The new period is <span>
2/3 π</span>
.
The period of the two elementary trig functions, <span>sin<span>(x)</span></span><span> and </span><span>cos<span>(x)</span></span><span> is </span><span>2π</span><span>.
</span>
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:
<span>cT=2π</span>
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:
<span>3T=2π</span>
<span>T=2/3 π</span>
Other method to solve this;
<span><span>sin3</span>x=<span>sin<span>(3x+2π)</span></span>=<span>sin<span>[3<span>(x+<span><span>2π/</span>3</span>)</span>]</span></span>=<span>sin3</span>x</span>
This means "after the arc rotating three time of <span>(x+<span>(2<span>π/3</span>)</span>)</span>, sin 3x comes back to its initial value"
So, the period of sin 3x is <span><span>2π/</span>3 or 2/3 </span>π.