График функции
0 -90 -80 -70 -60 -50 -40 -30 -20 -10 10 20 30 40 50 -200 200
Область определения функции
Точки, в которых функция точно неопределена:x 1 = − 0.785398163397448 x_{1} = -0.785398163397448 x 1 = − 0.785398163397448 x 2 = 2.35619449019234 x_{2} = 2.35619449019234 x 2 = 2.35619449019234
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( x ) sin ( x + π 4 ) = 0 \frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}} = 0 sin ( x + 4 π ) sin ( x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π x_{2} = \pi x 2 = π Численное решение x 1 = 59.6902604182061 x_{1} = 59.6902604182061 x 1 = 59.6902604182061 x 2 = − 97.3893722612836 x_{2} = -97.3893722612836 x 2 = − 97.3893722612836 x 3 = − 75.398223686155 x_{3} = -75.398223686155 x 3 = − 75.398223686155 x 4 = − 31.4159265358979 x_{4} = -31.4159265358979 x 4 = − 31.4159265358979 x 5 = 97.3893722612836 x_{5} = 97.3893722612836 x 5 = 97.3893722612836 x 6 = − 28.2743338823081 x_{6} = -28.2743338823081 x 6 = − 28.2743338823081 x 7 = 72.2566310325652 x_{7} = 72.2566310325652 x 7 = 72.2566310325652 x 8 = 47.1238898038469 x_{8} = 47.1238898038469 x 8 = 47.1238898038469 x 9 = 12.5663706143592 x_{9} = 12.5663706143592 x 9 = 12.5663706143592 x 10 = − 84.8230016469244 x_{10} = -84.8230016469244 x 10 = − 84.8230016469244 x 11 = 56.5486677646163 x_{11} = 56.5486677646163 x 11 = 56.5486677646163 x 12 = 0 x_{12} = 0 x 12 = 0 x 13 = 69.1150383789755 x_{13} = 69.1150383789755 x 13 = 69.1150383789755 x 14 = 81.6814089933346 x_{14} = 81.6814089933346 x 14 = 81.6814089933346 x 15 = 75.398223686155 x_{15} = 75.398223686155 x 15 = 75.398223686155 x 16 = − 40.8407044966673 x_{16} = -40.8407044966673 x 16 = − 40.8407044966673 x 17 = − 81.6814089933346 x_{17} = -81.6814089933346 x 17 = − 81.6814089933346 x 18 = 91.106186954104 x_{18} = 91.106186954104 x 18 = 91.106186954104 x 19 = − 94.2477796076938 x_{19} = -94.2477796076938 x 19 = − 94.2477796076938 x 20 = − 53.4070751110265 x_{20} = -53.4070751110265 x 20 = − 53.4070751110265 x 21 = 34.5575191894877 x_{21} = 34.5575191894877 x 21 = 34.5575191894877 x 22 = 40.8407044966673 x_{22} = 40.8407044966673 x 22 = 40.8407044966673 x 23 = − 62.8318530717959 x_{23} = -62.8318530717959 x 23 = − 62.8318530717959 x 24 = 21.9911485751286 x_{24} = 21.9911485751286 x 24 = 21.9911485751286 x 25 = 15.707963267949 x_{25} = 15.707963267949 x 25 = 15.707963267949 x 26 = − 15.707963267949 x_{26} = -15.707963267949 x 26 = − 15.707963267949 x 27 = − 21.9911485751286 x_{27} = -21.9911485751286 x 27 = − 21.9911485751286 x 28 = 18.8495559215388 x_{28} = 18.8495559215388 x 28 = 18.8495559215388 x 29 = − 3.14159265358979 x_{29} = -3.14159265358979 x 29 = − 3.14159265358979 x 30 = 3.14159265358979 x_{30} = 3.14159265358979 x 30 = 3.14159265358979 x 31 = 94.2477796076938 x_{31} = 94.2477796076938 x 31 = 94.2477796076938 x 32 = − 50.2654824574367 x_{32} = -50.2654824574367 x 32 = − 50.2654824574367 x 33 = 100.530964914873 x_{33} = 100.530964914873 x 33 = 100.530964914873 x 34 = − 78.5398163397448 x_{34} = -78.5398163397448 x 34 = − 78.5398163397448 x 35 = 31.4159265358979 x_{35} = 31.4159265358979 x 35 = 31.4159265358979 x 36 = − 43.9822971502571 x_{36} = -43.9822971502571 x 36 = − 43.9822971502571 x 37 = − 37.6991118430775 x_{37} = -37.6991118430775 x 37 = − 37.6991118430775 x 38 = − 59.6902604182061 x_{38} = -59.6902604182061 x 38 = − 59.6902604182061 x 39 = 37.6991118430775 x_{39} = 37.6991118430775 x 39 = 37.6991118430775 x 40 = − 100.530964914873 x_{40} = -100.530964914873 x 40 = − 100.530964914873 x 41 = 84.8230016469244 x_{41} = 84.8230016469244 x 41 = 84.8230016469244 x 42 = 87.9645943005142 x_{42} = 87.9645943005142 x 42 = 87.9645943005142 x 43 = − 12.5663706143592 x_{43} = -12.5663706143592 x 43 = − 12.5663706143592 x 44 = 62.8318530717959 x_{44} = 62.8318530717959 x 44 = 62.8318530717959 x 45 = − 69.1150383789755 x_{45} = -69.1150383789755 x 45 = − 69.1150383789755 x 46 = 9.42477796076938 x_{46} = 9.42477796076938 x 46 = 9.42477796076938 x 47 = − 18.8495559215388 x_{47} = -18.8495559215388 x 47 = − 18.8495559215388 x 48 = 50.2654824574367 x_{48} = 50.2654824574367 x 48 = 50.2654824574367 x 49 = − 6.28318530717959 x_{49} = -6.28318530717959 x 49 = − 6.28318530717959 x 50 = 43.9822971502571 x_{50} = 43.9822971502571 x 50 = 43.9822971502571 x 51 = 6.28318530717959 x_{51} = 6.28318530717959 x 51 = 6.28318530717959 x 52 = 53.4070751110265 x_{52} = 53.4070751110265 x 52 = 53.4070751110265 x 53 = − 56.5486677646163 x_{53} = -56.5486677646163 x 53 = − 56.5486677646163 x 54 = − 9.42477796076938 x_{54} = -9.42477796076938 x 54 = − 9.42477796076938 x 55 = − 87.9645943005142 x_{55} = -87.9645943005142 x 55 = − 87.9645943005142 x 56 = 25.1327412287183 x_{56} = 25.1327412287183 x 56 = 25.1327412287183 x 57 = − 25.1327412287183 x_{57} = -25.1327412287183 x 57 = − 25.1327412287183 x 58 = 65.9734457253857 x_{58} = 65.9734457253857 x 58 = 65.9734457253857 x 59 = 28.2743338823081 x_{59} = 28.2743338823081 x 59 = 28.2743338823081 x 60 = − 91.106186954104 x_{60} = -91.106186954104 x 60 = − 91.106186954104 x 61 = − 47.1238898038469 x_{61} = -47.1238898038469 x 61 = − 47.1238898038469 x 62 = − 72.2566310325652 x_{62} = -72.2566310325652 x 62 = − 72.2566310325652 x 63 = 78.5398163397448 x_{63} = 78.5398163397448 x 63 = 78.5398163397448 x 64 = − 65.9734457253857 x_{64} = -65.9734457253857 x 64 = − 65.9734457253857 x 65 = − 34.5575191894877 x_{65} = -34.5575191894877 x 65 = − 34.5575191894877
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 0 ) sin ( 0 + π 4 ) \frac{\sin{\left(0 \right)}}{\sin{\left(0 + \frac{\pi}{4} \right)}} sin ( 0 + 4 π ) sin ( 0 ) f ( 0 ) = 0 f{\left(0 \right)} = 0 f ( 0 ) = 0 (0, 0)
Экстремумы функции
Для того, чтобы найти экстремумы, нужно решить уравнениеd d x f ( x ) = 0 \frac{d}{d x} f{\left(x \right)} = 0 d x d f ( x ) = 0 d d x f ( x ) = \frac{d}{d x} f{\left(x \right)} = d x d f ( x ) = первая производная − sin ( x ) cos ( x + π 4 ) sin 2 ( x + π 4 ) + cos ( x ) sin ( x + π 4 ) = 0 - \frac{\sin{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}} + \frac{\cos{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}} = 0 − sin 2 ( x + 4 π ) sin ( x ) cos ( x + 4 π ) + sin ( x + 4 π ) cos ( x ) = 0 Решаем это уравнение
Точки перегибов
Найдем точки перегибов, для этого надо решить уравнениеd 2 d x 2 f ( x ) = 0 \frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0 d x 2 d 2 f ( x ) = 0 d 2 d x 2 f ( x ) = \frac{d^{2}}{d x^{2}} f{\left(x \right)} = d x 2 d 2 f ( x ) = вторая производная ( 1 + 2 cos 2 ( x + π 4 ) sin 2 ( x + π 4 ) ) sin ( x ) − sin ( x ) − 2 cos ( x ) cos ( x + π 4 ) sin ( x + π 4 ) sin ( x + π 4 ) = 0 \frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}} = 0 sin ( x + 4 π ) ( 1 + s i n 2 ( x + 4 π ) 2 c o s 2 ( x + 4 π ) ) sin ( x ) − sin ( x ) − s i n ( x + 4 π ) 2 c o s ( x ) c o s ( x + 4 π ) = 0 Решаем это уравнение x 1 = 25.9181393921158 x_{1} = 25.9181393921158 x 1 = 25.9181393921158 x 2 = − 96.6039740978861 x_{2} = -96.6039740978861 x 2 = − 96.6039740978861 x 3 = − 68.329640215578 x_{3} = -68.329640215578 x 3 = − 68.329640215578 x 4 = − 55.7632696012188 x_{4} = -55.7632696012188 x 4 = − 55.7632696012188 x 5 = − 21.2057504117311 x_{5} = -21.2057504117311 x 5 = − 21.2057504117311 x 6 = − 62.0464549083984 x_{6} = -62.0464549083984 x 6 = − 62.0464549083984 x 7 = 91.8915851175014 x_{7} = 91.8915851175014 x 7 = 91.8915851175014 x 8 = − 11.7809724509617 x_{8} = -11.7809724509617 x 8 = − 11.7809724509617 x 9 = 82.4668071567321 x_{9} = 82.4668071567321 x 9 = 82.4668071567321 x 10 = − 99.7455667514759 x_{10} = -99.7455667514759 x 10 = − 99.7455667514759 x 11 = 51.0508806208341 x_{11} = 51.0508806208341 x 11 = 51.0508806208341 x 12 = 88.7499924639117 x_{12} = 88.7499924639117 x 12 = 88.7499924639117 x 13 = − 27.4889357189107 x_{13} = -27.4889357189107 x 13 = − 27.4889357189107 x 14 = − 36.9137136796801 x_{14} = -36.9137136796801 x 14 = − 36.9137136796801 x 15 = 38.484510006475 x_{15} = 38.484510006475 x 15 = 38.484510006475 x 16 = − 93.4623814442964 x_{16} = -93.4623814442964 x 16 = − 93.4623814442964 x 17 = 44.7676953136546 x_{17} = 44.7676953136546 x 17 = 44.7676953136546 x 18 = − 90.3207887907066 x_{18} = -90.3207887907066 x 18 = − 90.3207887907066 x 19 = 13.3517687777566 x_{19} = 13.3517687777566 x 19 = 13.3517687777566 x 20 = − 40.0553063332699 x_{20} = -40.0553063332699 x 20 = − 40.0553063332699 x 21 = 3.92699081698724 x_{21} = 3.92699081698724 x 21 = 3.92699081698724 x 22 = 85.6083998103219 x_{22} = 85.6083998103219 x 22 = 85.6083998103219 x 23 = 95.0331777710912 x_{23} = 95.0331777710912 x 23 = 95.0331777710912 x 24 = 69.9004365423729 x_{24} = 69.9004365423729 x 24 = 69.9004365423729 x 25 = − 14.9225651045515 x_{25} = -14.9225651045515 x 25 = − 14.9225651045515 x 26 = − 71.4712328691678 x_{26} = -71.4712328691678 x 26 = − 71.4712328691678 x 27 = − 52.621676947629 x_{27} = -52.621676947629 x 27 = − 52.621676947629 x 28 = 41.6261026600648 x_{28} = 41.6261026600648 x 28 = 41.6261026600648 x 29 = 66.7588438887831 x_{29} = 66.7588438887831 x 29 = 66.7588438887831 x 30 = 16.4933614313464 x_{30} = 16.4933614313464 x 30 = 16.4933614313464 x 31 = − 18.0641577581413 x_{31} = -18.0641577581413 x 31 = − 18.0641577581413 x 32 = 22.776546738526 x_{32} = 22.776546738526 x 32 = 22.776546738526 x 33 = − 74.6128255227576 x_{33} = -74.6128255227576 x 33 = − 74.6128255227576 x 34 = − 49.4800842940392 x_{34} = -49.4800842940392 x 34 = − 49.4800842940392 x 35 = 35.3429173528852 x_{35} = 35.3429173528852 x 35 = 35.3429173528852 x 36 = 54.1924732744239 x_{36} = 54.1924732744239 x 36 = 54.1924732744239 x 37 = − 84.037603483527 x_{37} = -84.037603483527 x 37 = − 84.037603483527 x 38 = − 77.7544181763474 x_{38} = -77.7544181763474 x 38 = − 77.7544181763474 x 39 = 29.0597320457056 x_{39} = 29.0597320457056 x 39 = 29.0597320457056 x 40 = 57.3340659280137 x_{40} = 57.3340659280137 x 40 = 57.3340659280137 x 41 = − 24.3473430653209 x_{41} = -24.3473430653209 x 41 = − 24.3473430653209 x 42 = 60.4756585816035 x_{42} = 60.4756585816035 x 42 = 60.4756585816035 x 43 = 79.3252145031423 x_{43} = 79.3252145031423 x 43 = 79.3252145031423 x 44 = − 58.9048622548086 x_{44} = -58.9048622548086 x 44 = − 58.9048622548086 x 45 = 0.785398163397448 x_{45} = 0.785398163397448 x 45 = 0.785398163397448 x 46 = 19.6349540849362 x_{46} = 19.6349540849362 x 46 = 19.6349540849362 x 47 = − 43.1968989868597 x_{47} = -43.1968989868597 x 47 = − 43.1968989868597 x 48 = 98.174770424681 x_{48} = 98.174770424681 x 48 = 98.174770424681 x 49 = 101.316363078271 x_{49} = 101.316363078271 x 49 = 101.316363078271 x 50 = − 8.63937979737193 x_{50} = -8.63937979737193 x 50 = − 8.63937979737193 x 51 = 47.9092879672443 x_{51} = 47.9092879672443 x 51 = 47.9092879672443 x 52 = − 2.35619449019234 x_{52} = -2.35619449019234 x 52 = − 2.35619449019234 x 53 = 32.2013246992954 x_{53} = 32.2013246992954 x 53 = 32.2013246992954 x 54 = 63.6172512351933 x_{54} = 63.6172512351933 x 54 = 63.6172512351933 x 55 = 73.0420291959627 x_{55} = 73.0420291959627 x 55 = 73.0420291959627 x 56 = 10.2101761241668 x_{56} = 10.2101761241668 x 56 = 10.2101761241668 x 57 = − 33.7721210260903 x_{57} = -33.7721210260903 x 57 = − 33.7721210260903 x 58 = − 5.49778714378214 x_{58} = -5.49778714378214 x 58 = − 5.49778714378214 x 59 = − 65.1880475619882 x_{59} = -65.1880475619882 x 59 = − 65.1880475619882 x 60 = − 46.3384916404494 x_{60} = -46.3384916404494 x 60 = − 46.3384916404494 x 61 = 7.06858347057703 x_{61} = 7.06858347057703 x 61 = 7.06858347057703 x 62 = − 30.6305283725005 x_{62} = -30.6305283725005 x 62 = − 30.6305283725005 x 63 = − 87.1791961371168 x_{63} = -87.1791961371168 x 63 = − 87.1791961371168 x 64 = − 80.8960108299372 x_{64} = -80.8960108299372 x 64 = − 80.8960108299372 x 65 = 76.1836218495525 x_{65} = 76.1836218495525 x 65 = 76.1836218495525 x 1 = − 0.785398163397448 x_{1} = -0.785398163397448 x 1 = − 0.785398163397448 x 2 = 2.35619449019234 x_{2} = 2.35619449019234 x 2 = 2.35619449019234 lim x → − 0.78539816339744 8 − ( ( 1 + 2 cos 2 ( x + π 4 ) sin 2 ( x + π 4 ) ) sin ( x ) − sin ( x ) − 2 cos ( x ) cos ( x + π 4 ) sin ( x + π 4 ) sin ( x + π 4 ) ) = − − 1.41421356237309 cos 2 ( 0.785398163397448 − 0.25 π ) sin 2 ( 0.785398163397448 − 0.25 π ) + 1.4142135623731 cos ( 0.785398163397448 − 0.25 π ) sin ( 0.785398163397448 − 0.25 π ) sin ( 0.785398163397448 − 0.25 π ) \lim_{x \to -0.785398163397448^-}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = - \frac{- \frac{1.41421356237309 \cos^{2}{\left(0.785398163397448 - 0.25 \pi \right)}}{\sin^{2}{\left(0.785398163397448 - 0.25 \pi \right)}} + \frac{1.4142135623731 \cos{\left(0.785398163397448 - 0.25 \pi \right)}}{\sin{\left(0.785398163397448 - 0.25 \pi \right)}}}{\sin{\left(0.785398163397448 - 0.25 \pi \right)}} x → − 0.78539816339744 8 − lim sin ( x + 4 π ) ( 1 + s i n 2 ( x + 4 π ) 2 c o s 2 ( x + 4 π ) ) sin ( x ) − sin ( x ) − s i n ( x + 4 π ) 2 c o s ( x ) c o s ( x + 4 π ) = − sin ( 0.785398163397448 − 0.25 π ) − s i n 2 ( 0.785398163397448 − 0.25 π ) 1.41421356237309 c o s 2 ( 0.785398163397448 − 0.25 π ) + s i n ( 0.785398163397448 − 0.25 π ) 1.4142135623731 c o s ( 0.785398163397448 − 0.25 π ) Возьмём предел lim x → − 0.78539816339744 8 + ( ( 1 + 2 cos 2 ( x + π 4 ) sin 2 ( x + π 4 ) ) sin ( x ) − sin ( x ) − 2 cos ( x ) cos ( x + π 4 ) sin ( x + π 4 ) sin ( x + π 4 ) ) = − − 1.41421356237309 cos 2 ( 0.785398163397448 − 0.25 π ) sin 2 ( 0.785398163397448 − 0.25 π ) + 1.4142135623731 cos ( 0.785398163397448 − 0.25 π ) sin ( 0.785398163397448 − 0.25 π ) sin ( 0.785398163397448 − 0.25 π ) \lim_{x \to -0.785398163397448^+}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = - \frac{- \frac{1.41421356237309 \cos^{2}{\left(0.785398163397448 - 0.25 \pi \right)}}{\sin^{2}{\left(0.785398163397448 - 0.25 \pi \right)}} + \frac{1.4142135623731 \cos{\left(0.785398163397448 - 0.25 \pi \right)}}{\sin{\left(0.785398163397448 - 0.25 \pi \right)}}}{\sin{\left(0.785398163397448 - 0.25 \pi \right)}} x → − 0.78539816339744 8 + lim sin ( x + 4 π ) ( 1 + s i n 2 ( x + 4 π ) 2 c o s 2 ( x + 4 π ) ) sin ( x ) − sin ( x ) − s i n ( x + 4 π ) 2 c o s ( x ) c o s ( x + 4 π ) = − sin ( 0.785398163397448 − 0.25 π ) − s i n 2 ( 0.785398163397448 − 0.25 π ) 1.41421356237309 c o s 2 ( 0.785398163397448 − 0.25 π ) + s i n ( 0.785398163397448 − 0.25 π ) 1.4142135623731 c o s ( 0.785398163397448 − 0.25 π ) Возьмём предел lim x → 2.3561944901923 4 − ( ( 1 + 2 cos 2 ( x + π 4 ) sin 2 ( x + π 4 ) ) sin ( x ) − sin ( x ) − 2 cos ( x ) cos ( x + π 4 ) sin ( x + π 4 ) sin ( x + π 4 ) ) = 1.41421356237309 cos ( 0.25 π + 2.35619449019234 ) sin ( 0.25 π + 2.35619449019234 ) + 1.4142135623731 cos 2 ( 0.25 π + 2.35619449019234 ) sin 2 ( 0.25 π + 2.35619449019234 ) sin ( 0.25 π + 2.35619449019234 ) \lim_{x \to 2.35619449019234^-}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \frac{\frac{1.41421356237309 \cos{\left(0.25 \pi + 2.35619449019234 \right)}}{\sin{\left(0.25 \pi + 2.35619449019234 \right)}} + \frac{1.4142135623731 \cos^{2}{\left(0.25 \pi + 2.35619449019234 \right)}}{\sin^{2}{\left(0.25 \pi + 2.35619449019234 \right)}}}{\sin{\left(0.25 \pi + 2.35619449019234 \right)}} x → 2.3561944901923 4 − lim sin ( x + 4 π ) ( 1 + s i n 2 ( x + 4 π ) 2 c o s 2 ( x + 4 π ) ) sin ( x ) − sin ( x ) − s i n ( x + 4 π ) 2 c o s ( x ) c o s ( x + 4 π ) = sin ( 0.25 π + 2.35619449019234 ) s i n ( 0.25 π + 2.35619449019234 ) 1.41421356237309 c o s ( 0.25 π + 2.35619449019234 ) + s i n 2 ( 0.25 π + 2.35619449019234 ) 1.4142135623731 c o s 2 ( 0.25 π + 2.35619449019234 ) Возьмём предел lim x → 2.3561944901923 4 + ( ( 1 + 2 cos 2 ( x + π 4 ) sin 2 ( x + π 4 ) ) sin ( x ) − sin ( x ) − 2 cos ( x ) cos ( x + π 4 ) sin ( x + π 4 ) sin ( x + π 4 ) ) = 1.41421356237309 cos ( 0.25 π + 2.35619449019234 ) sin ( 0.25 π + 2.35619449019234 ) + 1.4142135623731 cos 2 ( 0.25 π + 2.35619449019234 ) sin 2 ( 0.25 π + 2.35619449019234 ) sin ( 0.25 π + 2.35619449019234 ) \lim_{x \to 2.35619449019234^+}\left(\frac{\left(1 + \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin^{2}{\left(x + \frac{\pi}{4} \right)}}\right) \sin{\left(x \right)} - \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \frac{\frac{1.41421356237309 \cos{\left(0.25 \pi + 2.35619449019234 \right)}}{\sin{\left(0.25 \pi + 2.35619449019234 \right)}} + \frac{1.4142135623731 \cos^{2}{\left(0.25 \pi + 2.35619449019234 \right)}}{\sin^{2}{\left(0.25 \pi + 2.35619449019234 \right)}}}{\sin{\left(0.25 \pi + 2.35619449019234 \right)}} x → 2.3561944901923 4 + lim sin ( x + 4 π ) ( 1 + s i n 2 ( x + 4 π ) 2 c o s 2 ( x + 4 π ) ) sin ( x ) − sin ( x ) − s i n ( x + 4 π ) 2 c o s ( x ) c o s ( x + 4 π ) = sin ( 0.25 π + 2.35619449019234 ) s i n ( 0.25 π + 2.35619449019234 ) 1.41421356237309 c o s ( 0.25 π + 2.35619449019234 ) + s i n 2 ( 0.25 π + 2.35619449019234 ) 1.4142135623731 c o s 2 ( 0.25 π + 2.35619449019234 ) Возьмём предел Интервалы выпуклости и вогнутости: [ 101.316363078271 , ∞ ) \left[101.316363078271, \infty\right) [ 101.316363078271 , ∞ ) ( − ∞ , − 99.7455667514759 ] \left(-\infty, -99.7455667514759\right] ( − ∞ , − 99.7455667514759 ]
Вертикальные асимптоты
Есть:x 1 = − 0.785398163397448 x_{1} = -0.785398163397448 x 1 = − 0.785398163397448 x 2 = 2.35619449019234 x_{2} = 2.35619449019234 x 2 = 2.35619449019234
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( sin ( x ) sin ( x + π 4 ) ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \left\langle -\infty, \infty\right\rangle x → − ∞ lim ( sin ( x + 4 π ) sin ( x ) ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩ lim x → ∞ ( sin ( x ) sin ( x + π 4 ) ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}\right) = \left\langle -\infty, \infty\right\rangle x → ∞ lim ( sin ( x + 4 π ) sin ( x ) ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(x)/sin(x + pi/4), делённой на x при x->+oo и x ->-oolim x → − ∞ ( sin ( x ) x sin ( x + π 4 ) ) = lim x → − ∞ ( sin ( x ) x sin ( x + π 4 ) ) \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{4} \right)}}\right) = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{4} \right)}}\right) x → − ∞ lim ( x sin ( x + 4 π ) sin ( x ) ) = x → − ∞ lim ( x sin ( x + 4 π ) sin ( x ) ) Возьмём предел y = x lim x → − ∞ ( sin ( x ) x sin ( x + π 4 ) ) y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{4} \right)}}\right) y = x x → − ∞ lim ( x sin ( x + 4 π ) sin ( x ) ) lim x → ∞ ( sin ( x ) x sin ( x + π 4 ) ) = lim x → ∞ ( sin ( x ) x sin ( x + π 4 ) ) \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{4} \right)}}\right) = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{4} \right)}}\right) x → ∞ lim ( x sin ( x + 4 π ) sin ( x ) ) = x → ∞ lim ( x sin ( x + 4 π ) sin ( x ) ) Возьмём предел y = x lim x → ∞ ( sin ( x ) x sin ( x + π 4 ) ) y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sin{\left(x + \frac{\pi}{4} \right)}}\right) y = x x → ∞ lim ( x sin ( x + 4 π ) sin ( x ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( x ) sin ( x + π 4 ) = − sin ( x ) cos ( x + π 4 ) \frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}} = - \frac{\sin{\left(x \right)}}{\cos{\left(x + \frac{\pi}{4} \right)}} sin ( x + 4 π ) sin ( x ) = − cos ( x + 4 π ) sin ( x ) sin ( x ) sin ( x + π 4 ) = sin ( x ) cos ( x + π 4 ) \frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}} = \frac{\sin{\left(x \right)}}{\cos{\left(x + \frac{\pi}{4} \right)}} sin ( x + 4 π ) sin ( x ) = cos ( x + 4 π ) sin ( x ) 
