PART I · PAPER 1 SECTION A第一部分 · 第一卷 A 节No calculator · short response · 21 marks不可使用计算器 · 简答题 · 21 分
Section A · Short ResponseA 节 · 简答题
Give exact values throughout. Quote the first-quadrant reference angle, then attach the CAST sign for the actual quadrant. When solving equations on a stated interval, list every solution explicitly; do not leave a "$+\,2k\pi$" tail. No calculator permitted.全程给出精确值。先写第一象限参考角，再按 CAST 给所在象限定符号。在给定区间上求解方程时，逐一列出每个解，不要遗留 "$+\,2k\pi$" 通项。不可使用计算器。
Solve $2 \sin x = \sqrt{3}$ for $x \in [0, 2\pi]$.在 $x \in [0, 2\pi]$ 上解 $2 \sin x = \sqrt{3}$。
(a)Isolate $\sin x$ and state the exact target value.分离 $\sin x$，写出精确目标值。[1]
(b)State the principal solution $\alpha = \arcsin(\sqrt{3}/2)$ and the second solution on $[0, 2\pi]$ (use the symmetry of the sine wave).写出主值 $\alpha = \arcsin(\sqrt{3}/2)$，并由正弦波对称性给出 $[0, 2\pi]$ 上的第二解。[3]
(c)Confirm that no further solutions lie in $[0, 2\pi]$ by considering the period of $\sin$.利用 $\sin$ 的周期，验证 $[0, 2\pi]$ 上无其他解。[1]
stating the values of $x$ for which both sides are defined.并写出使两边均有定义的 $x$ 值。
(a)Rewrite the numerator using a double-angle form of $\cos 2x$ that leaves only $\sin^{2} x$.用只含 $\sin^{2} x$ 的 $\cos 2x$ 倍角形式改写分子。[1]
(b)Rewrite the denominator using the double-angle form of $\sin 2x$.用 $\sin 2x$ 的倍角形式改写分母。[1]
(c)Simplify the resulting ratio to $\tan x$, justifying each cancellation.化简所得比值为 $\tan x$，每一次约分都说明依据。[3]
(d)State the excluded values of $x$ (where either side is undefined).写出 $x$ 的排除值（任一边无定义之处）。[1]
PART II · PAPER 1 SECTION B第二部分 · 第一卷 B 节No calculator · extended response · 11 marks不可使用计算器 · 长答题 · 11 分
Section B · Extended ResponseB 节 · 长答题
Before reading off a phase shift, factor the coefficient of $x$ out of the argument. When sketching a transformed sine wave, mark the midline, the maximum, and the minimum explicitly. Exact solutions only; decimal answers will not score.读相移之前，先把 $x$ 的系数从括号中提出。绘制变换后的正弦波时，要明确标出中线、最大值与最小值。仅给精确解，小数答案不得分。
(a)State the amplitude, period, phase shift, and vertical shift.写出振幅、周期、相移与纵移。[4]
(b)Hence state the range of $y$ over $x \in \mathbb{R}$.由此写出 $y$ 在 $x \in \mathbb{R}$ 上的值域。[2]
(c)Solve $y = \dfrac{5}{2}$ for $x \in [0, \pi]$, giving exact answers. (Hint: substitute $u = 2(x - \pi/4)$ and find every $u$ in the resulting interval that gives $\sin u = 1/2$.)在 $x \in [0, \pi]$ 上解 $y = \dfrac{5}{2}$，给出精确答案。（提示：代换 $u = 2(x - \pi/4)$，并求出所得区间内所有满足 $\sin u = 1/2$ 的 $u$。）[5]
PART III · PAPER 2第三部分 · 第二卷Calculator · mixed response · 16 marks可使用计算器 · 混合题型 · 16 分
Paper 2 · Calculator Permitted第二卷 · 允许使用计算器
A graphing calculator is required. Quote any numerical answer to 3 significant figures unless an exact form is requested. For modelling problems, give units and state the domain on which the model is being used.需要图形计算器（GDC）。除非题目要求精确形式，否则数值答案保留 3 位有效数字。建模题须给出单位，并说明模型适用的定义域。
Consider the function $y \;=\; -4 \cos\!\bigl(\pi x / 3\bigr)$ for $x \in \mathbb{R}$.考虑函数 $y \;=\; -4 \cos\!\bigl(\pi x / 3\bigr)$，$x \in \mathbb{R}$。
(a)State the amplitude.写出振幅。[1]
(b)Find the period.求周期。[2]
(c)State the maximum and minimum values, and give the smallest non-negative $x$ at which each occurs.写出最大值与最小值，并给出各自对应的最小非负 $x$。[4]
Q7HARDPaper 23.7 Modelling: Tide Height[9 marks]
The height of the tide in a harbour, in metres above the chart datum, is modelled by某港口潮位（单位：米，相对海图基准）的模型为
$$ H(t) \;=\; 5 + 3 \sin\!\bigl(\pi t / 6\bigr), \qquad 0 \le t \le 12, $$
where $t$ is the number of hours after midnight.其中 $t$ 为午夜后的小时数。
(a)State the mean tide height predicted by the model.写出模型预测的平均潮位。[1]
(b)Find the time $t \in [0, 12]$ at which the tide first reaches its maximum, and state that maximum height.求 $t \in [0, 12]$ 内潮位首次达到最大的时刻，并写出该最大高度。[3]
(c)Find, to 3 significant figures, the set of times $t \in [0, 12]$ for which $H(t) > 6$ metres. Express the answer as an interval.求 $t \in [0, 12]$ 内使 $H(t) > 6$ 米的时段，写成区间，结果保留 3 位有效数字。[5]
PART IV · PAPER 3第四部分 · 第三卷Calculator · HL extended exploration · 15 marks可使用计算器 · HL 长题探究 · 15 分
Paper 3 · HL Extended Problem第三卷 · HL 长题探究
A graphing calculator is required. Method marks dominate. State the compound-angle formula by name when you first invoke it, and keep $\sin x$, $\cos x$ as separate symbols throughout the algebra; only collect at the end.需要图形计算器。方法分占主导。首次引用复角公式时按名说明，全程把 $\sin x$、$\cos x$ 当作两个独立符号，最后再合并。
Throughout this question, assume $x \in \mathbb{R}$ is in any domain where the listed functions are defined.本题假设 $x \in \mathbb{R}$ 落在所列函数有定义的任意定义域内。
(a)Using $\sin 3x = \sin(2x + x)$, the compound-angle formula, the double-angle identities $\sin 2x = 2 \sin x \cos x$ and $\cos 2x = 1 - 2 \sin^{2} x$, and the Pythagorean identity, prove that利用 $\sin 3x = \sin(2x + x)$、复角公式、倍角恒等式 $\sin 2x = 2 \sin x \cos x$ 与 $\cos 2x = 1 - 2 \sin^{2} x$，以及毕氏恒等式，证明
$$ \sin 3x \;=\; 3 \sin x - 4 \sin^{3} x. $$
[4]
(b)By a similar method, prove that用类似方法证明
$$ \cos 3x \;=\; 4 \cos^{3} x - 3 \cos x. $$
[4]
(c)Hence find the exact value of $\sin(3\pi/10) - \sin(\pi/10)$ by setting $x = \pi/10$ in part (a). (You may use, without proof, that $\sin(3\pi/10) = \cos(\pi/5)$ and the identity $4 \cos(\pi/5) \sin(\pi/10) = 1$, which follows from $\sin(2\pi/5) = 2 \sin(\pi/5) \cos(\pi/5)$ and the double-angle / supplementary-angle relations.)由此，把 $x = \pi/10$ 代入 (a)，求 $\sin(3\pi/10) - \sin(\pi/10)$ 的精确值。（可不证地使用 $\sin(3\pi/10) = \cos(\pi/5)$，以及由 $\sin(2\pi/5) = 2 \sin(\pi/5) \cos(\pi/5)$ 与倍角/补角关系导出的恒等式 $4 \cos(\pi/5) \sin(\pi/10) = 1$。）[4]
(d)Starting from the Pythagorean identity $\sin^{2} x + \cos^{2} x = 1$, divide through twice to derive the two reciprocal-form identities $1 + \tan^{2} x = \sec^{2} x$ and $1 + \cot^{2} x = \csc^{2} x$. State the excluded $x$-values for each.由毕氏恒等式 $\sin^{2} x + \cos^{2} x = 1$ 出发，两次除项推导两个倒数形式恒等式 $1 + \tan^{2} x = \sec^{2} x$ 与 $1 + \cot^{2} x = \csc^{2} x$，并写出各自的 $x$ 排除值。[3]