The new period is 2/3 π.
The period of the two elementary trig functions,
sin(x) and cos(x) is 2π.
If we multiply the input variable by a constant has the effect of stretching or contracting the period. If the constant, c>1 then the period is stretched, if c<1 then the period is contracted.
We can see what change has been made to the period, T, by solving the equation:
cT=2π
What we are doing here is checking what new number, T, will effectively input the old period, 2π, to the function in light of the constant. So for our givens:
3T=2π
T=2/3 π
Other method to solve this;
sin3x=sin(3x+2π)=sin[3(x+2π/3)]=sin3x
This means "after the arc rotating three time of (x+(2π/3)), sin 3x comes back to its initial value"
So, the period of sin 3x is 2π/3 or 2/3 π.
