Answer
$$t\approx-2.73\times10^{-4}$$
Work Step by Step
The question asks to find the values of $t$ for which $P=0$ over $[0,0.005]$.
So we would use the formula of $P$ where $x=t$, which is $$P(t)=0.004\sin[2\pi(261.63)t+\frac{\pi}{7}]$$
where $P=0$, which means $$0.004\sin[2\pi(261.63)t+\frac{\pi}{7}]=0$$
The interval is $[0,0.005]$
Now we solve the equation:
$$0.004\sin[2\pi(261.63)t+\frac{\pi}{7}]=0$$
$$\sin[2\pi(261.63)t+\frac{\pi}{7}]=0\hspace{1cm}(1)$$
Over the interval $[0,0.005]$, there is one value which has sine equaling $0$, which is $0$ ($\sin0=0$)
Apply it to $(1)$:
$$2\pi(261.63)t+\frac{\pi}{7}=0$$
$$2\pi(261.63)t=-\frac{\pi}{7}$$
$$261.63t=-\frac{\pi}{7\times2\pi}=-\frac{1}{14}$$
$$t=-\frac{1}{14\times261.63}$$
$$t\approx-2.73\times10^{-4}$$
That is the value of $t$ needed to find.
