In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$2 x \sin{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 12.5663706143592$$
$$x_{2} = 78.5398163397448$$
$$x_{3} = 67.5516436614121$$
$$x_{4} = 86.3995849739529$$
$$x_{5} = -65.9734457253857$$
$$x_{6} = -86.3995849739529$$
$$x_{7} = -15.707963267949$$
$$x_{8} = 50.2654824574367$$
$$x_{9} = 81.6814089933346$$
$$x_{10} = 3.14159265358979$$
$$x_{11} = 20.4448034666183$$
$$x_{12} = 73.8341991854591$$
$$x_{13} = 23.5831433102848$$
$$x_{14} = 26.7222463741877$$
$$x_{15} = -306.306916073247$$
$$x_{16} = 14.1724320747999$$
$$x_{17} = -75.398223686155$$
$$x_{18} = -42.4232862577008$$
$$x_{19} = 58.1280655761511$$
$$x_{20} = -7.91705268466621$$
$$x_{21} = 56.5486677646163$$
$$x_{22} = 51.8459224452234$$
$$x_{23} = -17.3076405374146$$
$$x_{24} = -14.1724320747999$$
$$x_{25} = 15.707963267949$$
$$x_{26} = 45.5640665961997$$
$$x_{27} = 1.83659720315213$$
$$x_{28} = 89.5409746049841$$
$$x_{29} = 7.91705268466621$$
$$x_{30} = -73.8341991854591$$
$$x_{31} = -59.6902604182061$$
$$x_{32} = 21.9911485751286$$
$$x_{33} = 6.28318530717959$$
$$x_{34} = -87.9645943005142$$
$$x_{35} = -28.2743338823081$$
$$x_{36} = -67.5516436614121$$
$$x_{37} = -9.42477796076938$$
$$x_{38} = 42.4232862577008$$
$$x_{39} = 59.6902604182061$$
$$x_{40} = -83.2582106616487$$
$$x_{41} = 28.2743338823081$$
$$x_{42} = 94.2477796076938$$
$$x_{43} = -29.861872403816$$
$$x_{44} = -72.2566310325652$$
$$x_{45} = -84.8230016469244$$
$$x_{46} = 70.692907433161$$
$$x_{47} = -105.248104538899$$
$$x_{48} = -51.8459224452234$$
$$x_{49} = -4.81584231784594$$
$$x_{50} = -97.3893722612836$$
$$x_{51} = 37.6991118430775$$
$$x_{52} = -50.2654824574367$$
$$x_{53} = -94.2477796076938$$
$$x_{54} = 36.1421488970061$$
$$x_{55} = -37.6991118430775$$
$$x_{56} = -61.2692172687226$$
$$x_{57} = -64.410411962776$$
$$x_{58} = -95.8237937978449$$
$$x_{59} = -23.5831433102848$$
$$x_{60} = -20.4448034666183$$
$$x_{61} = -58.1280655761511$$
$$x_{62} = -81.6814089933346$$
$$x_{63} = 43.9822971502571$$
$$x_{64} = 48.7049516666752$$
$$x_{65} = -1.83659720315213$$
$$x_{66} = -36.1421488970061$$
$$x_{67} = -31.4159265358979$$
$$x_{68} = 0$$
$$x_{69} = -39.2826357527234$$
$$x_{70} = -21.9911485751286$$
$$x_{71} = 100.530964914873$$
$$x_{72} = 34.5575191894877$$
$$x_{73} = 278.032748190065$$
$$x_{74} = 65.9734457253857$$
$$x_{75} = 92.682377997352$$
$$x_{76} = -45.5640665961997$$
$$x_{77} = 80.1168534696549$$
$$x_{78} = -53.4070751110265$$
$$x_{79} = -89.5409746049841$$
$$x_{80} = 95.8237937978449$$
$$x_{81} = -6.28318530717959$$
$$x_{82} = 25.1327412287183$$
$$x_{83} = 64.410411962776$$
$$x_{84} = -43.9822971502571$$
$$x_{85} = 72.2566310325652$$
$$x_{86} = -80.1168534696549$$
$$x_{87} = 29.861872403816$$
$$x_{88} = 87.9645943005142$$
The values of the extrema at the points:
(12.566370614359172, 3.01544596183035e-30)
(78.53981633974483, 1.8941914820334e-29)
(67.5516436614121, 67.5479429919577)
(86.3995849739529, 86.3966915384367)
(-65.97344572538566, -6.34844983898999e-29)
(-86.3995849739529, -86.3966915384367)
(-15.707963267948966, -5.8895428941999e-30)
(50.26548245743669, 1.92988541557142e-28)
(81.68140899333463, 1.25601110053315e-27)
(3.141592653589793, 4.71163431535992e-32)
(20.4448034666183, 20.4325827297121)
(73.83419918545908, 73.8308133759219)
(23.583143310284843, 23.5725472811462)
(26.72224637418772, 26.7128941475173)
(-306.30691607324667, -306.306099900576)
(14.172432074799941, 14.1548141232633)
(-75.39822368615503, -6.51336327755355e-28)
(-42.423286257700816, -42.4173940862181)
(58.12806557615112, 58.1237650459065)
(-7.917052684666207, -7.88560072412753)
(56.548667764616276, 2.7478251327179e-28)
(51.84592244522343, 51.8411009136761)
(-17.307640537414635, -17.2932080946897)
(-14.172432074799941, -14.1548141232633)
(15.707963267948966, 5.8895428941999e-30)
(45.56406659619972, 45.5585804770373)
(1.8365972031521258, 1.70986852923209)
(89.54097460498406, 89.5381826741839)
(7.917052684666207, 7.88560072412753)
(-73.83419918545908, -73.8308133759219)
(-59.69026041820607, -8.97021321364436e-29)
(21.991148575128552, 1.61609057016845e-29)
(6.283185307179586, 3.76930745228793e-31)
(-87.96459430051421, -1.03429796490781e-27)
(-28.274333882308138, -3.43478141589738e-29)
(-67.5516436614121, -67.5479429919577)
(-9.42477796076938, -1.27214126514718e-30)
(42.423286257700816, 42.4173940862181)
(59.69026041820607, 8.97021321364436e-29)
(-83.25821066164869, -83.255208063081)
(28.274333882308138, 3.43478141589738e-29)
(94.2477796076938, 1.10977728956951e-27)
(-29.861872403816044, -29.853502870657)
(-72.25663103256524, -2.93139900017185e-27)
(-84.82300164692441, -3.99087542625273e-27)
(70.692907433161, 70.6893711873986)
(-105.24810453889911, -105.245729252817)
(-51.84592244522343, -51.8411009136761)
(-4.815842317845935, -4.76448393290203)
(-97.3893722612836, -4.58542475390885e-27)
(37.69911184307752, 8.14170409694193e-29)
(-50.26548245743669, -1.92988541557142e-28)
(-94.2477796076938, -1.10977728956951e-27)
(36.142148897006074, 36.135233089007)
(-37.69911184307752, -8.14170409694193e-29)
(-61.269217268722585, -61.2651371880071)
(-64.41041196277601, -64.4065308365988)
(-95.82379379784489, -95.8211849135206)
(-23.583143310284843, -23.5725472811462)
(-20.4448034666183, -20.4325827297121)
(-58.12806557615112, -58.1237650459065)
(-81.68140899333463, -1.25601110053315e-27)
(43.982297150257104, 1.29287245613476e-28)
(48.70495166667517, 48.6998192592491)
(-1.8365972031521258, -1.70986852923209)
(-36.142148897006074, -36.135233089007)
(-31.41592653589793, -4.71163431535992e-29)
(0, 0)
(-39.282635752723394, -39.2762726485285)
(-21.991148575128552, -1.61609057016845e-29)
(100.53096491487338, 1.54390833245714e-27)
(34.55751918948773, 1.68111309202325e-28)
(278.0327481900649, 278.031849018319)
(65.97344572538566, 6.34844983898999e-29)
(92.68237799735202, 92.6796806914592)
(-45.56406659619972, -45.5585804770373)
(80.11685346965491, 80.1137331491182)
(-53.40707511102649, -1.15535214562331e-28)
(-89.54097460498406, -89.5381826741839)
(95.82379379784489, 95.8211849135206)
(-6.283185307179586, -3.76930745228793e-31)
(25.132741228718345, 2.41235676946428e-29)
(64.41041196277601, 64.4065308365988)
(-43.982297150257104, -1.29287245613476e-28)
(72.25663103256524, 2.93139900017185e-27)
(-80.11685346965491, -80.1137331491182)
(29.861872403816044, 29.853502870657)
(87.96459430051421, 1.03429796490781e-27)
Intervals of increase and decrease of the function:Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = 12.5663706143592$$
$$x_{2} = 78.5398163397448$$
$$x_{3} = -86.3995849739529$$
$$x_{4} = 50.2654824574367$$
$$x_{5} = 81.6814089933346$$
$$x_{6} = 3.14159265358979$$
$$x_{7} = -306.306916073247$$
$$x_{8} = -42.4232862577008$$
$$x_{9} = -7.91705268466621$$
$$x_{10} = 56.5486677646163$$
$$x_{11} = -17.3076405374146$$
$$x_{12} = -14.1724320747999$$
$$x_{13} = 15.707963267949$$
$$x_{14} = -73.8341991854591$$
$$x_{15} = 21.9911485751286$$
$$x_{16} = 6.28318530717959$$
$$x_{17} = -67.5516436614121$$
$$x_{18} = 59.6902604182061$$
$$x_{19} = -83.2582106616487$$
$$x_{20} = 28.2743338823081$$
$$x_{21} = 94.2477796076938$$
$$x_{22} = -29.861872403816$$
$$x_{23} = -105.248104538899$$
$$x_{24} = -51.8459224452234$$
$$x_{25} = -4.81584231784594$$
$$x_{26} = 37.6991118430775$$
$$x_{27} = -61.2692172687226$$
$$x_{28} = -64.410411962776$$
$$x_{29} = -95.8237937978449$$
$$x_{30} = -23.5831433102848$$
$$x_{31} = -20.4448034666183$$
$$x_{32} = -58.1280655761511$$
$$x_{33} = 43.9822971502571$$
$$x_{34} = -1.83659720315213$$
$$x_{35} = -36.1421488970061$$
$$x_{36} = -39.2826357527234$$
$$x_{37} = 100.530964914873$$
$$x_{38} = 34.5575191894877$$
$$x_{39} = 65.9734457253857$$
$$x_{40} = -45.5640665961997$$
$$x_{41} = -89.5409746049841$$
$$x_{42} = 25.1327412287183$$
$$x_{43} = 72.2566310325652$$
$$x_{44} = -80.1168534696549$$
$$x_{45} = 87.9645943005142$$
Maxima of the function at points:
$$x_{45} = 67.5516436614121$$
$$x_{45} = 86.3995849739529$$
$$x_{45} = -65.9734457253857$$
$$x_{45} = -15.707963267949$$
$$x_{45} = 20.4448034666183$$
$$x_{45} = 73.8341991854591$$
$$x_{45} = 23.5831433102848$$
$$x_{45} = 26.7222463741877$$
$$x_{45} = 14.1724320747999$$
$$x_{45} = -75.398223686155$$
$$x_{45} = 58.1280655761511$$
$$x_{45} = 51.8459224452234$$
$$x_{45} = 45.5640665961997$$
$$x_{45} = 1.83659720315213$$
$$x_{45} = 89.5409746049841$$
$$x_{45} = 7.91705268466621$$
$$x_{45} = -59.6902604182061$$
$$x_{45} = -87.9645943005142$$
$$x_{45} = -28.2743338823081$$
$$x_{45} = -9.42477796076938$$
$$x_{45} = 42.4232862577008$$
$$x_{45} = -72.2566310325652$$
$$x_{45} = -84.8230016469244$$
$$x_{45} = 70.692907433161$$
$$x_{45} = -97.3893722612836$$
$$x_{45} = -50.2654824574367$$
$$x_{45} = -94.2477796076938$$
$$x_{45} = 36.1421488970061$$
$$x_{45} = -37.6991118430775$$
$$x_{45} = -81.6814089933346$$
$$x_{45} = 48.7049516666752$$
$$x_{45} = -31.4159265358979$$
$$x_{45} = -21.9911485751286$$
$$x_{45} = 278.032748190065$$
$$x_{45} = 92.682377997352$$
$$x_{45} = 80.1168534696549$$
$$x_{45} = -53.4070751110265$$
$$x_{45} = 95.8237937978449$$
$$x_{45} = -6.28318530717959$$
$$x_{45} = 64.410411962776$$
$$x_{45} = -43.9822971502571$$
$$x_{45} = 29.861872403816$$
Decreasing at intervals
$$\left[100.530964914873, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -306.306916073247\right]$$