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Resolver desigualdades/ 5sinx+cos2x≥-2

5sinx+cos2x≥-2 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
5*sin(x) + cos(2*x) >= -2
5sin⁡(x)+cos⁡(2x)≥−25 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2 
5*sin(x) + cos(2*x) >= -2
Solución detallada
Se da la desigualdad:
5sin⁡(x)+cos⁡(2x)≥−25 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
5sin⁡(x)+cos⁡(2x)=−25 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2
Resolvemos:
Tenemos la ecuación
5sin⁡(x)+cos⁡(2x)=−25 \sin{\left(x \right)} + \cos{\left(2 x \right)} = -2
cambiamos
5sin⁡(x)+cos⁡(2x)+2=05 \sin{\left(x \right)} + \cos{\left(2 x \right)} + 2 = 0
−2sin⁡2(x)+5sin⁡(x)+3=0- 2 \sin^{2}{\left(x \right)} + 5 \sin{\left(x \right)} + 3 = 0
Sustituimos
w=sin⁡(x)w = \sin{\left(x \right)}
Es la ecuación de la forma
a*w^2 + b*w + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=−2a = -2
b=5b = 5
c=3c = 3
, entonces
D = b^2 - 4 * a * c = 

(5)^2 - 4 * (-2) * (3) = 49

Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

o
w1=−12w_{1} = - \frac{1}{2}
w2=3w_{2} = 3
hacemos cambio inverso
sin⁡(x)=w\sin{\left(x \right)} = w
Tenemos la ecuación
sin⁡(x)=w\sin{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
sustituimos w:
x1=2πn+asin⁡(w1)x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
x1=2πn+asin⁡(−12)x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}
x1=2πn−π6x_{1} = 2 \pi n - \frac{\pi}{6}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(3)x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}
x2=2πn+asin⁡(3)x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(−12)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi
x3=2πn+7π6x_{3} = 2 \pi n + \frac{7 \pi}{6}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn+π−asin⁡(3)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}
x4=2πn+π−asin⁡(3)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}
x1=100.007366139275x_{1} = 100.007366139275
x2=−82.2050077689329x_{2} = -82.2050077689329
x3=68.5914396033772x_{3} = 68.5914396033772
x4=87.4409955249159x_{4} = 87.4409955249159
x5=−45867.7763411866x_{5} = -45867.7763411866
x6=−71.733032256967x_{6} = -71.733032256967
x7=41.3643032722656x_{7} = 41.3643032722656
x8=74.8746249105567x_{8} = 74.8746249105567
x9=12.0427718387609x_{9} = 12.0427718387609
x10=−65.4498469497874x_{10} = -65.4498469497874
x11=37.1755130674792x_{11} = 37.1755130674792
x12=110.479341651241x_{12} = 110.479341651241
x13=22.5147473507269x_{13} = 22.5147473507269
x14=30.8923277602996x_{14} = 30.8923277602996
x15=−15.1843644923507x_{15} = -15.1843644923507
x16=81.1578102177363x_{16} = 81.1578102177363
x17=62.3082542961976x_{17} = 62.3082542961976
x18=−59.1666616426078x_{18} = -59.1666616426078
x19=93.7241808320955x_{19} = 93.7241808320955
x20=47.6474885794452x_{20} = 47.6474885794452
x21=56.025068989018x_{21} = 56.025068989018
x22=91.6297857297023x_{22} = 91.6297857297023
x23=79.0634151153431x_{23} = 79.0634151153431
x24=18.3259571459405x_{24} = 18.3259571459405
x25=66.497044500984x_{25} = 66.497044500984
x26=−75.9218224617533x_{26} = -75.9218224617533
x27=−239.284640448423x_{27} = -239.284640448423
x28=−31.9395253114962x_{28} = -31.9395253114962
x29=−46.6002910282486x_{29} = -46.6002910282486
x30=−101.054563690472x_{30} = -101.054563690472
x31=28.7979326579064x_{31} = 28.7979326579064
x32=53.9306738866248x_{32} = 53.9306738866248
x33=−90.5825881785057x_{33} = -90.5825881785057
x34=−8.90117918517108x_{34} = -8.90117918517108
x35=−6.80678408277789x_{35} = -6.80678408277789
x36=−69.6386371545737x_{36} = -69.6386371545737
x37=60.2138591938044x_{37} = 60.2138591938044
x38=−44.5058959258554x_{38} = -44.5058959258554
x39=−94.7713783832921x_{39} = -94.7713783832921
x40=−27.7507351067098x_{40} = -27.7507351067098
x41=−25.6563400043166x_{41} = -25.6563400043166
x42=−34.0339204138894x_{42} = -34.0339204138894
x43=−50.789081233035x_{43} = -50.789081233035
x44=−13.0899693899575x_{44} = -13.0899693899575
x45=3.66519142918809x_{45} = 3.66519142918809
x46=−88.4881930761125x_{46} = -88.4881930761125
x47=85.3466004225227x_{47} = 85.3466004225227
x48=−84.2994028713261x_{48} = -84.2994028713261
x49=16.2315620435473x_{49} = 16.2315620435473
x50=−52.8834763354282x_{50} = -52.8834763354282
x51=−643.502895210309x_{51} = -643.502895210309
x52=−40.317105721069x_{52} = -40.317105721069
x53=−0.523598775598299x_{53} = -0.523598775598299
x54=97.9129710368819x_{54} = 97.9129710368819
x55=49.7418836818384x_{55} = 49.7418836818384
x56=35.081117965086x_{56} = 35.081117965086
x57=791.15774992903x_{57} = 791.15774992903
x58=−96.8657734856853x_{58} = -96.8657734856853
x59=24.60914245312x_{59} = 24.60914245312
x60=5.75958653158129x_{60} = 5.75958653158129
x61=−21.4675497995303x_{61} = -21.4675497995303
x62=−63.3554518473942x_{62} = -63.3554518473942
x63=−78.0162175641465x_{63} = -78.0162175641465
x64=−38.2227106186758x_{64} = -38.2227106186758
x65=72.7802298081635x_{65} = 72.7802298081635
x66=43.4586983746588x_{66} = 43.4586983746588
x67=−2.61799387799149x_{67} = -2.61799387799149
x68=9.94837673636768x_{68} = 9.94837673636768
x69=−57.0722665402146x_{69} = -57.0722665402146
x70=112.573736753634x_{70} = 112.573736753634
x71=−19.3731546971371x_{71} = -19.3731546971371
x1=100.007366139275x_{1} = 100.007366139275
x2=−82.2050077689329x_{2} = -82.2050077689329
x3=68.5914396033772x_{3} = 68.5914396033772
x4=87.4409955249159x_{4} = 87.4409955249159
x5=−45867.7763411866x_{5} = -45867.7763411866
x6=−71.733032256967x_{6} = -71.733032256967
x7=41.3643032722656x_{7} = 41.3643032722656
x8=74.8746249105567x_{8} = 74.8746249105567
x9=12.0427718387609x_{9} = 12.0427718387609
x10=−65.4498469497874x_{10} = -65.4498469497874
x11=37.1755130674792x_{11} = 37.1755130674792
x12=110.479341651241x_{12} = 110.479341651241
x13=22.5147473507269x_{13} = 22.5147473507269
x14=30.8923277602996x_{14} = 30.8923277602996
x15=−15.1843644923507x_{15} = -15.1843644923507
x16=81.1578102177363x_{16} = 81.1578102177363
x17=62.3082542961976x_{17} = 62.3082542961976
x18=−59.1666616426078x_{18} = -59.1666616426078
x19=93.7241808320955x_{19} = 93.7241808320955
x20=47.6474885794452x_{20} = 47.6474885794452
x21=56.025068989018x_{21} = 56.025068989018
x22=91.6297857297023x_{22} = 91.6297857297023
x23=79.0634151153431x_{23} = 79.0634151153431
x24=18.3259571459405x_{24} = 18.3259571459405
x25=66.497044500984x_{25} = 66.497044500984
x26=−75.9218224617533x_{26} = -75.9218224617533
x27=−239.284640448423x_{27} = -239.284640448423
x28=−31.9395253114962x_{28} = -31.9395253114962
x29=−46.6002910282486x_{29} = -46.6002910282486
x30=−101.054563690472x_{30} = -101.054563690472
x31=28.7979326579064x_{31} = 28.7979326579064
x32=53.9306738866248x_{32} = 53.9306738866248
x33=−90.5825881785057x_{33} = -90.5825881785057
x34=−8.90117918517108x_{34} = -8.90117918517108
x35=−6.80678408277789x_{35} = -6.80678408277789
x36=−69.6386371545737x_{36} = -69.6386371545737
x37=60.2138591938044x_{37} = 60.2138591938044
x38=−44.5058959258554x_{38} = -44.5058959258554
x39=−94.7713783832921x_{39} = -94.7713783832921
x40=−27.7507351067098x_{40} = -27.7507351067098
x41=−25.6563400043166x_{41} = -25.6563400043166
x42=−34.0339204138894x_{42} = -34.0339204138894
x43=−50.789081233035x_{43} = -50.789081233035
x44=−13.0899693899575x_{44} = -13.0899693899575
x45=3.66519142918809x_{45} = 3.66519142918809
x46=−88.4881930761125x_{46} = -88.4881930761125
x47=85.3466004225227x_{47} = 85.3466004225227
x48=−84.2994028713261x_{48} = -84.2994028713261
x49=16.2315620435473x_{49} = 16.2315620435473
x50=−52.8834763354282x_{50} = -52.8834763354282
x51=−643.502895210309x_{51} = -643.502895210309
x52=−40.317105721069x_{52} = -40.317105721069
x53=−0.523598775598299x_{53} = -0.523598775598299
x54=97.9129710368819x_{54} = 97.9129710368819
x55=49.7418836818384x_{55} = 49.7418836818384
x56=35.081117965086x_{56} = 35.081117965086
x57=791.15774992903x_{57} = 791.15774992903
x58=−96.8657734856853x_{58} = -96.8657734856853
x59=24.60914245312x_{59} = 24.60914245312
x60=5.75958653158129x_{60} = 5.75958653158129
x61=−21.4675497995303x_{61} = -21.4675497995303
x62=−63.3554518473942x_{62} = -63.3554518473942
x63=−78.0162175641465x_{63} = -78.0162175641465
x64=−38.2227106186758x_{64} = -38.2227106186758
x65=72.7802298081635x_{65} = 72.7802298081635
x66=43.4586983746588x_{66} = 43.4586983746588
x67=−2.61799387799149x_{67} = -2.61799387799149
x68=9.94837673636768x_{68} = 9.94837673636768
x69=−57.0722665402146x_{69} = -57.0722665402146
x70=112.573736753634x_{70} = 112.573736753634
x71=−19.3731546971371x_{71} = -19.3731546971371
Las raíces dadas
x5=−45867.7763411866x_{5} = -45867.7763411866
x51=−643.502895210309x_{51} = -643.502895210309
x27=−239.284640448423x_{27} = -239.284640448423
x30=−101.054563690472x_{30} = -101.054563690472
x58=−96.8657734856853x_{58} = -96.8657734856853
x39=−94.7713783832921x_{39} = -94.7713783832921
x33=−90.5825881785057x_{33} = -90.5825881785057
x46=−88.4881930761125x_{46} = -88.4881930761125
x48=−84.2994028713261x_{48} = -84.2994028713261
x2=−82.2050077689329x_{2} = -82.2050077689329
x63=−78.0162175641465x_{63} = -78.0162175641465
x26=−75.9218224617533x_{26} = -75.9218224617533
x6=−71.733032256967x_{6} = -71.733032256967
x36=−69.6386371545737x_{36} = -69.6386371545737
x10=−65.4498469497874x_{10} = -65.4498469497874
x62=−63.3554518473942x_{62} = -63.3554518473942
x18=−59.1666616426078x_{18} = -59.1666616426078
x69=−57.0722665402146x_{69} = -57.0722665402146
x50=−52.8834763354282x_{50} = -52.8834763354282
x43=−50.789081233035x_{43} = -50.789081233035
x29=−46.6002910282486x_{29} = -46.6002910282486
x38=−44.5058959258554x_{38} = -44.5058959258554
x52=−40.317105721069x_{52} = -40.317105721069
x64=−38.2227106186758x_{64} = -38.2227106186758
x42=−34.0339204138894x_{42} = -34.0339204138894
x28=−31.9395253114962x_{28} = -31.9395253114962
x40=−27.7507351067098x_{40} = -27.7507351067098
x41=−25.6563400043166x_{41} = -25.6563400043166
x61=−21.4675497995303x_{61} = -21.4675497995303
x71=−19.3731546971371x_{71} = -19.3731546971371
x15=−15.1843644923507x_{15} = -15.1843644923507
x44=−13.0899693899575x_{44} = -13.0899693899575
x34=−8.90117918517108x_{34} = -8.90117918517108
x35=−6.80678408277789x_{35} = -6.80678408277789
x67=−2.61799387799149x_{67} = -2.61799387799149
x53=−0.523598775598299x_{53} = -0.523598775598299
x45=3.66519142918809x_{45} = 3.66519142918809
x60=5.75958653158129x_{60} = 5.75958653158129
x68=9.94837673636768x_{68} = 9.94837673636768
x9=12.0427718387609x_{9} = 12.0427718387609
x49=16.2315620435473x_{49} = 16.2315620435473
x24=18.3259571459405x_{24} = 18.3259571459405
x13=22.5147473507269x_{13} = 22.5147473507269
x59=24.60914245312x_{59} = 24.60914245312
x31=28.7979326579064x_{31} = 28.7979326579064
x14=30.8923277602996x_{14} = 30.8923277602996
x56=35.081117965086x_{56} = 35.081117965086
x11=37.1755130674792x_{11} = 37.1755130674792
x7=41.3643032722656x_{7} = 41.3643032722656
x66=43.4586983746588x_{66} = 43.4586983746588
x20=47.6474885794452x_{20} = 47.6474885794452
x55=49.7418836818384x_{55} = 49.7418836818384
x32=53.9306738866248x_{32} = 53.9306738866248
x21=56.025068989018x_{21} = 56.025068989018
x37=60.2138591938044x_{37} = 60.2138591938044
x17=62.3082542961976x_{17} = 62.3082542961976
x25=66.497044500984x_{25} = 66.497044500984
x3=68.5914396033772x_{3} = 68.5914396033772
x65=72.7802298081635x_{65} = 72.7802298081635
x8=74.8746249105567x_{8} = 74.8746249105567
x23=79.0634151153431x_{23} = 79.0634151153431
x16=81.1578102177363x_{16} = 81.1578102177363
x47=85.3466004225227x_{47} = 85.3466004225227
x4=87.4409955249159x_{4} = 87.4409955249159
x22=91.6297857297023x_{22} = 91.6297857297023
x19=93.7241808320955x_{19} = 93.7241808320955
x54=97.9129710368819x_{54} = 97.9129710368819
x1=100.007366139275x_{1} = 100.007366139275
x12=110.479341651241x_{12} = 110.479341651241
x70=112.573736753634x_{70} = 112.573736753634
x57=791.15774992903x_{57} = 791.15774992903
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x5x_{0} \leq x_{5}
Consideremos, por ejemplo, el punto
x0=x5−110x_{0} = x_{5} - \frac{1}{10}
=
−45867.7763411866+−110-45867.7763411866 + - \frac{1}{10}
=
−45867.8763411866-45867.8763411866
lo sustituimos en la expresión
5sin⁡(x)+cos⁡(2x)≥−25 \sin{\left(x \right)} + \cos{\left(2 x \right)} \geq -2
5sin⁡(−45867.8763411866)+cos⁡((−45867.8763411866)2)≥−25 \sin{\left(-45867.8763411866 \right)} + \cos{\left(\left(-45867.8763411866\right) 2 \right)} \geq -2
-2.60182118651354 >= -2

pero
-2.60182118651354 < -2

Entonces
x≤−45867.7763411866x \leq -45867.7763411866
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x≥−45867.7763411866∧x≤−643.502895210309x \geq -45867.7763411866 \wedge x \leq -643.502895210309
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       x5      x51      x27      x30      x58      x39      x33      x46      x48      x2      x63      x26      x6      x36      x10      x62      x18      x69      x50      x43      x29      x38      x52      x64      x42      x28      x40      x41      x61      x71      x15      x44      x34      x35      x67      x53      x45      x60      x68      x9      x49      x24      x13      x59      x31      x14      x56      x11      x7      x66      x20      x55      x32      x21      x37      x17      x25      x3      x65      x8      x23      x16      x47      x4      x22      x19      x54      x1      x12      x70      x57

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≥−45867.7763411866∧x≤−643.502895210309x \geq -45867.7763411866 \wedge x \leq -643.502895210309
x≥−239.284640448423∧x≤−101.054563690472x \geq -239.284640448423 \wedge x \leq -101.054563690472
x≥−96.8657734856853∧x≤−94.7713783832921x \geq -96.8657734856853 \wedge x \leq -94.7713783832921
x≥−90.5825881785057∧x≤−88.4881930761125x \geq -90.5825881785057 \wedge x \leq -88.4881930761125
x≥−84.2994028713261∧x≤−82.2050077689329x \geq -84.2994028713261 \wedge x \leq -82.2050077689329
x≥−78.0162175641465∧x≤−75.9218224617533x \geq -78.0162175641465 \wedge x \leq -75.9218224617533
x≥−71.733032256967∧x≤−69.6386371545737x \geq -71.733032256967 \wedge x \leq -69.6386371545737
x≥−65.4498469497874∧x≤−63.3554518473942x \geq -65.4498469497874 \wedge x \leq -63.3554518473942
x≥−59.1666616426078∧x≤−57.0722665402146x \geq -59.1666616426078 \wedge x \leq -57.0722665402146
x≥−52.8834763354282∧x≤−50.789081233035x \geq -52.8834763354282 \wedge x \leq -50.789081233035
x≥−46.6002910282486∧x≤−44.5058959258554x \geq -46.6002910282486 \wedge x \leq -44.5058959258554
x≥−40.317105721069∧x≤−38.2227106186758x \geq -40.317105721069 \wedge x \leq -38.2227106186758
x≥−34.0339204138894∧x≤−31.9395253114962x \geq -34.0339204138894 \wedge x \leq -31.9395253114962
x≥−27.7507351067098∧x≤−25.6563400043166x \geq -27.7507351067098 \wedge x \leq -25.6563400043166
x≥−21.4675497995303∧x≤−19.3731546971371x \geq -21.4675497995303 \wedge x \leq -19.3731546971371
x≥−15.1843644923507∧x≤−13.0899693899575x \geq -15.1843644923507 \wedge x \leq -13.0899693899575
x≥−8.90117918517108∧x≤−6.80678408277789x \geq -8.90117918517108 \wedge x \leq -6.80678408277789
x≥−2.61799387799149∧x≤−0.523598775598299x \geq -2.61799387799149 \wedge x \leq -0.523598775598299
x≥3.66519142918809∧x≤5.75958653158129x \geq 3.66519142918809 \wedge x \leq 5.75958653158129
x≥9.94837673636768∧x≤12.0427718387609x \geq 9.94837673636768 \wedge x \leq 12.0427718387609
x≥16.2315620435473∧x≤18.3259571459405x \geq 16.2315620435473 \wedge x \leq 18.3259571459405
x≥22.5147473507269∧x≤24.60914245312x \geq 22.5147473507269 \wedge x \leq 24.60914245312
x≥28.7979326579064∧x≤30.8923277602996x \geq 28.7979326579064 \wedge x \leq 30.8923277602996
x≥35.081117965086∧x≤37.1755130674792x \geq 35.081117965086 \wedge x \leq 37.1755130674792
x≥41.3643032722656∧x≤43.4586983746588x \geq 41.3643032722656 \wedge x \leq 43.4586983746588
x≥47.6474885794452∧x≤49.7418836818384x \geq 47.6474885794452 \wedge x \leq 49.7418836818384
x≥53.9306738866248∧x≤56.025068989018x \geq 53.9306738866248 \wedge x \leq 56.025068989018
x≥60.2138591938044∧x≤62.3082542961976x \geq 60.2138591938044 \wedge x \leq 62.3082542961976
x≥66.497044500984∧x≤68.5914396033772x \geq 66.497044500984 \wedge x \leq 68.5914396033772
x≥72.7802298081635∧x≤74.8746249105567x \geq 72.7802298081635 \wedge x \leq 74.8746249105567
x≥79.0634151153431∧x≤81.1578102177363x \geq 79.0634151153431 \wedge x \leq 81.1578102177363
x≥85.3466004225227∧x≤87.4409955249159x \geq 85.3466004225227 \wedge x \leq 87.4409955249159
x≥91.6297857297023∧x≤93.7241808320955x \geq 91.6297857297023 \wedge x \leq 93.7241808320955
x≥97.9129710368819∧x≤100.007366139275x \geq 97.9129710368819 \wedge x \leq 100.007366139275
x≥110.479341651241∧x≤112.573736753634x \geq 110.479341651241 \wedge x \leq 112.573736753634
x≥791.15774992903x \geq 791.15774992903
Solución de la desigualdad en el gráfico
0-60-50-40-30-20-10102030405060-1010
Respuesta rápida [src] 
  /   /             7*pi\     /11*pi                \\
Or|And|0 <= x, x <= ----|, And|----- <= x, x <= 2*pi||
  \   \              6  /     \  6                  //
(0≤x∧x≤7π6)∨(11π6≤x∧x≤2π)\left(0 \leq x \wedge x \leq \frac{7 \pi}{6}\right) \vee \left(\frac{11 \pi}{6} \leq x \wedge x \leq 2 \pi\right) 
((0 <= x)∧(x <= 7*pi/6))∨((11*pi/6 <= x)∧(x <= 2*pi))
Respuesta rápida 2 [src] 
    7*pi     11*pi       
[0, ----] U [-----, 2*pi]
     6         6
x in [0,7π6]∪[11π6,2π]x\ in\ \left[0, \frac{7 \pi}{6}\right] \cup \left[\frac{11 \pi}{6}, 2 \pi\right] 
x in Union(Interval(0, 7*pi/6), Interval(11*pi/6, 2*pi))
    © Sr Examen – Калькулятор