PART I · SHORT RESPONSE第一部分 · 短答题SAT-style MCQ + ON/BC/AB short answer · 18 marksSAT 风格选择题 + 安/卑/阿省考短答 · 共 18 分
Section A · Short ResponseA 部分 · 短答题
Mix of multiple-choice and short-answer items focused on the unit-circle foundations. For MCQs, circle the letter and leave enough margin work that a marker could verify. State exact values in simplified radical form (e.g. $\tfrac{\sqrt{3}}{2}$, not the decimal $0.866\ldots$). No calculator on Q1–Q5.本节为选择题与短答题混合，聚焦单位圆基础。选择题请圈出字母答案，并在空白处写下足以让阅卷人核对的过程。精确值请以最简根式形式作答（例如 $\tfrac{\sqrt{3}}{2}$，而非小数 $0.866\ldots$）。Q1–Q5 不可使用计算器。
The terminal arm of an angle $\theta$ in standard position lies in Quadrant III, and the reference angle is $\dfrac{\pi}{4}$. What is the value of $\cos\theta$?标准位置上的角 $\theta$ 终边位于第三象限，参考角为 $\dfrac{\pi}{4}$。$\cos\theta$ 的值是多少？
Given that $\sin\theta = -\dfrac{3}{5}$ and $\theta$ lies in Quadrant IV, determine the exact values of:已知 $\sin\theta = -\dfrac{3}{5}$ 且 $\theta$ 位于第四象限，求下列各量的精确值：
(a)$\cos\theta$ (use the Pythagorean identity and the quadrant sign).$\cos\theta$（使用毕达哥拉斯恒等式与象限符号判定）。[2]
(b)$\csc\theta$ and $\sec\theta$ in lowest terms.$\csc\theta$ 与 $\sec\theta$，以最简形式作答。[2]
Q5MEDIUM中🇨🇦 BC卑🇨🇦 AB阿AB Math 30-1 style阿省 Math 30-1 风格Honors / Pre-Calc荣誉级 / 微积分预备§1 Coterminal + Arc Length共终边角 + 弧长 · AB Math 30-1 Trig GO 1阿省 Math 30-1 三角 GO 1[5 marks][5 分]
An angle in standard position measures $\theta = \dfrac{11\pi}{3}$ radians.标准位置上的角度数为 $\theta = \dfrac{11\pi}{3}$ 弧度。
(a)Find the coterminal angle $\theta'$ that satisfies $0 \le \theta' < 2\pi$.求满足 $0 \le \theta' < 2\pi$ 的共终边角 $\theta'$。[2]
(b)State the quadrant in which the terminal arm lies, and give the reference angle.写出终边所在的象限，并给出参考角。[2]
(c)A circle has radius $r = 6$ cm. State the arc length subtended by the coterminal angle $\theta'$ from part (a), using the formula $s = r\theta'$.某圆半径为 $r = 6$ cm。利用公式 $s = r\theta'$，写出 (a) 中所得共终边角 $\theta'$ 所截弧长。[1]
PART II · EXTENDED RESPONSE第二部分 · 简答题AP-feeder FRQ + honors · 35 marksAP 衔接简答题 + 荣誉级 · 共 35 分
Section B · Extended ResponseB 部分 · 简答题
Show every step. State each exact value in simplified radical form. For graph questions, label amplitude $A$, period $T$, phase shift $C$, and vertical shift $D$ explicitly before sketching or solving. No calculator on Q6–Q9.每一步都要写出。所有精确值请以最简根式形式作答。图像题在作图或求解前，请先显式标注振幅 $A$、周期 $T$、相位（phase shift）$C$ 与垂直位移 $D$。Q6–Q9 不可使用计算器。
Evaluate each expression using unit-circle exact values. State the reference angle and the quadrant in every case, then read the sign and magnitude from the unit circle. Give exact values in simplified form.使用单位圆精确值求下列各表达式的值。每题都先写出参考角与象限，再从单位圆读取符号与大小。结果以最简形式给出精确值。
(a) $\cos\!\left(\dfrac{7\pi}{6}\right)$ [2]
(b) $\tan\!\left(\dfrac{5\pi}{4}\right)$ [2]
(c) $\sec\!\left(\dfrac{4\pi}{3}\right)$ [2]
(d)$\csc\!\left(-\dfrac{\pi}{3}\right)$ (use the coterminal angle $2\pi - \tfrac{\pi}{3} = \tfrac{5\pi}{3}$, or read $\sin$ as an odd function).$\csc\!\left(-\dfrac{\pi}{3}\right)$（可使用共终边角 $2\pi - \tfrac{\pi}{3} = \tfrac{5\pi}{3}$，或利用 $\sin$ 为奇函数这一性质）。[2]
Consider the function $f(x) = \sin x$ on the closed interval $[0, 2\pi]$.考虑闭区间 $[0, 2\pi]$ 上的函数 $f(x) = \sin x$。
(a)List the five "key points" of one full period at $x = 0, \tfrac{\pi}{2}, \pi, \tfrac{3\pi}{2}, 2\pi$ with their function values $\bigl(x, \sin x\bigr)$.列出一个完整周期内 $x = 0, \tfrac{\pi}{2}, \pi, \tfrac{3\pi}{2}, 2\pi$ 处的五个"关键点"及其函数值 $\bigl(x, \sin x\bigr)$。[2]
(b)State the amplitude, period, and range of $f$.写出 $f$ 的振幅、周期与值域。[3]
(c)Solve $\sin x = \dfrac{1}{2}$ on $[0, 2\pi]$ by reading off the unit circle, and state all solutions exactly.通过读取单位圆，在 $[0, 2\pi]$ 上求解 $\sin x = \dfrac{1}{2}$，写出所有精确解。[2]
(d)State the value of $\sin(-x)$ in terms of $\sin x$, and name the symmetry property this expresses (odd or even).用 $\sin x$ 表示 $\sin(-x)$，并指出这体现了哪种对称性（奇函数或偶函数）。[1]
Q8HARD难🇨🇦 BC卑BC Provincial-style卑诗省考风格Honors / Pre-Calc荣誉级 / 微积分预备§7 Transformations函数变换 · BC PC 12[8 marks][8 分]
(a)Identify $A$, $B$, $C$, and $D$ in the transformation form $y = A\sin\bigl(B(x - C)\bigr) + D$.在变换形式 $y = A\sin\bigl(B(x - C)\bigr) + D$ 中辨识 $A$、$B$、$C$、$D$。[1]
(b)State the amplitude, period, phase shift, and vertical shift, with appropriate signs and direction (e.g. "phase shift $\tfrac{\pi}{4}$ to the right").写出振幅、周期、相位（phase shift）与垂直位移，标明符号与方向（例如"相位向右平移 $\tfrac{\pi}{4}$"）。[3]
(c)State the maximum value, minimum value, and the equation of the midline (sinusoidal axis).写出最大值、最小值与中线（正弦曲线轴）的方程。[3]
(d)State the range of $g$ in interval notation.用区间记号写出 $g$ 的值域。[1]
Q9HARD难Honors荣誉级🇺🇸 US美🇨🇦 BC卑AP-feeder FRQAP 衔接简答题§7 Inverse Modeling反向建模 · HSF-TF.B.5 / BC PC 12 & AB Math 30-1 GO 4.9[11 marks][11 分]
A sinusoidal curve passes through the maximum point $\left(\dfrac{\pi}{6}, 7\right)$ and the next minimum point $\left(\dfrac{7\pi}{6}, -3\right)$. The curve is to be written in the form $y = A\sin\!\bigl(B(x - C)\bigr) + D$ with $A > 0$ and $B > 0$.某正弦曲线（sinusoid）经过最大值点 $\left(\dfrac{\pi}{6}, 7\right)$ 与紧邻的最小值点 $\left(\dfrac{7\pi}{6}, -3\right)$。需将该曲线写成形式 $y = A\sin\!\bigl(B(x - C)\bigr) + D$，其中 $A > 0$ 且 $B > 0$。
(a)Determine the midline equation $y = D$ from the average of the max and min heights, and state $D$.由最大值与最小值高度的平均数确定中线方程 $y = D$，并写出 $D$。[1]
(b)Determine the amplitude $A$ from the half-distance between max and min.由最大值与最小值之间距离的一半确定振幅 $A$。[1]
(c)The distance from a max to the next min is half a period. Use this to determine the period $T$, then the angular frequency $B = \tfrac{2\pi}{T}$.从最大值到下一个最小值的距离为半个周期。据此确定周期 $T$，再求角频率 $B = \tfrac{2\pi}{T}$。[3]
(d)The parent function $\sin$ reaches its max at $x = \tfrac{\pi}{2}$. For the transformed curve, the max occurs where $B(x - C) = \tfrac{\pi}{2}$. Use the given max coordinate to solve for $C$.母函数 $\sin$ 在 $x = \tfrac{\pi}{2}$ 处取最大值。对变换后的曲线，最大值出现在 $B(x - C) = \tfrac{\pi}{2}$ 处。利用给定的最大值横坐标求 $C$。[3]
(e)Write the final equation $y = A\sin\!\bigl(B(x - C)\bigr) + D$ with your values from (a)–(d). Verify by substituting $x = \tfrac{\pi}{6}$ and confirming $y = 7$.代入 (a)–(d) 所得值写出最终方程 $y = A\sin\!\bigl(B(x - C)\bigr) + D$。代入 $x = \tfrac{\pi}{6}$ 验证 $y = 7$。[3]
PART III · MODELING / APPLIED第三部分 · 建模与应用Universal sinusoidal modelling · 28 marks通用正弦型建模 · 共 28 分
Section C · Modeling and ApplicationsC 部分 · 建模与应用
Name your variables (with units) before writing equations. For each model, identify $A$, $B$, $C$, $D$ from the situation, then write the function and answer the contextual question. Calculator permitted throughout Part III, but exact-value answers in radical form are preferred where reasonable.在写方程前，先定义变量名（含单位）。每个模型都需先从情境中辨识 $A$、$B$、$C$、$D$，再写出函数并回答情境问题。第三部分全程可用计算器，但在合理情况下优先以根式形式给出精确值。
A Ferris wheel has a diameter of $40$ m. Its lowest point is $2$ m above the ground (the boarding platform), and it completes one full rotation every $80$ seconds. Riders board at the lowest point at time $t = 0$ and the wheel rotates at a constant rate. Let $h(t)$ be the height of a rider above the ground, in metres, $t$ seconds after boarding.某摩天轮直径为 $40$ 米。最低点距地面 $2$ 米（即登舱平台），每 $80$ 秒转一圈。乘客在 $t = 0$ 时于最低点登舱，转轮以恒定速率旋转。设 $h(t)$ 为登舱 $t$ 秒后乘客距地面的高度（米）。
(a)State the centre height (midline $D$), the amplitude $A$, and the period $T$ of $h(t)$.写出 $h(t)$ 的中心高度（中线 $D$）、振幅 $A$ 与周期 $T$。[3]
(b)Write $h(t)$ in the form $h(t) = -A\cos\!\bigl(B t\bigr) + D$ (the negative cosine form starts at the minimum at $t = 0$, matching the boarding condition). State $B$ exactly.将 $h(t)$ 写成 $h(t) = -A\cos\!\bigl(B t\bigr) + D$ 形式（负余弦形式在 $t = 0$ 时从最低点出发，与登舱条件吻合）。精确给出 $B$。[2]
(c)Find the height of the rider $30$ seconds after boarding. Round to the nearest tenth of a metre.求登舱 $30$ 秒后乘客的高度，四舍五入到 $0.1$ 米。[2]
(d)At what time during the first rotation does the rider first reach a height of $30$ m on the way up? Round to the nearest second.在首圈中，乘客上升过程中首次达到 $30$ 米高度是何时？四舍五入到秒。[2]
A coastal harbour records a high tide of $4.8$ m and a low tide of $0.6$ m. Successive high tides occur $12$ hours apart. The first high tide of the day occurs at $t = 3$ hours (where $t$ is hours after midnight). Let $H(t)$ be the water depth at the harbour at time $t$, in metres.某沿海港口记录到高潮位为 $4.8$ 米、低潮位为 $0.6$ 米。两次相邻高潮相隔 $12$ 小时。当日第一次高潮出现在 $t = 3$ 小时（$t$ 为零时之后的小时数）。设 $H(t)$ 为时刻 $t$ 港口水深（米）。
(a)Compute the midline $D$, the amplitude $A$, and the period $T$ of $H(t)$.计算 $H(t)$ 的中线 $D$、振幅 $A$ 与周期 $T$。[3]
(b)The cosine function naturally peaks at its argument $= 0$. Write $H(t)$ in the form $H(t) = A\cos\!\bigl(B(t - C)\bigr) + D$ using the high-tide time as the phase shift $C$. State $B$ and $C$ explicitly.余弦函数自然在自变量 $= 0$ 时取最大值。将 $H(t)$ 写成 $H(t) = A\cos\!\bigl(B(t - C)\bigr) + D$，以高潮时刻作为相位 $C$，并显式给出 $B$ 与 $C$。[3]
(c)Compute the water depth at $t = 7$ hours (i.e. 7 a.m.). Round to two decimal places.计算 $t = 7$ 小时（即上午 7 时）的水深，保留两位小数。[2]
(d)A small ferry requires a minimum water depth of $2.0$ m to dock safely. Without solving, explain whether the ferry can dock at the time computed in part (c).某小型渡轮需至少 $2.0$ 米水深方可安全停靠。不另行求解，说明在 (c) 中所得时刻该渡轮能否停靠。[1]
Q12HARD难Honors荣誉级🇨🇦 BC卑🇨🇦 AB阿BC Provincial-style卑诗省考风格§7 Temperature Model气温模型 · BC PC 12 / AB Math 30-1 GO 4.9[10 marks][10 分]
The mean daily temperature in a Northern Alberta town, in degrees Celsius, is modelled by a sinusoidal function of the day of the year $d$ (where $d = 1$ on January 1 and $d = 365$ on December 31). Records show the warmest mean daily temperature of $24^{\circ}\text{C}$ occurs on day $d = 200$, and the coldest of $-22^{\circ}\text{C}$ occurs roughly half a year later. Let $T(d)$ denote the modelled temperature.阿尔伯塔北部某城镇的日均气温（摄氏度）由一年中第 $d$ 天的正弦型函数建模（$d = 1$ 为 1 月 1 日，$d = 365$ 为 12 月 31 日）。记录显示最高日均气温 $24^{\circ}\text{C}$ 出现在 $d = 200$，最低气温 $-22^{\circ}\text{C}$ 出现在大约半年之后。设 $T(d)$ 表示建模温度。
(a)State the midline $D$, the amplitude $A$, and the period (in days).写出中线 $D$、振幅 $A$ 与周期（以天为单位）。[3]
(c)Write $T(d) = A\cos\!\bigl(B(d - C)\bigr) + D$ using the maximum-day as the phase shift $C$. State the explicit equation.以最高气温所在天数作为相位 $C$，写出 $T(d) = A\cos\!\bigl(B(d - C)\bigr) + D$，并给出显式方程。[2]
(d)Compute the modelled temperature on April 30 ($d = 120$). Round to the nearest degree.计算 4 月 30 日（$d = 120$）的建模气温，四舍五入到整数度。[2]
(e)Without solving algebraically, identify two days of the year (one in spring, one in fall) on which the modelled temperature equals the midline $D$. Justify briefly using the geometry of the cosine curve.不进行代数求解，指出一年中分别位于春、秋的两个日期，使建模气温等于中线 $D$。利用余弦曲线的几何性质简要说明理由。[2]
🇨🇦 Ontario安大略MCR3U Strand D · Trigonometric Functions (degree-mode sinusoidal models) · MHF4U Strand B · Trigonometric Functions (radians, unit circle, $\sin/\cos/\tan$ graphs and transformations)MCR3U 单元 D · 三角函数（角度制正弦型模型）· MHF4U 单元 B · 三角函数（弧度制、单位圆、$\sin/\cos/\tan$ 图像及变换）
🇨🇦 British Columbia不列颠哥伦比亚PC 11 angles in standard position (degree mode) · PC 12 radians + unit circle + reference / coterminal / special angles · PC 12 graphing primary trig functions, incl. transformationsPC 11 标准位置上的角（角度制）· PC 12 弧度制 + 单位圆 + 参考角 / 共终边角 / 特殊角 · PC 12 基本三角函数作图，含变换
🇨🇦 Alberta阿尔伯塔Math 30-1 Trig GO 1 (radians 1.1–1.9) · GO 2 (unit-circle equation 2.1–2.2) · GO 3 (exact values at $30^{\circ}/45^{\circ}/60^{\circ}$ multiples, 3.2) · GO 4 (graphs + write $y = a\sin b(x - c) + d$, 4.1–4.9)Math 30-1 三角 GO 1（弧度 1.1–1.9）· GO 2（单位圆方程 2.1–2.2）· GO 3（$30^{\circ}/45^{\circ}/60^{\circ}$ 倍数处的精确值，3.2）· GO 4（图像 + 写出 $y = a\sin b(x - c) + d$，4.1–4.9）
Full 4-column Syllabus Map lives in ../Study Guides/Unit_8_Unit-Circle_Trig_and_Trigonometric_Functions.html.完整的四列大纲对照表见 ../Study Guides/Unit_8_Unit-Circle_Trig_and_Trigonometric_Functions.html。