Maximize the Integral

Similar solution with @Guilherme Niedu 's, slightly shorter.

$\begin{aligned} I & = \int_{-\frac \pi 3}^{\frac {2\pi}3} \sin (x+a) \ dx \\ & = - \cos (x+a) \bigg|_{-\frac \pi 3}^{\frac {2\pi}3} \\ & = {\color{#3D99F6}- \cos \left( -\frac {2\pi}3 + a\right)} +{\color{#D61F06} \cos \left(-\frac \pi 3 + a\right)} & \small \color{#3D99F6} \text{Note that }\cos(\pi -x) = - \cos x \\ & = {\color{#3D99F6} \cos \left(\frac \pi3 - a\right)} + {\color{#D61F06} \cos \left(\frac \pi3 - a\right)} & \small \color{#D61F06} \text{and that }\cos(-x) = \cos x \\ & = 2 \cos \left(\frac \pi3 - a\right) \end{aligned}$

$I$ is maximum, when $\cos \left(\frac \pi3 - a\right) = 1$ , its maximum, or $a = \frac \pi 3$ , the minimum value. $\implies \lfloor 1000a \rfloor = \boxed{1047}$ .

$\large \displaystyle I = \int_{-\pi/3}^{2\pi/3} \sin(x+a)dx$

$\large \displaystyle I = \cos(x+a) \Bigg |_{2\pi/3}^{-\pi/3}$

$\large \displaystyle I = [ \cos(a) \cos(\pi/3) + \sin(a) \sin(\pi/3) ] - [- \cos(a) \cos(2 \pi/3) - \sin(a) \sin(2 \pi/3)]$

Since $\cos(x) = -\cos(\pi - x)$ and $\sin(x) = -\sin(\pi - x)$ :

$\large \displaystyle I = [ \cos(a) \cos(\pi/3) + \sin(a) \sin(\pi/3) ] + [\cos(a) \cos(\pi/3) + \sin(a) \sin(\pi/3)]$

$\large \displaystyle I = 2[ \cos(a) \cos(\pi/3) + \sin(a) \sin(\pi/3) ]$

$\color{#20A900} \boxed{\large \displaystyle I = 2 \cos(a - \pi/3) }$

For $I$ to be maximum:

$\large \displaystyle \cos(a - \pi/3) = 1$

$\color{#20A900} \boxed{ \large \displaystyle a = 2k\pi + \pi/3, a \in \mathbf{Z} }$ .

For $a$ to be minimum, since $a$ is positive, $k=0$ . So:

$\color{#3D99F6} a = \pi/3, \boxed{\large \displaystyle \lfloor 1000a \rfloor = 1047}$