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CBSE Class 12-science Answered

In a triangle, if 
 
 
       1                   1                    1
 
   1+sin A          1+sin B            1+sinC        =0
 
  sinA+Sin2A    sinB+Sin2B     SinC+Sin2C
 
 
 
then using the property of determinant prove that the triangle is  an   isosceles triangle
Asked by ppratim02 | 05 Jul, 2016, 09:30: PM
answered-by-expert Expert Answer
begin mathsize 12px style open vertical bar table row 1 1 1 row cell 1 plus sinA end cell cell 1 plus sinB end cell cell 1 plus sinC end cell row cell sinA plus sin squared straight A end cell cell space space space space space space sinB plus sin squared straight B end cell cell space space space space space space space sinC plus sin squared straight C end cell end table close vertical bar equals 0
straight C subscript 2 rightwards arrow straight C subscript 2 minus straight C subscript 1
rightwards double arrow open vertical bar table row 1 0 1 row cell 1 plus sinA end cell cell sinB minus sinA end cell cell 1 plus sinC end cell row cell sinA plus sin squared straight A end cell cell space space space space space space sinB plus sin squared straight B minus sinA minus sin squared straight A end cell cell space space space space space space space sinC plus sin squared straight C end cell end table close vertical bar equals 0
straight C subscript 3 rightwards arrow straight C subscript 3 minus straight C subscript 1
rightwards double arrow open vertical bar table row 1 0 0 row cell 1 plus sinA end cell cell sinB minus sinA end cell cell sinC minus sinA end cell row cell sinA plus sin squared straight A end cell cell space space space space space space sinB plus sin squared straight B minus sinA minus sin squared straight A end cell cell space space space space space space space sinC plus sin squared straight C minus sinA minus sin squared straight A end cell end table close vertical bar equals 0
rightwards double arrow open vertical bar table row 1 0 0 row cell 1 plus sinA end cell cell sinB minus sinA end cell cell sinC minus sinA end cell row cell sinA plus sin squared straight A end cell cell space space space space space space left parenthesis sinB plus sinA right parenthesis left parenthesis sinB minus sinA right parenthesis plus left parenthesis sinB minus sinA right parenthesis end cell cell space space space space space space space left parenthesis sinC plus sinA right parenthesis left parenthesis sinC minus sinA right parenthesis plus left parenthesis sinC minus sinA right parenthesis end cell end table close vertical bar equals 0
rightwards double arrow left parenthesis sinB minus sinA right parenthesis left parenthesis sinC minus sinA right parenthesis open vertical bar table row 1 0 0 row cell 1 plus sinA end cell 1 1 row cell sinA plus sin squared straight A end cell cell space space space space space space left parenthesis sinB plus sinA right parenthesis plus 1 end cell cell space space space space space space space left parenthesis sinC plus sinA right parenthesis plus 1 end cell end table close vertical bar equals 0
rightwards double arrow left parenthesis sinB minus sinA right parenthesis left parenthesis sinC minus sinA right parenthesis left square bracket space left parenthesis sinC plus sinA right parenthesis plus 1 minus space left parenthesis sinB plus sinA right parenthesis minus 1 right square bracket equals 0
rightwards double arrow left parenthesis sinB minus sinA right parenthesis left parenthesis sinC minus sinA right parenthesis left square bracket sinC plus sinA right parenthesis minus space sinB minus sinA right square bracket equals 0
rightwards double arrow left parenthesis sinB minus sinA right parenthesis left parenthesis sinC minus sinA right parenthesis left parenthesis sinC minus space sinB right parenthesis equals 0
rightwards double arrow sinB minus sinA equals 0 space space or space space sinC minus sinA equals 0 space space or space space sinC minus space sinB equals 0
rightwards double arrow sinB equals sinA space space space or space space sinC equals sinA space space space or space space sinC equals space sinB
rightwards double arrow straight B equals straight A space or space straight C equals straight A space or space straight C equals straight B
Hence comma space increment ABC space is space an space isosceles space triangle.
end style
Answered by Rebecca Fernandes | 07 Jul, 2016, 10:29: AM
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ANSWERED BY EXPERT ANSWERED BY EXPERT
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open vertical bar table row cell left parenthesis straight b plus straight c right parenthesis hat 2 end cell ab ca row ab cell left parenthesis straight a plus straight c right parenthesis hat 2 end cell bc row ac bc cell left parenthesis straight a plus straight b right parenthesis hat 2 end cell end table close vertical bar equals 2 abc left parenthesis straight a plus straight b plus straight c right parenthesis hat 3
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