$Evaluate space the space determinant increment space space equals space space open vertical bar table row 1 cell space space space space sin space straight theta end cell 1 row cell negative sin end cell cell space 1 end cell cell space space sin space straight theta end cell row cell negative 1 end cell cell negative sin end cell 1 end table close vertical bar space comma space Also comma space prove space that space 2 space less than space straight A space less than space 4.$

increment space space equals space space open vertical bar table row 1 cell space space space space sin space straight theta end cell 1 row cell negative sin end cell cell space 1 end cell cell space space sin space straight theta end cell row cell negative 1 end cell cell negative sin end cell 1 end table close vertical bar space space space space space space space equals space 1 space cross times space open vertical bar table row 1 cell space sin end cell row cell negative sin space straight theta end cell 1 end table close vertical bar space minus sin space straight theta space space space open vertical bar table row cell negative sin space straight theta end cell cell space sin space straight theta end cell row cell negative 1 end cell 1 end table close vertical bar space plus 1 space cross times space open vertical bar table row cell negative sin space straight theta end cell cell space space 1 end cell row cell negative 1 end cell cell space space space minus sin space straight theta end cell end table close vertical bar space space space space space space equals space 1 space cross times space left parenthesis 1 plus sin squared space straight theta right parenthesis minus sin space straight theta space left parenthesis negative sin space straight theta plus sin space straight theta vertical line right parenthesis plus 1 space left parenthesis sin squared space straight theta space plus 1 right parenthesis space space space space space space equals space 2 plus 2 space sin squared space straight theta space equals 2 space left parenthesis 1 plus sin space squared space straight theta right parenthesis we space know space chat space space space space space space space space space minus 1 space less or equal than space sin space straight theta space less or equal than space 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta therefore space space space space space space space space space 0 space less or equal than space sin squared space straight theta space less or equal than space 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta rightwards double arrow space space space space space space space 1 plus 0 space less or equal than space 1 space plus sin space squared space straight theta space space space less or equal than 1 plus 1 space space space space space space space space space space space for space all space straight theta rightwards double arrow space space space space space space space 1 space space less or equal than space 1 space space sin squared space straight theta space less or equal than space 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta rightwards double arrow space space space space space space space 2 space space space less or equal than space 2 space left parenthesis 1 plus sin squared space straight theta right parenthesis space space less or equal than space 4 space space space space space space space space space space space space space space space space space space for space all space straight theta rightwards double arrow space space space space space space space 2 space space space less or equal than space increment space less or equal than space space 4 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space for space all space straight theta

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Determinants

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