<s>[INST] The sum of the solutions of $\sin\left(2x\right)=\frac{\sqrt{3}}{2}$ over the interval$ [–π, d]$ is$–π.$ find the value of $d$ The sum of the solutions of $\sin\left(2x\right)=\frac{\sqrt{3}}{2}$ over the interval$  [–π, d]$ is$–π.$ find the value of $d$
I think the first thing I must know is the number of solutions. I know that $x=\:\frac{\pi }{6},\frac{\pi }{3}$, but how can I deduce how many more? How can I utilize $-\pi$ is the domain? What is the general approach to these kinds of questions?
Multiple choice options:
The options are
$a) o$,
$b)\frac{\pi }{6}$,
$c)\frac{3\pi }{4}$,
$d)\frac{7\pi }{6}$,
$e) \frac{3\pi }{2}$
 [/INST] The solutions in
$$[-\pi,\infty)$$ are
$$-\frac{5\pi}6,-\frac{2\pi}3,\frac\pi6,\frac\pi3,\frac{7\pi}6,\frac{4\pi}3,\cdots$$
and if you accumulate them you obtain
$$-\frac{5\pi}6,-\frac{3\pi}2,-\frac{4\pi}3,-\pi,\frac\pi6,\frac{3\pi}2\cdots$$
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