先将方程 $\tan(x-60^\circ)=2\cot x$ 表示为关于 $\tan x$ 的一元二次方程，再解此方程，其中 $0^\circ\le x\le180^\circ$。

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。

按题意先化成正切的二次方程。解出两个正切值后，要在第二象限补回负正切值所对应的角。

The original equation excludes $x=0^\circ,180^\circ$ and $x=150^\circ$.

Using the tangent subtraction formula and writing $T=\tan x$,

The value $x=90^\circ$, where $T$ is undefined, is not a solution of the original equation. For the remaining allowed angles, multiplying through gives

Hence

Solving,

The positive root gives $x=79.8^\circ$. The negative root gives a second-quadrant angle $x=160.2^\circ$. Both are defined in the original equation.

Thus