(a) 通过画出合适的两个函数图像的草图，证明方程 $\operatorname{cosec}\frac12x=\frac12(x+3)$ 在区间 $0\le x\le\pi$ 内恰有一个根。

(b) 通过计算证明这个根位于 $1$ 与 $1.2$ 之间。

(c) 证明：若由迭代公式

给出的实数数列收敛，则它收敛于 8(a) 中方程的根。

(d) 使用 8(c) 中的迭代公式计算这个根，答案精确到小数点后 $3$ 位。每次迭代的结果保留小数点后 $5$ 位。

Answer all questions. Write your answers on the answer booklet. Follow the instructions on the front cover of the answer booklet. If you need additional answer paper, ask the invigilator for a continuation booklet.

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。在答题册上作答，并遵守答题册封面的指示。如需额外答题纸，请向监考人员索取续答册。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。

在区间内，余割函数从正无穷递减至 $1$，而直线递增。因此图像恰有一个交点。

Sketch $y=\operatorname{cosec}(x/2)$ for $0<x\le\pi$, with vertical asymptote $x=0$ and endpoint $(\pi,1)$. Sketch also $y=(x+3)/2$, starting at $(0,3/2)$.

The cosecant curve decreases continuously from $+\infty$ to $1$, while the line increases from $3/2$ to $(\pi+3)/2>1$. Therefore the graphs cross exactly once in the interval, giving exactly one root. The endpoint $x=0$ is excluded by the cosecant function’s domain.

计算余割函数与直线的高度差。区间内连续且端点符号不同，所以根在两端点之间。

Let $f(x)=\operatorname{cosec}(x/2)-(x+3)/2$. With angles in radians,

Since $f$ is continuous on $[1,1.2]$, the root lies between $1$ and $1.2$.

若迭代收敛，前后两项趋于同一极限。对固定点方程取正弦并整理，即得到原方程。

If the sequence converges to $L$, the recurrence and continuity give

For any sequence defined through all its iterations, the outputs lie in $[-\pi,\pi]$. A subsequent input in this interval must satisfy $x_n\ge-1$ for the inverse sine to be real. The next output is then positive and at most $\pi$. Hence the limit lies in $[0,\pi]$. Taking sine in the fixed-point equation gives

The fixed-point equation excludes $L=0$. Thus $L$ is the unique root from part (a).

从 $x_0=1$ 开始按公式迭代。保留内部精度，并将各项写成五位小数，直到三位小数稳定。

Using radians and $x_0=1$, the iterations are

The root is $1.037$ to three decimal places.