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  • The positive Integral value of n >3 and satisfying the equation : 1/sin (pi/n) =1/sin (2pi/n) + 1/sin (3pi /n ) , is

The positive Integral value of n >3 and satisfying the equation : 1/sin (pi/n) =1/sin (2pi/n) + 1/sin (3pi /n ) , is

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Solution:    Put    fracpin= heta .

               herefore           frac1sin 2 heta =frac1sin heta -frac1sin 3 heta

              Rightarrow            frac12sin heta cdot cos heta =fracsin 3 heta -sin heta sin heta cdot sin 3 heta

              Rightarrow            frac12cos heta =frac2cos 2 heta cdot sin heta sin 3 heta

              Rightarrow           sin 3 heta =sin 4 heta =sin (pi -4 heta )

              Rightarrow           7 heta =piRightarrow heta =fracpi7    

                herefore         fracpin=fracpi7Rightarrow n=7.

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Deependra Verma

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