The position of two particles on the $x$-axis are $x_1 = \sin t$ and $x_2 = \sin \left(t + \frac{\pi}{3}\right)$ 
The position of two particles on the $x$-axis are $x_1 = \sin t$ and $x_2 = \sin \left( t + \frac{\pi}{3}\right)$
(a) At what time(s) in the interval $[0,2\pi]$ do the particles meet?
(b) What is the farthest apart that the particles every get?
(c) When in the interval $[0,2\pi]$ is the distance between the particles changing the fastest?

I am studying for the AP Calculus BC exam and this is a problem out of the Calculus Problem book. I know the answers: (a) $\frac{\pi}{3}, \, \frac{4\pi}{3}$ (b) 1 (c) $\frac{\pi}{3}, \, \frac{4\pi}{3}$
I am have trouble with (a)
I rewrote
$$\sin\left(t + \frac{\pi}{3}\right) \Rightarrow \frac{\sqrt{3}}{2}\sin t + \frac{1}{2}\cos t$$
I get this which I can't seem to solve for the answer given:
$$\left(2 - \sqrt{3}\right)\sin t = \cos t $$
For part (b), I would maximized the distance formula. $d(x)=\sqrt{x_1^2 + x_2^2}$
Set the 1st derivative to zero and use the 2nd derivative test to find the max.
for part (c) I would find the maximum of $d^{\prime}(x)$

(a)
For coincidence, $\sin(t+\frac{\pi}3)=\sin t$
So, $t+\frac{\pi}3=n\pi+(-1)^nt$ where $n$ is any intgere.
If $n=2m$(even), $t+\frac{\pi}3=2m\pi+t\implies 2m\pi=\frac{\pi}3$ which is impossible.
If  $n=2m+1$(odd), $t+\frac{\pi}3=(2m+1)\pi-t\implies t=(6m+2)\frac{\pi}6=\frac{(3m+1)\pi}3$
Putting $m=0,t=\frac{\pi}3$
Putting $m=1,t=\frac{4\pi}3$
(b) We need to maximize  $\mid\sin(t+\frac{\pi}3)-\sin t\mid$
Now,  $\sin(t+\frac{\pi}3)-\sin t=\sin t(\cos \frac{\pi}3-1 )+\cos t \sin\frac{\pi}3=\sin(\frac{\pi}3-t) $
So, the distance will be maximum if $\sin(\frac{\pi}3-t)$ is minimum/maximum. 
But $-1\le \sin(\frac{\pi}3-t)\le 1$
So, the distance will be maximum $(=1)$
if $\sin(\frac{\pi}3-t)=\pm1\implies \cos(\frac{\pi}3-t)=0\implies \frac{\pi}3-t=(2r+1)\frac{\pi}2,t=\frac{\pi}3-(2r+1)\frac{\pi}2$ where $r$ is any integer.
(c)The change of distance=$$\mid\frac{d\{\sin t-\sin(t+\frac{\pi}3)\}}{dt}\mid=\mid\cos t-\cos(t+\frac{\pi}3)\mid$$
Now, $\cos t-\cos(t+\frac{\pi}3)=\cos t(1-\cos\frac{\pi}3)-\sin t\sin \frac{\pi}3=\cos(t+\frac{\pi}3) $
For the distance between the particles changing the fastest,
$\cos(t+\frac{\pi}3)=\pm 1$ as  $-1\le \cos(\frac{\pi}3-t)\le 1$
$\implies \sin(t+\frac{\pi}3)=0$
$\implies t+\frac{\pi}3=s\pi$ where $s$ is any integer. 
So, $t=s\pi-\frac{\pi}3$
Putting $s=0,t=-\frac{\pi}3$ which is not  $[0,2\pi]$ 
Putting $s=1,t=\frac{2\pi}3$ 
Putting $s=2,t=\frac{5\pi}3$ 
Putting $s=3,t=\frac{8\pi}3>2\pi$ so is not acceptable.
