(a) 在阿尔冈图的草图上，用阴影表示由满足不等式 $|z+2|\le2$ 和 $\operatorname{Im}z\ge1$ 的复数 $z$ 所对应的点组成的区域。

(b) 求阴影区域内各点所对应的 $\arg z$ 的最大值。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

第一个条件是圆心 $(-2,0)$、半径 $2$ 的闭圆盘，第二个条件要求在水平线 $y=1$ 上方，取交集。

Write $z=x+y\mathrm{i}$. The inequalities become

Draw the circle with centre $(-2,0)$ and radius $2$, and the line $y=1$. Shade the upper cap of the disc, including the circle arc and the horizontal chord.

The chord endpoints are $(-2-\sqrt3,1)$ and $(-2+\sqrt3,1)$.

区域完全在第二象限，最大辐角在最靠近负实轴的方向。沿每一水平线先取最左点，再比较高度，得到弦的左端点。

For $1\le y\le2$, the leftmost point in the disc has

At a fixed height, this point has the largest argument. Its reference-angle tangent is

As $y$ increases from $1$ to $2$, the numerator increases and the positive denominator decreases, so this ratio increases. Thus the greatest argument occurs at the smallest height, $y=1$, at $(-2-\sqrt3,1)$.

Hence

Since $\tan(\pi/12)=2-\sqrt3$,