(a) 在阿尔冈图上，用阴影表示由满足不等式 $-\frac13\pi\le\arg(z-1-2\mathrm{i})\le\frac13\pi$ 和 $\operatorname{Re}z\le3$ 的复数 $z$ 所对应的点组成的区域。

(b) 求 (a) 问区域内各点所对应的 $\arg z$ 的最小值。答案用弧度表示，保留 $3$ 位小数。

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。

辐角是以 $(1,2)$ 为起点量度的，画出向右上、右下各 $60^\circ$ 的射线。与直线 $x=3$ 左侧取交集，得到三角形区域，但辐角起点本身无定义。

Write $z=x+y\mathrm{i}$. Mark $P=(1,2)$.

The argument condition describes the wedge between the rays

Both rays are restricted to $x\ge1$.

Draw the vertical line $x=3$. Shade the triangular region inside the wedge and to the left of this line. Its vertices are

Include the bounding rays and the vertical segment, but exclude $P=(1,2)$ because $\arg0$ is undefined there.

相对于原点，最小辐角出现在三角形最右下方的顶点。可用下边界上的 $y/x$ 随 $x$ 递减来说明。

For each fixed $x$ in the region, the least argument is on the lower edge

Since $x>0$, minimising the argument is equivalent to minimising $y/x$. Along this edge,

which decreases as $x$ increases. Thus the least argument is at $(3,2-2\sqrt3)$.

Therefore

to 3 decimal places.