(a) 曲线 $y=\sin x$ 经过变换得到曲线 $y=4\sin\big(\frac12x-30^\circ\big)$。

完整描述一组组合起来实现这一变换的变换，并明确说明这些变换的实施顺序。

(b) 求方程 $4\sin\big(\frac12x-30^\circ\big)=2\sqrt2$ 在 $0^\circ\le x\le360^\circ$ 范围内的精确解。

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。

先将角度写成 $(x-60^\circ)/2$。这个形式直接给出先横向拉伸、再右移的一组顺序。

Stretch the graph parallel to the $x$-axis by factor $2$, giving $y=\sin(x/2)$. Translate it by $60$ units in the positive $x$-direction, giving $y=\sin((x-60^\circ)/2)$. Then stretch parallel to the $y$-axis by factor $4$ to obtain the required graph.

也可以先将正弦图像右移 $30$。随后拉伸时，括号中的整个输入由 $x$ 换成 $x/2$。

Translate by $30$ units in the positive $x$-direction to obtain $y=\sin(x-30^\circ)$. Stretch parallel to the $x$-axis by factor $2$ to obtain $y=\sin(x/2-30^\circ)$. Finally stretch parallel to the $y$-axis by factor $4$.

先把复合角设为一个变量并换算它的范围。找到两个角后再解回 $x$。

Let $u=x/2-30^\circ$. Then $-30^\circ\le u\le150^\circ$ and $\sin u=\sqrt2/2$. The two possible values are $45^\circ$ and $135^\circ$. Solving $x=2(u+30^\circ)$ gives $x=150^\circ$ or $330^\circ$.