422x_{18} = 17.8023583703422
x34=19.8967534727354x_{34} = 19.8967534727354
x62=24.0855436775217x_{62} = 24.0855436775217
x53=26.1799387799149x_{53} = 26.1799387799149
x17=30.3687289847013x_{17} = 30.3687289847013
x66=32.4631240870945x_{66} = 32.4631240870945
x12=36.6519142918809x_{12} = 36.6519142918809
x1=38.7463093942741x_{1} = 38.7463093942741
x22=42.9350995990605x_{22} = 42.9350995990605
x42=45.0294947014537x_{42} = 45.0294947014537
x16=49.2182849062401x_{16} = 49.2182849062401
x41=51.3126800086333x_{41} = 51.3126800086333
x39=55.5014702134197x_{39} = 55.5014702134197
x28=57.5958653158129x_{28} = 57.5958653158129
x64=61.7846555205993x_{64} = 61.7846555205993
x21=63.8790506229925x_{21} = 63.8790506229925
x43=68.0678408277789x_{43} = 68.0678408277789
x31=70.162235930172x_{31} = 70.162235930172
x48=74.3510261349584x_{48} = 74.3510261349584
x36=76.4454212373516x_{36} = 76.4454212373516
x50=80.634211442138x_{50} = 80.634211442138
x57=82.7286065445312x_{57} = 82.7286065445312
x61=86.9173967493176x_{61} = 86.9173967493176
x8=89.0117918517108x_{8} = 89.0117918517108
x14=93.2005820564972x_{14} = 93.2005820564972
x65=95.2949771588904x_{65} = 95.2949771588904
x51=99.4837673636768x_{51} = 99.4837673636768
x38=1651.43053823704x_{38} = 1651.43053823704
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0≤x52x_{0} \leq x_{52}
For example, let's take the point
x0=x52−110x_{0} = x_{52} - \frac{1}{10}
=
−359.188760060433+−110-359.188760060433 + - \frac{1}{10}
=
−359.288760060433-359.288760060433
substitute to the expression
cos⁡(t)≤12\cos{\left(t \right)} \leq \frac{1}{2}
cos⁡(t)≤12\cos{\left(t \right)} \leq \frac{1}{2}
cos(t) <= 1/2

Then
x≤−359.188760060433x \leq -359.188760060433
no execute
one of the solutions of our inequality is:
x≥−359.188760060433∧x≤−225.147473507269x \geq -359.188760060433 \wedge x \leq -225.147473507269
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       x52      x23      x20      x15      x24      x56      x7      x45      x59      x35      x60      x13      x11      x54      x6      x47      x4      x9      x63      x44      x29      x49      x32      x19      x5      x33      x30      x46      x25      x58      x27      x67      x40      x37      x10      x55      x26      x3      x2      x18      x34      x62      x53      x17      x66      x12      x1      x22      x42      x16      x41      x39      x28      x64      x21      x43      x31      x48      x36      x50      x57      x61      x8      x14      x65      x51      x38

Other solutions will get with the changeover to the next point
etc.
The answer:
x≥−359.188760060433∧x≤−225.147473507269x \geq -359.188760060433 \wedge x \leq -225.147473507269
x≥−99.4837673636768∧x≤−95.2949771588904x \geq -99.4837673636768 \wedge x \leq -95.2949771588904
x≥−93.2005820564972∧x≤−89.0117918517108x \geq -93.2005820564972 \wedge x \leq -89.0117918517108
x≥−86.9173967493176∧x≤−82.7286065445312x \geq -86.9173967493176 \wedge x \leq -82.7286065445312
x≥−80.634211442138∧x≤−76.4454212373516x \geq -80.634211442138 \wedge x \leq -76.4454212373516
x≥−74.3510261349584∧x≤−70.162235930172x \geq -74.3510261349584 \wedge x \leq -70.162235930172
x≥−68.0678408277789∧x≤−63.8790506229925x \geq -68.0678408277789 \wedge x \leq -63.8790506229925
x≥−61.7846555205993∧x≤−57.5958653158129x \geq -61.7846555205993 \wedge x \leq -57.5958653158129
x≥−55.5014702134197∧x≤−51.3126800086333x \geq -55.5014702134197 \wedge x \leq -51.3126800086333
x≥−49.2182849062401∧x≤−45.0294947014537x \geq -49.2182849062401 \wedge x \leq -45.0294947014537
x≥−42.9350995990605∧x≤−38.7463093942741x \geq -42.9350995990605 \wedge x \leq -38.7463093942741
x≥−36.6519142918809∧x≤−32.4631240870945x \geq -36.6519142918809 \wedge x \leq -32.4631240870945
x≥−30.3687289847013∧x≤−26.1799387799149x \geq -30.3687289847013 \wedge x \leq -26.1799387799149
x≥−24.0855436775217∧x≤−19.8967534727354x \geq -24.0855436775217 \wedge x \leq -19.8967534727354
x≥−17.8023583703422∧x≤−13.6135681655558x \geq -17.8023583703422 \wedge x \leq -13.6135681655558
x≥−11.5191730631626∧x≤−7.33038285837618x \geq -11.5191730631626 \wedge x \leq -7.33038285837618
x≥−5.23598775598299∧x≤−1.0471975511966x \geq -5.23598775598299 \wedge x \leq -1.0471975511966
x≥1.0471975511966∧x≤5.23598775598299x \geq 1.0471975511966 \wedge x \leq 5.23598775598299
x≥7.33038285837618∧x≤11.5191730631626x \geq 7.33038285837618 \wedge x \leq 11.5191730631626
x≥13.6135681655558∧x≤17.8023583703422x \geq 13.6135681655558 \wedge x \leq 17.8023583703422
x≥19.8967534727354∧x≤24.0855436775217x \geq 19.8967534727354 \wedge x \leq 24.0855436775217
x≥26.1799387799149∧x≤30.3687289847013x \geq 26.1799387799149 \wedge x \leq 30.3687289847013
x≥32.4631240870945∧x≤36.6519142918809x \geq 32.4631240870945 \wedge x \leq 36.6519142918809
x≥38.7463093942741∧x≤42.9350995990605x \geq 38.7463093942741 \wedge x \leq 42.9350995990605
x≥45.0294947014537∧x≤49.2182849062401x \geq 45.0294947014537 \wedge x \leq 49.2182849062401
x≥51.3126800086333∧x≤55.5014702134197x \geq 51.3126800086333 \wedge x \leq 55.5014702134197
x≥57.5958653158129∧x≤61.7846555205993x \geq 57.5958653158129 \wedge x \leq 61.7846555205993
x≥63.8790506229925∧x≤68.0678408277789x \geq 63.8790506229925 \wedge x \leq 68.0678408277789
x≥70.162235930172∧x≤74.3510261349584x \geq 70.162235930172 \wedge x \leq 74.3510261349584
x≥76.4454212373516∧x≤80.634211442138x \geq 76.4454212373516 \wedge x \leq 80.634211442138
x≥82.7286065445312∧x≤86.9173967493176x \geq 82.7286065445312 \wedge x \leq 86.9173967493176
x≥89.0117918517108∧x≤93.2005820564972x \geq 89.0117918517108 \wedge x \leq 93.2005820564972
x≥95.2949771588904∧x≤99.4837673636768x \geq 95.2949771588904 \wedge x \leq 99.4837673636768
x≥1651.43053823704x \geq 1651.43053823704
Rapid solution 2 [src] 
 pi  5*pi 
[--, ----]
 3    3
x in [π3,5π3]x\ in\ \left[\frac{\pi}{3}, \frac{5 \pi}{3}\right] 
x in Interval(pi/3, 5*pi/3)
Rapid solution [src] 
   /pi            5*pi\
And|-- <= t, t <= ----|
   \3              3  /
π3≤t∧t≤5π3\frac{\pi}{3} \leq t \wedge t \leq \frac{5 \pi}{3} 
(pi/3 <= t)∧(t <= 5*pi/3)
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