How to determine period of sinx

[AAQIB IQBAL]

How to determine period of sin and cos functions?
I use:
sin(x + p) = sin(x)
=> sin(x + p) - sin(x) = 0
=> 2*cos(x+p/2)*sin(p/2) = 0
=> either p/2 = k(pi) => p = 2k(pi)
or x + p/2 = (2k+1)(pi)/2 => p = (2k+1)(pi) - 2x
NOW I DON'T KNOW HOW TO FIND THE SOLUTION WHICH SATISFIES BOTH.. THOUGH I KNOW THAT PERIOD IS 2(PI) BUT I HAVE TO PROVE IT WITHOUT GRAPH.

 

[tiny-tim]

Hi AAQIB IQBAL!

sin(x + p) = sin(x)
=> sin(x + p) - sin(x) = 0
=> 2*cos(x+p/2)*sin(p/2) = 0

Fine until here.

Now that must be true for any x (for the same p) …

which can only happen if sin(p/2) = 0.

(in other words: you can ignore the other part)