Question

find the general solution of √3.cosx-sinx=1

Answer 1

√3  Cos x  - Sin x = 1

divide the equation by 2:

=> √3/2  Cos x - 1/2  sin x = 1/2
=> Sin π/3 Cos x - Cos π/3 Sin x = Sin π/6
=> Sin (π/3 - x) = Sin π/6
=> Sin (x - π/3) = Sin (-π/6)  or Sin (π + π/6)

=>    x - π/3 =    2 n π - π/6  or  2 n π + 7 π/6
       x = 2 n π + π/6    or  2 n π + 3 π / 2

   this 2 n π + 3 π/2 can also be expressed as:  2 n π + 2π - π/2
             ie.,  2 m π - π / 2

Hence, x = 2 n π + π/6    and x =  2 n π - π/2

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another way:

√3 Cos x - Sin x = 1
=> √3/2 Cos x  - 1/2 Sin x = 1/2
=>  Cos π/6 Cos x - Sin π/6 Sin x = Cos π/3
=>  Cos (x + π/6)  = Cos π/3

     we have general solution for Cos X = Cos Y,  X = 2 n π + Y  or  2 n π  - Y
 
=>  x + π/6  =  2 n π  + π/3  or  2 n π - π/3
=>  x = 2 n π + π/6    or    2 n π - π/2

Answer 2

I hope its help you