If A,B,C are the angles of a triangle, the system of equations, $$sinAx+y+z=cosA$$,$$x+sinBy+z=cosB$$,$$x+y+sinCz=1$$−cosC has

Question

If A,B,C are the angles of a triangle, the system of equations,  $$sinAx+y+z=cosA$$,$$x+sinBy+z=cosB$$,$$x+y+sinCz=1$$−cosC has

Infinity Learn · Accepted Answer

We haveΔ$$=sinA111sinB111sinC$$          $$C2$$→$$C2-C3$$        $$=sinA$$             $$0$$                        $$1$$−sinA   $$1$$                  sinB−$$1$$             $$0$$    $$1$$                $$1$$−sinC         sinC−$$1$$        $$=sinAsinB$$−$$1sinC$$−$$1$$−$$0+1$$−$$sinA1$$−sinc−$$sinB+1$$        $$=sinAsinBsinC-sinAsinB-sinAsinC+sinA$$       $$+2+-sinB-sinC-2sinA+sinAsinB+sinAsinC$$ $$=2+sinAsinBsinC-sinA-sinB-sinC$$ >$$0i$$.e., Δ>$$0$$⇒The system has unique solution.

If A,B,C are the angles of a triangle, the system of equations, $(\sin A) x + y + z = \cos A$ , $x + (\sin B) y + z = \cos B$ , $x + y + (\sin C) z = 1 - \cos C$ has

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If A,B,C are the angles of a triangle, the system of equations, sinAx+y+z=cosA,x+sinBy+z=cosB,x+y+sinCz=1−cosC has

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If A,B,C are the angles of a triangle, the system of equations, $(\sin A) x + y + z = \cos A$ , $x + (\sin B) y + z = \cos B$ , $x + y + (\sin C) z = 1 - \cos C$ has