High School Math

Unit-Circle Trigonometry and Trigonometric Functions单位圆三角学与三角函数

Right-triangle trig defines $\sin, \cos, \tan$ only for acute angles. The unit circle , a circle of radius $1$ centred at the origin , extends those three ratios to every real angle by reading $\cos\theta = x$ and $\sin\theta = y$ off the point where the angle's terminal arm meets the circle. Radian measure, $\pi = 180^{\circ}$, makes calculus possible. Then the same picture, rolled out horizontally, gives the periodic graphs of $\sin x$, $\cos x$, $\tan x$ and the transformation toolkit $y = A \sin(B(x - C)) + D$ that physics, signal processing, and AP Pre-Calc all rely on.直角三角形三角学只为锐角定义 $\sin, \cos, \tan$。单位圆（unit circle，半径为 $1$、以原点为圆心的圆）通过让角的终边与圆相交于点 $(x, y)$，并令 $\cos\theta = x$、$\sin\theta = y$，把这三个三角比扩展到所有实数角。弧度制（radians）以 $\pi = 180^{\circ}$ 为换算桥梁，是后续微积分的基础。把单位圆沿水平方向展开，便得到 $\sin x$、$\cos x$、$\tan x$ 的周期图像，以及 $y = A \sin(B(x - C)) + D$ 这一变换工具组 , 物理、信号处理与 AP Pre-Calc 都离不开它。

7 sections7 节内容 US Common Core · ON · BC · ABUS 共同核心 · ON · BC · AB Gateway to AP Calc & IB Math AA HL通往 AP Calc 与 IB Math AA HL
Read me first先读这里

How to use this guide如何使用本指南

The unit circle sits at a higher grade level than right-triangle trig in every curriculum we map to. In the US it lives in the HSF-TF domain , the AP-feeder Pre-Calc territory, not Geometry. In Ontario, sinusoidal modelling appears in Grade 11 MCR3U (degree mode), but radian-mode unit-circle work and the formal graphs $y = \sin x / \cos x / \tan x$ wait for Grade 12 MHF4U. In BC, angles in standard position are introduced in Pre-Calc 11 (degree mode), and the full unit-circle + radians + transformations package arrives in Pre-Calc 12. In Alberta the entire package is Math 30-1 (Grade 12), with separate outcomes for radians, the unit-circle equation, exact values, and graphing. The seven-row table below tells you which sections are on your syllabus right now.在我们对照的所有大纲中，单位圆所在的年级都高于直角三角形三角学。美国共同核心把它放在 HSF-TF 域 , 属于 AP 衔接 Pre-Calc 范围，而非几何课。安大略 11 年级 MCR3U（角度制）已涉及正弦型建模，但弧度制单位圆与正式 $y = \sin x / \cos x / \tan x$ 图像要等 12 年级 MHF4U。BC 在 PC 11 引入标准位置上的角（角度制），PC 12 才提供"单位圆 + 弧度 + 变换"完整包。阿尔伯塔则把整包内容集中在 Math 30-1（12 年级），并为弧度、单位圆方程、精确值、图像各设独立目标。下面的七行表告诉你当前大纲下应重点学习哪些节。

If you are in…如果你在… Focus on these sections重点学习 Defer / skip可推迟 Source依据
🇺🇸 US Grade 10 (Geometry)美国 10 年级（几何） None , this unit is HSF-TF, not HSG-SRT. Stay on right-triangle trig.无 , 本单元属 HSF-TF，不在 HSG-SRT。继续直角三角形三角学。 All seven sections defer to Pre-Calc全 7 节推迟至 Pre-Calc ccssm_hs_math.pdf , the HSF-TF domain (unit circle, radian measure, sinusoidal modelling) is separate from the HSG-SRT Geometry standard for right-triangle ratios, HSF-TF 域（单位圆、弧度、正弦建模）与几何 HSG-SRT（直角三角形）分属不同标准
🇺🇸 US AP-feeder (Pre-Calc / Honors)美国 AP 衔接（Pre-Calc / 荣誉） All seven. Especially §1 (radians + reference angles), §2 (unit circle), and §7 (transformations) , these are the AP Pre-Calc “Unit 3 Trigonometric and Polar Functions” prerequisites in concentrated form全部 7 节。尤其 §1（弧度与参考角）、§2（单位圆）、§7（变换）, 这是 AP Pre-Calc "Unit 3 三角与极坐标函数"前置知识的浓缩 Nothing , AP Calc and IB Math AA HL both assume mastery of all seven无 , AP Calc 与 IB Math AA HL 都默认你完全掌握 7 节 ccssm_hs_math.pdf , HSF-TF.A.1 (radians from arc length), HSF-TF.A.2 (extend trig to all reals), HSF-TF.B.5 (model with $A \sin(B(x - C)) + D$), HSF-TF.A.1（由弧长定义弧度）、A.2（把三角扩展到全体实数）、B.5（用 $A \sin(B(x - C)) + D$ 建模）
🇨🇦 ON Grade 11 , MCR3U安大略 11 年级 , MCR3U §1 in degree mode (skip radians), §2 (unit circle in degrees), §4-§5 (sine and cosine graphs from a degree perspective , MCR3U covers sinusoidal models)§1 只用角度制（跳过弧度），§2（角度版单位圆），§4-§5（从角度视角看 sin、cos 图象 , MCR3U 涵盖正弦型建模） Radians (§1 second half) and the formal $y = \tan x$ asymptote work (§6) wait for MHF4U. Reciprocal functions (§3) also formalised in Grade 12.弧度部分（§1 后半）与正式的 $y = \tan x$ 渐近线（§6）留到 MHF4U；倒数函数（§3）也在 12 年级才正式化。 math_grades_11-12.pdf , MCR3U Strand D Trigonometric Functions (right + oblique triangle trig and sinusoidal models in degree mode), MCR3U 单元 D 三角函数（直角与斜三角形三角学；角度制下的正弦型建模）
🇨🇦 ON Grade 12 , MHF4U安大略 12 年级 , MHF4U All seven. MHF4U Strand B introduces radian measure, unit-circle definition for all real angles, all six ratios (incl. reciprocals), and graphs of $\sin x, \cos x, \tan x$ with full transformation work全部 7 节。MHF4U 单元 B 引入弧度、向全体实数扩展单位圆定义、六种比（含倒数）、$\sin x, \cos x, \tan x$ 图像及完整变换 Nothing , treat this unit as MHF4U Strand B in concentrated form无 , 视本单元为 MHF4U 单元 B 的浓缩 math_grades_11-12.pdf , MHF4U Strand B Trigonometric Functions is the Grade 12 home for radians + unit-circle + sinusoidal graphs + transformations, MHF4U 单元 B 三角函数：12 年级"弧度 + 单位圆 + 正弦图像 + 变换"的对应单元
🇨🇦 BC Grade 11 , PC 11BC 11 年级 , PC 11 §1 in degree mode only (PC 11 introduces "angles in standard position" but not radians), §2 (unit-circle exact values in degrees)§1 仅角度制（PC 11 引入"标准位置上的角"但不含弧度）；§2（角度制下的单位圆精确值） §3 reciprocals, §4-§7 (radians, graphs, transformations) all in PC 12§3 倒数函数、§4-§7（弧度、图像、变换）均在 PC 12 pc11_elab.pdf , Content: "trigonometry: non-right triangles and angles in standard position", 内容："三角学：非直角三角形与标准位置上的角"
🇨🇦 BC Grade 12 , PC 12BC 12 年级 , PC 12 All seven. PC 12 explicitly lists radians + degrees, unit circle + reference / coterminal + special angles, and graphing primary trig functions with transformations全部 7 节。PC 12 明确列出弧度与角度、单位圆 + 参考 / 共终边 + 特殊角、含变换的基本三角函数图像 Nothing , PC 12 covers this whole unit verbatim无 , PC 12 完整覆盖本单元 pc12_elab.pdf , Content elaborations: "examining angles in standard position in both radians and degrees"; "exploring unit circle, reference and coterminal angles, special angles"; "graphing primary trigonometric functions, including transformations", 内容："在弧度与角度两种单位下考察标准位置上的角"；"探索单位圆、参考与共终边角、特殊角"；"绘制基本三角函数图像，含变换"
🇨🇦 AB Grade 12 , Math 30-1阿尔伯塔 12 年级 , Math 30-1 All seven, with strong emphasis on indicators 1.1-1.9 (radians) for §1, indicators 2.1-2.3 (unit-circle equation) for §2, indicator 3.2 (exact values at $30^{\circ} / 45^{\circ} / 60^{\circ}$ multiples) across §2, and indicators 4.1-4.9 (graph then write equation from graph) across §4-§7全部 7 节，重点对应：§1 用指标 1.1-1.9（弧度）；§2 用指标 2.1-2.3（单位圆方程）；§2 中用指标 3.2（$30^{\circ} / 45^{\circ} / 60^{\circ}$ 倍数处精确值）；§4-§7 用指标 4.1-4.9（先画图再由图象写方程） Nothing , Math 30-1 is the dedicated home for the unit circle and trig graphs in AB无 , Math 30-1 是阿尔伯塔单位圆与三角图像的对应课程 pos_10-12_indicators.pdf , Math 30-1 Trigonometry General Outcome with Specific Outcomes 1 (standard-position angles in degrees + radians), 2 (unit-circle equation), 3 (six trig ratios with exact values for standard multiples), 4 (graph and analyze sine, cosine, tangent with $a, b, c, d$), Math 30-1 三角学总目标，含具体目标 1（标准位置上的角，角度+弧度）、2（单位圆方程）、3（六种三角比及标准倍数处精确值）、4（用 $a, b, c, d$ 绘制并分析 sin、cos、tan）

Once you have located your row, use the two cards below for the speed at which you should work through the recommended sections.找到所在行后，用下面两张卡片决定推进速度。

!
If you are cramming the night before如果你在临阵磨枪

Memorise six things: $\pi = 180^{\circ}$; the unit-circle picture $\cos\theta = x$, $\sin\theta = y$ at the terminal point; the sign chart for each ratio by quadrant (ASTC, “All Students Take Calculus”); the exact values at $0, \pi/6, \pi/4, \pi/3, \pi/2$; the period of each function ($2\pi$ for sin/cos, $\pi$ for tan); and the four-parameter form $y = A\sin(B(x - C)) + D$ with amplitude $|A|$, period $2\pi/|B|$, phase shift $C$, vertical shift $D$. Read every cram-cheat box. Skip the going-deeper derivations.背熟六件事：$\pi = 180^{\circ}$；单位圆图示 $\cos\theta = x$、$\sin\theta = y$；各象限符号表（ASTC，"All Students Take Calculus"）；$0, \pi/6, \pi/4, \pi/3, \pi/2$ 处精确值；各函数周期（sin/cos 为 $2\pi$，tan 为 $\pi$）；以及四参数形式 $y = A\sin(B(x - C)) + D$，其中振幅 $|A|$、周期 $2\pi/|B|$、相位移 $C$、竖直平移 $D$。读每个速记框，跳过深入推导。

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If you are going for the top mark如果你目标顶分

Practise reading a unit-circle picture: from $(x, y)$ on the circle you should reach $\sin\theta, \cos\theta, \tan\theta, \csc\theta, \sec\theta, \cot\theta$ without writing a formula down. Learn to switch between degree and radian mode mentally ($30^{\circ} \leftrightarrow \pi/6$, etc.). For graphs, practise the four-step routine: identify $A, B, C, D$; mark the midline $y = D$; mark amplitude $|A|$ above and below it; compute period $2\pi/|B|$ and plot one cycle. AB Math 30-1 indicator 4.9 expects you to reverse this: read $A, B, C, D$ off a given graph.练习"看图说话"：从单位圆上 $(x, y)$ 出发，不用公式就能给出 $\sin\theta, \cos\theta, \tan\theta, \csc\theta, \sec\theta, \cot\theta$。心算度-弧度互换（$30^{\circ} \leftrightarrow \pi/6$ 等）。图像部分练四步法：识别 $A, B, C, D$；画中线 $y = D$；上下偏移 $|A|$ 标振幅；算周期 $2\pi/|B|$ 并画一个周期。AB Math 30-1 指标 4.9 要求反向操作：从给定图象读出 $A, B, C, D$。

Section 1 · Angles, radians, reference angles第 1 节 · 角度、弧度、参考角

Angles in Standard Position: Degrees, Radians, Coterminal, and Reference Pre-Calc for US Alg 2标准位置上的角：角度、弧度、共终边、参考 US Alg 2 之上

Four definitions that unlock the unit circle.打开单位圆的四个定义。
  • Standard position.标准位置。 Vertex at the origin, initial side along the positive $x$-axis, measured counter-clockwise (positive) or clockwise (negative). The other side is the terminal side (or terminal arm).顶点在原点，初始边沿正 $x$ 轴；逆时针为正，顺时针为负。另一边称为终边（terminal arm）。
  • Radian measure.弧度制。 CCSSM HSF-TF.A.1: a radian is the central angle whose arc on the unit circle has length $1$. Conversion: $\pi$ radians $= 180^{\circ}$.CCSSM HSF-TF.A.1：弧度为单位圆上所截弧长等于 $1$ 时对应的圆心角。换算：$\pi$ 弧度 $= 180^{\circ}$。
  • Coterminal angles.共终边角。 Share the same terminal side. Add or subtract $360^{\circ}$ (or $2\pi$) any whole number of times. AB Math 30-1 indicator 1.8 names "the general form of the measures of all angles that are coterminal."终边相同的角。加减任意 $360^{\circ}$（或 $2\pi$）的整数倍即可。AB Math 30-1 指标 1.8 称之为"所有共终边角度量的一般形式"。
  • Reference angle.参考角。 The acute angle (between $0$ and $\pi/2$) that the terminal side makes with the $x$-axis. Trig values at $\theta$ have the same magnitudes as at the reference angle; only the signs change by quadrant.终边与 $x$ 轴所成的锐角（介于 $0$ 与 $\pi/2$）。$\theta$ 处三角值的绝对值等于参考角处的对应值；只有符号随象限变化。
$$ \text{Degrees} \to \text{radians:} \;\; \theta_{\mathrm{rad}} = \theta_{\mathrm{deg}} \cdot \frac{\pi}{180}, \qquad \text{Radians} \to \text{degrees:} \;\; \theta_{\mathrm{deg}} = \theta_{\mathrm{rad}} \cdot \frac{180}{\pi}. $$ AB Math 30-1 names this explicitly: indicator 1.5 "express the measure of an angle in radians, given its measure in degrees" and 1.6 "express the measure of an angle in degrees, given its measure in radians." Indicators 1.7-1.8 do the coterminal work.AB Math 30-1 明确点名：指标 1.5"由角度求弧度"；1.6"由弧度求角度"；1.7-1.8 处理共终边角。
Worked Example 1 · Degree-radian conversion and reference angle例题 1 · 度-弧度换算与参考角

Convert $\theta = 210^{\circ}$ to radians as an exact multiple of $\pi$, identify the quadrant of its terminal side, find the reference angle (in radians), and list two positive and one negative coterminal angles (in degrees).把 $\theta = 210^{\circ}$ 化为 $\pi$ 的精确倍数；指出终边所在象限；求参考角（用弧度）；给出两个正的与一个负的共终边角（用角度）。

Convert.换算。

$$ 210^{\circ} \cdot \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6}. $$

Quadrant.象限。 $180^{\circ} < 210^{\circ} < 270^{\circ}$, so the terminal side is in Quadrant III.$180^{\circ} < 210^{\circ} < 270^{\circ}$，故终边在第 III 象限。

Reference angle.参考角。 In QIII, reference $= \theta - 180^{\circ} = 30^{\circ} = \pi/6$.QIII 中，参考 $= \theta - 180^{\circ} = 30^{\circ} = \pi/6$。

Coterminal angles.共终边角。 Add $360^{\circ}$ or $720^{\circ}$ for two positives: $570^{\circ}$ and $930^{\circ}$. Subtract $360^{\circ}$ for a negative: $-150^{\circ}$. (Sanity-check: all three have terminal sides that land in the same QIII direction.)加 $360^{\circ}$ 或 $720^{\circ}$ 得两正：$570^{\circ}$ 与 $930^{\circ}$；减 $360^{\circ}$ 得一负：$-150^{\circ}$。（验证：三者终边都指向 QIII 同一方向。）

Convert $\theta = 135^{\circ}$ to radians as an exact multiple of $\pi$.把 $\theta = 135^{\circ}$ 化为 $\pi$ 的精确倍数。
§1 · Q1
$2\pi/3$
$5\pi/6$
$3\pi/4$
$7\pi/4$
$135^{\circ} \cdot \pi/180 = 135\pi/180 = 3\pi/4$. Sanity check: $135^{\circ} < 180^{\circ} = \pi$, so the answer must be less than $\pi$. $3\pi/4$ qualifies.$135^{\circ} \cdot \pi/180 = 135\pi/180 = 3\pi/4$。验证：$135^{\circ} < 180^{\circ} = \pi$，答案应小于 $\pi$，$3\pi/4$ 符合。
Multiply by $\pi/180$ and reduce. $135/180 = 3/4$, so $3\pi/4$.乘 $\pi/180$ 后化简。$135/180 = 3/4$，故 $3\pi/4$。
For $\theta = 5\pi/3$, find the reference angle (in radians) and the quadrant of the terminal side.$\theta = 5\pi/3$ 的参考角（弧度）与终边象限为？
§1 · Q2
Reference $\pi/6$, Quadrant III参考 $\pi/6$，第 III 象限
Reference $\pi/3$, Quadrant IV参考 $\pi/3$，第 IV 象限
Reference $2\pi/3$, Quadrant II参考 $2\pi/3$，第 II 象限
Reference $\pi/3$, Quadrant I参考 $\pi/3$，第 I 象限
$5\pi/3 = 300^{\circ}$, in $(270^{\circ}, 360^{\circ})$, so QIV. The reference angle is $360^{\circ} - 300^{\circ} = 60^{\circ} = \pi/3$. Equivalently in radians: $2\pi - 5\pi/3 = \pi/3$.$5\pi/3 = 300^{\circ}$，在 $(270^{\circ}, 360^{\circ})$ 内，故 QIV。参考 $= 360^{\circ} - 300^{\circ} = 60^{\circ} = \pi/3$；弧度直接 $2\pi - 5\pi/3 = \pi/3$。
Convert to degrees first: $5\pi/3 \cdot 180/\pi = 300^{\circ}$. Then identify QIV and take reference $= 360^{\circ} - 300^{\circ}$.先化为角度 $300^{\circ}$，再判 QIV，取参考 $= 360^{\circ} - 300^{\circ}$。
Section 2 · The unit circle第 2 节 · 单位圆

The Unit Circle: Exact Values in All Four Quadrants Pre-Calc for US Alg 2单位圆：四象限精确值 US Alg 2 之上

The unit-circle definition.单位圆定义。

The unit circle is $x^{2} + y^{2} = 1$ (centre origin, radius $1$). Let $\theta$ be in standard position. If the terminal side meets the unit circle at point $P(x, y)$, then:单位圆为 $x^{2} + y^{2} = 1$（以原点为圆心、半径为 $1$）。设 $\theta$ 处于标准位置，终边与单位圆交点为 $P(x, y)$，则：

$$ \cos\theta = x, \qquad \sin\theta = y, \qquad \tan\theta = \frac{y}{x} \;\; (x \ne 0). $$
  • Equation derivation.方程推导。 AB Math 30-1 indicator 2.1: "derive the equation of the unit circle from the Pythagorean theorem." On the unit circle $x^{2} + y^{2} = 1$, so $\cos^{2}\theta + \sin^{2}\theta = 1$ , the Pythagorean identity is just the equation of the circle in trig clothing.AB Math 30-1 指标 2.1："由勾股定理推出单位圆方程"。单位圆上 $x^{2} + y^{2} = 1$，故 $\cos^{2}\theta + \sin^{2}\theta = 1$ , 勾股恒等式正是单位圆方程的三角形式。
  • Sign by quadrant (ASTC):分象限符号（ASTC）： QI all positive; QII only $\sin$ positive; QIII only $\tan$ positive; QIV only $\cos$ positive. Mnemonic "All Students Take Calculus."QI 全正；QII 仅 $\sin$ 正；QIII 仅 $\tan$ 正；QIV 仅 $\cos$ 正。口诀 "All Students Take Calculus"。
  • Exact values via reference angle.借参考角求精确值。 AB Math 30-1 indicator 3.2: "determine, using a unit circle or reference triangle, the exact value of a trigonometric ratio for angles expressed in degrees that are multiples of $0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}$, or $90^{\circ}$." Strategy: compute the value at the reference angle from the 30-60-90 or 45-45-90 triangle, then attach the QI / II / III / IV sign.AB Math 30-1 指标 3.2："用单位圆或参考三角形，求 $0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}$ 倍数角处三角比的精确值"。策略：用 30-60-90 或 45-45-90 三角形算出参考角处的值，再按象限加符号。
First-quadrant exact-value table (drilled from §2 of the Right-Triangle Trigonometry guide)第 I 象限精确值表（沿用直角三角形三角学指南 §2）
$$ \begin{array}{c|cccccc}\theta & 0 & \pi/6 & \pi/4 & \pi/3 & \pi/2 \\ \hline \sin\theta & 0 & 1/2 & \sqrt{2}/2 & \sqrt{3}/2 & 1 \\ \cos\theta & 1 & \sqrt{3}/2 & \sqrt{2}/2 & 1/2 & 0 \\ \tan\theta & 0 & \sqrt{3}/3 & 1 & \sqrt{3} & \text{undef.} \end{array} $$
Worked Example 2 · Exact values across all four quadrants例题 2 · 四象限精确值

Find the exact values of $\sin\theta, \cos\theta, \tan\theta$ for (a) $\theta = 5\pi/6$, (b) $\theta = 7\pi/6$, and (c) $\theta = 11\pi/6$.求下列 $\theta$ 的 $\sin\theta, \cos\theta, \tan\theta$ 精确值：(a) $5\pi/6$，(b) $7\pi/6$，(c) $11\pi/6$。

(a) $\theta = 5\pi/6 = 150^{\circ}$, QII.(a) $\theta = 5\pi/6 = 150^{\circ}$，QII。 Reference $= \pi - 5\pi/6 = \pi/6$. From the table $\sin(\pi/6) = 1/2, \cos(\pi/6) = \sqrt{3}/2, \tan(\pi/6) = \sqrt{3}/3$. QII signs: $\sin +$, $\cos -$, $\tan -$.参考 $= \pi - 5\pi/6 = \pi/6$。表中 $\sin(\pi/6) = 1/2, \cos(\pi/6) = \sqrt{3}/2, \tan(\pi/6) = \sqrt{3}/3$。QII 符号 $\sin +$、$\cos -$、$\tan -$。

$$ \sin(5\pi/6) = 1/2, \quad \cos(5\pi/6) = -\sqrt{3}/2, \quad \tan(5\pi/6) = -\sqrt{3}/3. $$

(b) $\theta = 7\pi/6 = 210^{\circ}$, QIII.(b) $\theta = 7\pi/6 = 210^{\circ}$，QIII。 Reference $= 7\pi/6 - \pi = \pi/6$. QIII signs: $\sin -$, $\cos -$, $\tan +$.参考 $= 7\pi/6 - \pi = \pi/6$。QIII 符号 $\sin -$、$\cos -$、$\tan +$。

$$ \sin(7\pi/6) = -1/2, \quad \cos(7\pi/6) = -\sqrt{3}/2, \quad \tan(7\pi/6) = \sqrt{3}/3. $$

(c) $\theta = 11\pi/6 = 330^{\circ}$, QIV.(c) $\theta = 11\pi/6 = 330^{\circ}$，QIV。 Reference $= 2\pi - 11\pi/6 = \pi/6$. QIV signs: $\sin -$, $\cos +$, $\tan -$.参考 $= 2\pi - 11\pi/6 = \pi/6$。QIV 符号 $\sin -$、$\cos +$、$\tan -$。

$$ \sin(11\pi/6) = -1/2, \quad \cos(11\pi/6) = \sqrt{3}/2, \quad \tan(11\pi/6) = -\sqrt{3}/3. $$

Sanity-check.合理性核验。 All three angles have the same reference $\pi/6$, so all three magnitudes match the QI table. Only the signs change , exactly what the ASTC chart predicts.三角参考都是 $\pi/6$，绝对值与 QI 表一致；变化的只有符号 , 与 ASTC 表预测完全相符。

Find the exact value of $\cos(2\pi/3)$.求 $\cos(2\pi/3)$ 的精确值。
§2 · Q1
$\sqrt{3}/2$
$1/2$
$-\sqrt{3}/2$
$-1/2$
$2\pi/3 = 120^{\circ}$, QII. Reference $= \pi - 2\pi/3 = \pi/3$. $\cos(\pi/3) = 1/2$. QII makes cosine negative, so $\cos(2\pi/3) = -1/2$.$2\pi/3 = 120^{\circ}$，QII。参考 $= \pi - 2\pi/3 = \pi/3$。$\cos(\pi/3) = 1/2$。QII 中余弦为负，故 $\cos(2\pi/3) = -1/2$。
Identify the quadrant first (QII), then compute the reference-angle value ($1/2$) and apply the QII cosine-negative sign.先判象限（QII），再算参考角值（$1/2$），最后按 QII 余弦为负加符号。
If the terminal side of $\theta$ in standard position passes through $(-3, 4)$, find $\sin\theta$ as an exact fraction.设标准位置上 $\theta$ 的终边过点 $(-3, 4)$，求 $\sin\theta$ 的精确分数值。
§2 · Q2
$4/5$
$-3/5$
$-4/5$
$3/5$
For any point $P(x, y)$ on the terminal side, $r = \sqrt{x^{2} + y^{2}}$ and $\sin\theta = y/r, \cos\theta = x/r$ (AB Math 30-1 indicator 3.4). Here $r = \sqrt{9 + 16} = 5$, so $\sin\theta = 4/5$. (The terminal side is in QII where $\sin$ is positive , consistent.)终边上任一点 $P(x, y)$ 满足 $r = \sqrt{x^{2} + y^{2}}$，$\sin\theta = y/r$、$\cos\theta = x/r$（AB Math 30-1 指标 3.4）。此处 $r = \sqrt{9 + 16} = 5$，故 $\sin\theta = 4/5$。（终边位于 QII，$\sin$ 为正 , 与结果一致。）
Compute the radius $r = \sqrt{x^{2} + y^{2}}$, then $\sin\theta = y/r$. Check the QII sign of sine ($+$).先算 $r = \sqrt{x^{2} + y^{2}}$，再 $\sin\theta = y/r$，并按 QII（$\sin +$）核验符号。
Section 3 · Reciprocal ratios第 3 节 · 倒数三角比

Reciprocal Functions: cosecant, secant, cotangent倒数函数：余割、正割、余切

Three reciprocals , three definitions to memorise.三个倒数 , 三个定义要背熟。 $$ \csc\theta = \frac{1}{\sin\theta}, \qquad \sec\theta = \frac{1}{\cos\theta}, \qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}. $$
  • Pairing.配对。 $\sin \leftrightarrow \csc$, $\cos \leftrightarrow \sec$, $\tan \leftrightarrow \cot$. Notice $\sec$ pairs with $\cos$ (not $\sin$) , the “co-” prefix is misleading here.$\sin \leftrightarrow \csc$、$\cos \leftrightarrow \sec$、$\tan \leftrightarrow \cot$。注意 $\sec$ 配 $\cos$（而非 $\sin$）, 词首 "co-" 易误导。
  • Undefined where the denominator hits zero.分母为零处无定义。 $\csc\theta$ is undefined when $\sin\theta = 0$ (i.e., $\theta = k\pi$). $\sec\theta$ is undefined when $\cos\theta = 0$ (i.e., $\theta = \pi/2 + k\pi$). $\cot\theta$ is undefined when $\sin\theta = 0$.$\csc\theta$ 在 $\sin\theta = 0$ 时（$\theta = k\pi$）无定义；$\sec\theta$ 在 $\cos\theta = 0$ 时（$\theta = \pi/2 + k\pi$）无定义；$\cot\theta$ 在 $\sin\theta = 0$ 时无定义。
  • Same-sign by quadrant.象限同号。 A reciprocal has the same sign as its primary in each quadrant. The QIII rule "tan positive, cos / sin negative" carries over to "cot positive, sec / csc negative."每个象限中，倒数与其本原同号。QIII 的"tan 正、cos / sin 负"延续为"cot 正、sec / csc 负"。
  • Unit-circle reading.单位圆读法。 For the terminal-side point $P(x, y)$ on the unit circle: $\csc\theta = 1/y$, $\sec\theta = 1/x$, $\cot\theta = x/y$. AB Math 30-1 indicator 2.2 names this for all six ratios.单位圆终边点 $P(x, y)$ 满足 $\csc\theta = 1/y$、$\sec\theta = 1/x$、$\cot\theta = x/y$。AB Math 30-1 指标 2.2 对六种比都给出此定义。
Worked Example 3 · Reciprocal values at a standard angle例题 3 · 标准角处的倒数值

Find the exact values of $\csc(7\pi/6), \sec(7\pi/6), \cot(7\pi/6)$.求 $\csc(7\pi/6), \sec(7\pi/6), \cot(7\pi/6)$ 的精确值。

Use the primary values from §2.借用 §2 的本原值。 From §2(b): $\sin(7\pi/6) = -1/2$, $\cos(7\pi/6) = -\sqrt{3}/2$, $\tan(7\pi/6) = \sqrt{3}/3$.由 §2(b)：$\sin(7\pi/6) = -1/2$、$\cos(7\pi/6) = -\sqrt{3}/2$、$\tan(7\pi/6) = \sqrt{3}/3$。

Take reciprocals.取倒数。

$$ \csc(7\pi/6) = \frac{1}{-1/2} = -2. $$ $$ \sec(7\pi/6) = \frac{1}{-\sqrt{3}/2} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}. $$ $$ \cot(7\pi/6) = \frac{1}{\sqrt{3}/3} = \frac{3}{\sqrt{3}} = \sqrt{3}. $$

Sanity-check.合理性核验。 QIII: $\sin -, \cos -, \tan +$, so $\csc -, \sec -, \cot +$. Signs $-, -, +$ match. ✓QIII 中 $\sin -, \cos -, \tan +$，故 $\csc -, \sec -, \cot +$。所得符号 $-, -, +$ 与之一致 ✓。

Find the exact value of $\sec(\pi/4)$.求 $\sec(\pi/4)$ 的精确值。
§3 · Q1
$\sqrt{2}$
$1/\sqrt{2}$
$2/\sqrt{3}$
$2$
$\sec(\pi/4) = 1/\cos(\pi/4) = 1/(\sqrt{2}/2) = 2/\sqrt{2} = \sqrt{2}$.$\sec(\pi/4) = 1/\cos(\pi/4) = 1/(\sqrt{2}/2) = 2/\sqrt{2} = \sqrt{2}$。
$\sec$ pairs with $\cos$, not $\sin$. Take $1/\cos(\pi/4)$ and rationalise.$\sec$ 配 $\cos$ 而非 $\sin$。取 $1/\cos(\pi/4)$ 并有理化分母。
At which of the following angles is $\csc\theta$ undefined?下列哪个角处 $\csc\theta$ 无定义？
§3 · Q2
$\theta = \pi/2$
$\theta = \pi/4$
$\theta = \pi$
$\theta = \pi/3$
$\csc\theta = 1/\sin\theta$ is undefined when $\sin\theta = 0$, i.e., at $\theta = k\pi$ for integer $k$. $\theta = \pi$ qualifies; the other three give nonzero sines.$\csc\theta = 1/\sin\theta$ 在 $\sin\theta = 0$ 时无定义，即 $\theta = k\pi$（整数 $k$）。$\theta = \pi$ 满足；其余三角 $\sin$ 均非零。
$\csc$ shares its zeros with $\sin$. $\sin\pi = 0$, so $\csc\pi$ is undefined. ($\sec$, not $\csc$, dies at $\pi/2$.)$\csc$ 的奇点与 $\sin$ 的零点相同。$\sin\pi = 0$，故 $\csc\pi$ 无定义。（$\pi/2$ 处死掉的是 $\sec$，不是 $\csc$。）
Section 4 · Graph of $y = \sin x$第 4 节 · $y = \sin x$ 图像

The Sine Curve: Period $2\pi$, Amplitude $1$, Key Points Pre-Calc for US Alg 2正弦曲线：周期 $2\pi$、振幅 $1$、关键点 US Alg 2 之上

Five points name the curve.五点定形。

$y = \sin x$ on $[0, 2\pi]$ passes through five key points (read off as the $y$-coordinate of the terminal-side point as $\theta$ travels round the unit circle):$y = \sin x$ 在 $[0, 2\pi]$ 上有五个关键点（即 $\theta$ 绕单位圆运动时终边点的 $y$ 坐标）：

$$ (0, 0), \;\; (\pi/2, 1), \;\; (\pi, 0), \;\; (3\pi/2, -1), \;\; (2\pi, 0). $$
  • Domain:定义域： all real $x$.全体实数 $x$。
  • Range:值域： $[-1, 1]$.
  • Period:周期： $2\pi$. $\sin(x + 2\pi) = \sin x$ for every $x$.对任意 $x$，$\sin(x + 2\pi) = \sin x$。
  • Amplitude:振幅： $1$. Half of (max $-$ min).即（最大 $-$ 最小）的一半。
  • Symmetry:对称性： $y = \sin x$ is odd: $\sin(-x) = -\sin x$. Symmetric about the origin.$y = \sin x$ 为奇函数：$\sin(-x) = -\sin x$。关于原点对称。
  • Zeros:零点： $x = k\pi$ for integer $k$.$x = k\pi$（整数 $k$）。
AB Math 30-1 indicators 4.1, 4.2: sketch the graph and determine "characteristics (amplitude, asymptotes, domain, period, range and zeros)." Sine has no asymptote , that is a tangent feature.AB Math 30-1 指标 4.1、4.2：作图并确定"特征（振幅、渐近线、定义域、周期、值域、零点）"。正弦无渐近线 , 那是正切的特征。
Worked Example 4 · Sketch $y = \sin x$ on $[-2\pi, 2\pi]$ and locate zeros例题 4 · 在 $[-2\pi, 2\pi]$ 上作 $y = \sin x$ 并定位零点

List the zeros of $y = \sin x$ in $[-2\pi, 2\pi]$ and the $x$-coordinates of the maxima and minima in the same interval.在 $[-2\pi, 2\pi]$ 中列出 $y = \sin x$ 的零点，以及最大与最小值的 $x$ 坐标。

Zeros at multiples of $\pi$.零点为 $\pi$ 的倍数。

$$ x = -2\pi, \;\; -\pi, \;\; 0, \;\; \pi, \;\; 2\pi. $$

Maxima ($y = 1$) at $\pi/2 + 2k\pi$.最大值（$y = 1$）在 $\pi/2 + 2k\pi$。

$$ x = -3\pi/2, \;\; \pi/2 \quad (\text{within } [-2\pi, 2\pi]). $$

Minima ($y = -1$) at $-\pi/2 + 2k\pi$.最小值（$y = -1$）在 $-\pi/2 + 2k\pi$。

$$ x = -\pi/2, \;\; 3\pi/2 \quad (\text{within } [-2\pi, 2\pi]). $$

Cycle count.周期数。 The interval has length $4\pi$, which is $2$ full periods. So expect $5$ zeros, $2$ maxima, $2$ minima , matches.区间长 $4\pi$，等于 $2$ 个周期。预期 $5$ 零点、$2$ 最大、$2$ 最小 , 与所得一致。

What is the range of $y = \sin x$?$y = \sin x$ 的值域为？
§4 · Q1
$[0, 1]$
$(-\infty, \infty)$
$[-1, 1]$
$[-2\pi, 2\pi]$
$\sin\theta = y$ on the unit circle, and $y$ ranges from $-1$ (south pole) to $1$ (north pole). So $\sin x \in [-1, 1]$ for every real $x$.单位圆上 $\sin\theta = y$，而 $y$ 取值由 $-1$（最下）到 $1$（最上）。故对任意实 $x$，$\sin x \in [-1, 1]$。
Domain is all reals, range is bounded by the unit circle's vertical extent $[-1, 1]$.定义域为全体实数；值域由单位圆竖直范围 $[-1, 1]$ 给出。
Which symmetry does $y = \sin x$ have?$y = \sin x$ 具有何种对称？
§4 · Q2
Even: $\sin(-x) = \sin x$偶函数：$\sin(-x) = \sin x$
Odd: $\sin(-x) = -\sin x$奇函数：$\sin(-x) = -\sin x$
Neither两者皆非
Symmetric about $x = \pi/2$ only仅关于 $x = \pi/2$ 对称
On the unit circle, reflecting through the origin sends $(x, y) \to (-x, -y)$, so the angle $-\theta$ has $y$-coordinate $-\sin\theta$. Hence $\sin(-x) = -\sin x$ , odd. (CCSSM HSF-TF.B.4 (+) names odd / even symmetry of trig functions via the unit circle explicitly.)单位圆上关于原点反射 $(x, y) \to (-x, -y)$，故角 $-\theta$ 的 $y$ 坐标为 $-\sin\theta$，即 $\sin(-x) = -\sin x$ , 奇函数。（CCSSM HSF-TF.B.4 (+) 明确点名通过单位圆解释三角函数奇偶性。）
Use the unit-circle reflection $\theta \to -\theta$: the $y$-coordinate flips. So sin is odd.用单位圆反射 $\theta \to -\theta$：$y$ 坐标取反，故 sin 为奇函数。
Section 5 · Graph of $y = \cos x$第 5 节 · $y = \cos x$ 图像

The Cosine Curve: The Sine Curve Shifted Left by $\pi/2$ Pre-Calc for US Alg 2余弦曲线：正弦曲线向左平移 $\pi/2$ US Alg 2 之上

Cosine = sine, phase-shifted.余弦即正弦的相位平移。 $$ \cos x = \sin(x + \pi/2). $$

Geometric proof: cosine is the $x$-coordinate on the unit circle; sine is the $y$-coordinate. Rotating the picture $\pi/2$ counter-clockwise swaps roles. Equivalently, the cosine curve is the sine curve shifted left $\pi/2$.几何理解：余弦是单位圆 $x$ 坐标、正弦是 $y$ 坐标；将图像逆时针旋转 $\pi/2$ 后角色互换。等价地，余弦曲线即正弦曲线左移 $\pi/2$。

  • Five key points on $[0, 2\pi]$:$[0, 2\pi]$ 五关键点： $(0, 1), \;\; (\pi/2, 0), \;\; (\pi, -1), \;\; (3\pi/2, 0), \;\; (2\pi, 1)$.
  • Domain:定义域： all real $x$.全体实数 $x$。 Range:值域： $[-1, 1]$.
  • Period:周期： $2\pi$.
  • Amplitude:振幅： $1$.
  • Symmetry:对称性： $y = \cos x$ is even: $\cos(-x) = \cos x$. Symmetric about the $y$-axis.$y = \cos x$ 为偶函数：$\cos(-x) = \cos x$。关于 $y$ 轴对称。
  • Zeros:零点： $x = \pi/2 + k\pi$ for integer $k$.$x = \pi/2 + k\pi$（整数 $k$）。
The co-function identity from the Right-Triangle Trigonometry guide (right-triangle co-angles, CCSSM HSG-SRT.C.7) extends to the whole real line: $\sin x = \cos(\pi/2 - x)$ for every $x$.直角三角形三角学指南中的余角恒等式（CCSSM HSG-SRT.C.7）扩展至全体实数：对任意 $x$，$\sin x = \cos(\pi/2 - x)$。
Worked Example 5 · Compare $\sin$ and $\cos$ at standard angles例题 5 · 标准角处比较 $\sin$ 与 $\cos$

Verify the shift relation $\cos x = \sin(x + \pi/2)$ at $x = 0, \pi/4, \pi/2, \pi$, and explain why $y = \cos x$ has the same amplitude and period as $y = \sin x$.在 $x = 0, \pi/4, \pi/2, \pi$ 处验证平移关系 $\cos x = \sin(x + \pi/2)$，并说明 $y = \cos x$ 为何与 $y = \sin x$ 振幅、周期相同。

Check the four values.逐点核验。

$x = 0$: $\cos 0 = 1$ and $\sin(\pi/2) = 1$. ✓$\cos 0 = 1$，$\sin(\pi/2) = 1$ ✓。

$x = \pi/4$: $\cos(\pi/4) = \sqrt{2}/2$ and $\sin(3\pi/4) = \sqrt{2}/2$. ✓$\cos(\pi/4) = \sqrt{2}/2$，$\sin(3\pi/4) = \sqrt{2}/2$ ✓。

$x = \pi/2$: $\cos(\pi/2) = 0$ and $\sin\pi = 0$. ✓$\cos(\pi/2) = 0$，$\sin\pi = 0$ ✓。

$x = \pi$: $\cos\pi = -1$ and $\sin(3\pi/2) = -1$. ✓$\cos\pi = -1$，$\sin(3\pi/2) = -1$ ✓。

Same amplitude and period.同振幅、同周期。 A horizontal shift moves a graph left or right but does not stretch it vertically and does not change the period. So $\cos x = \sin(x + \pi/2)$ has the same range $[-1, 1]$ and the same period $2\pi$ as $\sin x$.水平平移不改变竖直缩放，也不改变周期，故 $\cos x = \sin(x + \pi/2)$ 与 $\sin x$ 值域同为 $[-1, 1]$、周期同为 $2\pi$。

Which symmetry does $y = \cos x$ have?$y = \cos x$ 具有何种对称？
§5 · Q1
Odd: $\cos(-x) = -\cos x$奇函数：$\cos(-x) = -\cos x$
Neither两者皆非
Even: $\cos(-x) = \cos x$偶函数：$\cos(-x) = \cos x$
Symmetric about $x = \pi$ only仅关于 $x = \pi$ 对称
On the unit circle, reflecting through the $x$-axis sends $(x, y) \to (x, -y)$; the $x$-coordinate is unchanged. So $\cos(-\theta) = \cos\theta$ , cosine is even, symmetric about the $y$-axis.单位圆上关于 $x$ 轴反射 $(x, y) \to (x, -y)$，$x$ 坐标不变。故 $\cos(-\theta) = \cos\theta$ , 余弦为偶函数，关于 $y$ 轴对称。
Reflecting the unit circle across the $x$-axis (i.e. $\theta \to -\theta$) leaves the $x$-coordinate alone, so $\cos$ is even.单位圆关于 $x$ 轴反射（$\theta \to -\theta$）保持 $x$ 坐标，故 $\cos$ 为偶。
List the zeros of $y = \cos x$ on $[0, 2\pi]$.列出 $y = \cos x$ 在 $[0, 2\pi]$ 上的零点。
§5 · Q2
$x = 0, \pi, 2\pi$
$x = \pi/2, 3\pi/2$
$x = \pi/4, 5\pi/4$
$x = \pi/3, 5\pi/3$
$\cos x = 0$ when the terminal-side $x$-coordinate is zero, i.e., when the terminal side points straight up or straight down. On $[0, 2\pi]$ that gives $x = \pi/2$ and $x = 3\pi/2$.$\cos x = 0$ 当终边 $x$ 坐标为 $0$，即终边竖直向上或向下。$[0, 2\pi]$ 上即 $x = \pi/2$、$x = 3\pi/2$。
Cosine is zero where the terminal-side $x$-coordinate is zero , at $\pi/2 + k\pi$.余弦在终边 $x$ 坐标为 $0$ 处为零 , 即 $\pi/2 + k\pi$。
Section 6 · Graph of $y = \tan x$第 6 节 · $y = \tan x$ 图像

The Tangent Curve: Period $\pi$, Asymptotes at $\pi/2 + k\pi$ Pre-Calc for US Alg 2正切曲线：周期 $\pi$、渐近线 $\pi/2 + k\pi$ US Alg 2 之上

Tangent is the slope of the terminal side.正切即终边斜率。 $$ \tan x = \frac{\sin x}{\cos x} = \frac{y}{x}. $$
  • Domain:定义域： all real $x$ except $x = \pi/2 + k\pi$ (where $\cos x = 0$).所有实数 $x$，但排除 $x = \pi/2 + k\pi$（此处 $\cos x = 0$）。
  • Range:值域： all real numbers, $(-\infty, \infty)$.全体实数 $(-\infty, \infty)$。
  • Period:周期： $\pi$ (half the sine / cosine period , the unit-circle slope repeats every $\pi$ because $(x, y) \to (-x, -y)$ gives the same ratio $y/x$).（正余弦周期之半 , 单位圆斜率每 $\pi$ 重复一次，因为 $(x, y) \to (-x, -y)$ 比值 $y/x$ 不变）。
  • Vertical asymptotes:竖直渐近线： $x = \pi/2 + k\pi$. The graph approaches $+\infty$ from the left and $-\infty$ from the right of each asymptote (or vice versa).每条渐近线左侧趋向 $+\infty$、右侧趋向 $-\infty$（或反之）。
  • Zeros:零点： $x = k\pi$ (same as sine).$x = k\pi$（与正弦相同）。
  • Symmetry:对称性： $\tan(-x) = -\tan x$ , tangent is odd.$\tan(-x) = -\tan x$ , 正切为奇函数。
  • No amplitude.无振幅。 The unbounded range means “amplitude” is not defined for $y = \tan x$. AB Math 30-1 indicator 4.2 names "asymptotes" as a tangent-specific characteristic instead.值域无界，故 $y = \tan x$ 无"振幅"。AB Math 30-1 指标 4.2 用"渐近线"作为正切特征。
Worked Example 6 · Asymptotes and one cycle of $y = \tan x$例题 6 · $y = \tan x$ 的渐近线与一个周期

List the vertical asymptotes of $y = \tan x$ in $(-\pi, \pi)$, the zero in $(-\pi/2, \pi/2)$, and explain why the period is $\pi$ (not $2\pi$).列出 $y = \tan x$ 在 $(-\pi, \pi)$ 中的竖直渐近线、$(-\pi/2, \pi/2)$ 中的零点，并说明周期为何是 $\pi$（而非 $2\pi$）。

Asymptotes.渐近线。 $\cos x = 0$ at $x = -\pi/2$ and $x = \pi/2$ , the two asymptotes in $(-\pi, \pi)$.$\cos x = 0$ 处 $x = -\pi/2$、$x = \pi/2$ , 即 $(-\pi, \pi)$ 内两条渐近线。

Zero.零点。 $\sin x = 0$ at $x = 0$ inside $(-\pi/2, \pi/2)$. So $\tan 0 = 0$.$(-\pi/2, \pi/2)$ 内 $\sin x = 0$ 处 $x = 0$，故 $\tan 0 = 0$。

Why the period is $\pi$.为何周期 $\pi$。 Adding $\pi$ to $\theta$ sends the terminal-side point $(x, y)$ to $(-x, -y)$ on the unit circle. The ratio $y/x$ becomes $(-y)/(-x) = y/x$ , unchanged. So $\tan(x + \pi) = \tan x$.$\theta$ 加 $\pi$ 把终边点 $(x, y)$ 送到 $(-x, -y)$。比值 $y/x$ 化为 $(-y)/(-x) = y/x$，不变。故 $\tan(x + \pi) = \tan x$。

What is the period of $y = \tan x$?$y = \tan x$ 的周期为？
§6 · Q1
$2\pi$
$\pi/2$
$\pi/4$
$\pi$
$\tan(x + \pi) = \tan x$ because the terminal-side ratio $y/x$ is unchanged when the point is reflected through the origin. So the period is $\pi$ , half the period of sine and cosine.$\tan(x + \pi) = \tan x$（终边点关于原点反射后 $y/x$ 不变），故周期 $\pi$ , 正余弦周期之半。
Tangent is the unit-circle slope $y/x$. Reflecting through the origin keeps the ratio the same, so the period is $\pi$, not $2\pi$.正切为单位圆斜率 $y/x$。关于原点反射后比值不变，故周期 $\pi$ 而非 $2\pi$。
Which of the following is a vertical asymptote of $y = \tan x$?下列哪个是 $y = \tan x$ 的竖直渐近线？
§6 · Q2
$x = 0$
$x = \pi$
$x = 3\pi/2$
$x = \pi/4$
Asymptotes occur where $\cos x = 0$, i.e., $x = \pi/2 + k\pi$. $x = 3\pi/2 = \pi/2 + \pi$ qualifies. The other three give finite $\tan x$ values ($0, 0, 1$).渐近线发生在 $\cos x = 0$ 处，即 $x = \pi/2 + k\pi$。$x = 3\pi/2 = \pi/2 + \pi$ 满足。其余三个 $\tan x$ 有限（$0, 0, 1$）。
Asymptotes of $\tan x$ are the zeros of $\cos x$ , the odd multiples of $\pi/2$.$\tan x$ 的渐近线即 $\cos x$ 的零点 , $\pi/2$ 的奇数倍。
Section 7 · Transformations第 7 节 · 函数变换

Transformations: $y = A\sin(B(x - C)) + D$ Pre-Calc for US Alg 2变换：$y = A\sin(B(x - C)) + D$ US Alg 2 之上

The four-parameter form , what each letter does.四参数形式 , 各字母含义。 $$ y = A \sin\!\big(B(x - C)\big) + D. $$
  • $|A|$ = amplitude.= 振幅。 Half of $(\max - \min)$. If $A < 0$, the graph is reflected vertically.$(\max - \min)$ 的一半。若 $A < 0$，图像沿竖直方向反射。
  • $|B|$ = horizontal compression factor.= 水平压缩因子。 Period $= 2\pi/|B|$ for sine and cosine; period $= \pi/|B|$ for tangent.正余弦周期 $= 2\pi/|B|$；正切周期 $= \pi/|B|$。
  • $C$ = phase shift (horizontal shift).= 相位移（水平平移）。 $C > 0$ shifts right; $C < 0$ shifts left. Note the form requires $B$ factored out: $B(x - C)$ shifts the basic curve by $C$, not by $C/B$.$C > 0$ 右移；$C < 0$ 左移。注意形式要求 $B$ 已提到括号外：$B(x - C)$ 把基本曲线移动 $C$，而非 $C/B$。
  • $D$ = vertical shift (midline).= 竖直平移（中线）。 The new midline is $y = D$. The max is $D + |A|$ and the min is $D - |A|$.新中线为 $y = D$。最大值 $D + |A|$、最小值 $D - |A|$。
AB Math 30-1 indicator 4.7 says verbatim: "sketch, without technology, graphs of the form $y = a \sin b(x - c) + d$ or $y = a \cos b(x - c) + d$, using transformations." Indicator 4.8 names "amplitude, asymptotes, domain, period, phase shift, range and zeros" as the seven characteristics. Indicator 4.9 reverses the task , from a given graph, write the equation with the right $a, b, c, d$.AB Math 30-1 指标 4.7 原文："不借助技术，使用变换作 $y = a \sin b(x - c) + d$ 或 $y = a \cos b(x - c) + d$ 的图象"。指标 4.8 列出七项特征："振幅、渐近线、定义域、周期、相位移、值域、零点"。指标 4.9 反向操作 , 由给定图象写出 $a, b, c, d$ 对应的方程。
Worked Example 7 · Read $A, B, C, D$ off the equation; sketch one cycle例题 7 · 由方程读出 $A, B, C, D$ 并画一个周期

For $y = 3 \sin\!\big(2(x - \pi/4)\big) + 1$, find the amplitude, period, phase shift, vertical shift, midline, max, and min. Sketch one full cycle by listing the five key points.对 $y = 3 \sin\!\big(2(x - \pi/4)\big) + 1$，求振幅、周期、相位移、竖直平移、中线、最大、最小，并通过列出五个关键点画出一个完整周期。

Read parameters directly.直接读参数。

$$ A = 3, \;\; B = 2, \;\; C = \pi/4, \;\; D = 1. $$

Compute features.求各特征。

Amplitude $|A| = 3$. Period $= 2\pi/|B| = 2\pi/2 = \pi$. Phase shift $C = \pi/4$ (right). Vertical shift $D = 1$. Midline $y = 1$. Max $= 1 + 3 = 4$. Min $= 1 - 3 = -2$.振幅 $|A| = 3$；周期 $= 2\pi/|B| = 2\pi/2 = \pi$；相位移 $C = \pi/4$（右移）；竖直平移 $D = 1$；中线 $y = 1$；最大 $1 + 3 = 4$；最小 $1 - 3 = -2$。

Five key points.五个关键点。 Start the cycle at $x = C = \pi/4$ (where the basic sine starts at $(0, 0)$). One period covers $[\pi/4, \pi/4 + \pi] = [\pi/4, 5\pi/4]$. Divide into quarters of length $\pi/4$:周期起点 $x = C = \pi/4$（对应基本正弦的 $(0, 0)$）。一个周期覆盖 $[\pi/4, \pi/4 + \pi] = [\pi/4, 5\pi/4]$，每四分之一长 $\pi/4$：

$$ (\pi/4, 1), \;\; (\pi/2, 4), \;\; (3\pi/4, 1), \;\; (\pi, -2), \;\; (5\pi/4, 1). $$

Sanity-check.合理性核验。 All five $y$-values are on or between $-2$ and $4$ , consistent with max $4$ and min $-2$. The midline value $1$ recurs every $\pi/2$ , consistent with period $\pi$ (midline is hit twice per period).五个 $y$ 值均在 $-2$ 到 $4$ 之间 , 与最大 $4$、最小 $-2$ 一致。中线值 $1$ 每 $\pi/2$ 出现一次 , 与周期 $\pi$ 内每周期穿越中线两次一致。

What is the period of $y = 2 \cos(3x) - 5$?$y = 2 \cos(3x) - 5$ 的周期为？
§7 · Q1
$2\pi$
$\pi$
$2\pi/3$
$3\pi$
$B = 3$, so period $= 2\pi/|B| = 2\pi/3$. (The $A = 2$ stretches the curve vertically; the $-5$ shifts down; neither changes the period.)$B = 3$，故周期 $= 2\pi/|B| = 2\pi/3$。（$A = 2$ 竖直拉伸；$-5$ 下移；两者皆不改变周期。）
Only $B$ controls the period: period $= 2\pi/|B|$ for sine and cosine.仅 $B$ 决定周期：正余弦周期 $= 2\pi/|B|$。
The midline of $y = -4 \sin(x - \pi/6) + 7$ is which line?$y = -4 \sin(x - \pi/6) + 7$ 的中线为哪条直线？
§7 · Q2
$y = -4$
$y = 7$
$y = 4$
$y = \pi/6$
The midline is $y = D$, where $D$ is the vertical-shift constant. Here $D = 7$, so the midline is $y = 7$. (Amplitude is $|A| = 4$; max $= 11$, min $= 3$, both centred on $7$.)中线即 $y = D$，$D$ 为竖直平移常数。此处 $D = 7$，故中线 $y = 7$。（振幅 $|A| = 4$；最大 $11$、最小 $3$，均围绕 $7$ 对称。）
Midline = the constant $D$ in $y = A \sin(B(x - C)) + D$. The negative sign on $A$ flips the curve but does not change the midline.中线为 $y = A \sin(B(x - C)) + D$ 中的常数 $D$。$A$ 的负号只反射曲线，不改变中线。
Exam Tactics考试策略

Exam Strategy and Common Pitfalls考试策略与常见陷阱

Degree-radian discipline度-弧度纪律
  • Match the calculator mode to the question.计算器模式与题目一致。 If the angle reads $30^{\circ}$, use DEG. If it reads $\pi/6$, use RAD. Switching modes between MCR3U (degree) and MHF4U / Math 30-1 (radian) is the single most common error.题目写 $30^{\circ}$ 用 DEG；写 $\pi/6$ 用 RAD。MCR3U（角度）与 MHF4U / Math 30-1（弧度）之间切换是最常见的错误来源。
  • Convert via $\pi = 180^{\circ}$.用 $\pi = 180^{\circ}$ 换算。 Multiply by $\pi/180$ for degrees$\to$radians; multiply by $180/\pi$ for radians$\to$degrees. AB Math 30-1 indicators 1.5 / 1.6 expect exact values when the input is a standard multiple.度$\to$弧度乘 $\pi/180$；弧度$\to$度乘 $180/\pi$。AB Math 30-1 指标 1.5 / 1.6 要求标准倍数输入下给出精确值。
  • Coterminal works in both units.共终边在两种单位下都适用。 Add multiples of $360^{\circ}$ (or $2\pi$) to land any negative or out-of-range angle in $[0^{\circ}, 360^{\circ})$ (or $[0, 2\pi)$) first.先加 $360^{\circ}$（或 $2\pi$）的倍数把负角或超界角化到 $[0^{\circ}, 360^{\circ})$（或 $[0, 2\pi)$）。
Exact-value moves (§2-§3)精确值套路（§2-§3）
  • Three-step exact value.三步求精确值。 (1) Identify the quadrant. (2) Compute the reference angle. (3) Apply ASTC to attach a sign. AB Math 30-1 indicator 3.2 expects exactly this procedure.(1) 判象限；(2) 算参考角；(3) 用 ASTC 加符号。AB Math 30-1 指标 3.2 就要求此流程。
  • Reciprocals share signs with primaries.倒数与本原同号。 $\csc$ matches $\sin$; $\sec$ matches $\cos$; $\cot$ matches $\tan$. Computing the primary first and then taking the reciprocal avoids sign errors.$\csc$ 与 $\sin$ 同号；$\sec$ 与 $\cos$ 同号；$\cot$ 与 $\tan$ 同号。先求本原再取倒数可避免符号错误。
  • Pythagorean identity always works.勾股恒等式永远适用。 If you know one of $\sin / \cos$ and the quadrant, the other follows from $\sin^{2}\theta + \cos^{2}\theta = 1$ , no calculator needed.已知 $\sin / \cos$ 之一与象限时，另一个由 $\sin^{2}\theta + \cos^{2}\theta = 1$ 直接得出 , 无需计算器。
Graphing the four-parameter form (§4-§7)四参数图象（§4-§7）
  • Factor $B$ before reading the phase shift.读相位移前先把 $B$ 提出。 $y = \sin(2x - \pi/3)$ is not shifted by $\pi/3$. Rewrite as $y = \sin(2(x - \pi/6))$ to read $C = \pi/6$. Forgetting this step is the most common $C$ error.$y = \sin(2x - \pi/3)$ 不是右移 $\pi/3$；改写为 $y = \sin(2(x - \pi/6))$ 才知 $C = \pi/6$。漏掉此步是 $C$ 最常见错误。
  • Period formula is $2\pi/|B|$ (sine, cosine), $\pi/|B|$ (tangent).周期公式：sin、cos 为 $2\pi/|B|$；tan 为 $\pi/|B|$。 $B$ inside the function compresses the cycle, so $|B| > 1$ shortens the period.函数内的 $B$ 压缩周期，$|B| > 1$ 时周期变短。
  • Five key points = quartered period.五点 = 周期四等分。 Once you know period $T = 2\pi/|B|$ and start $x = C$, draw a small table $C, C + T/4, C + T/2, C + 3T/4, C + T$ and assign midline / max / midline / min / midline (or the cosine variant).已知周期 $T = 2\pi/|B|$ 与起点 $x = C$ 后，列表 $C, C + T/4, C + T/2, C + 3T/4, C + T$，对应中线 / 最大 / 中线 / 最小 / 中线（或余弦变体）。
  • Reverse the task.反向操作。 AB Math 30-1 indicator 4.9 expects you to read $A, B, C, D$ off a given graph: amplitude $= |A|$ from (max $-$ min)/$2$, $D$ from midline, $B$ from $2\pi/$period, $C$ from horizontal shift of the “start of cycle.”AB Math 30-1 指标 4.9 要求由图象读出 $A, B, C, D$：$|A| = (\max - \min)/2$、$D$ = 中线、$B = 2\pi/$周期、$C$ = "周期起点"的水平位移。
Sanity-check moves合理性核验套路
  • Pythagorean identity check.勾股恒等式验证。 After computing $\sin$ and $\cos$ at any angle, verify $\sin^{2} + \cos^{2} = 1$.求得 $\sin$、$\cos$ 后，验证 $\sin^{2} + \cos^{2} = 1$。
  • Sign by quadrant.分象限符号。 Before submitting any answer involving a non-QI angle, look at the unit-circle picture and ask: is the $x$ negative? is the $y$ negative? Does my answer's sign agree?对非 QI 角作答前，看单位圆问：$x$ 是否为负？$y$ 是否为负？答案符号是否一致？
  • Range of sin / cos.sin / cos 取值范围。 If a computation gives $|\sin\theta| > 1$ or $|\cos\theta| > 1$, you made a slip (almost always a degree-radian mix-up, or copying a side ratio for a non-unit-radius circle without dividing by $r$).若算出 $|\sin\theta| > 1$ 或 $|\cos\theta| > 1$，必有差错（多为度-弧度混淆，或在非单位圆上未除以 $r$）。
Active Recall主动复习

Flashcards闪卡

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Degree-radian conversion?度-弧度换算？
$$\pi \text{ rad} = 180^{\circ}$$ $$\theta_{\mathrm{rad}} = \theta_{\mathrm{deg}} \cdot \frac{\pi}{180}$$
Unit-circle definition of $\sin\theta, \cos\theta$?单位圆定义 $\sin\theta, \cos\theta$？
$$\cos\theta = x, \;\; \sin\theta = y$$ at the terminal-side point $(x, y)$ on the unit circle单位圆上终边点 $(x, y)$ 处
Sign by quadrant (ASTC)?象限符号（ASTC）？
QI all $+$; QII $\sin$ $+$; QIII $\tan$ $+$; QIV $\cos$ $+$. ("All Students Take Calculus.")QI 全 $+$；QII $\sin$ $+$；QIII $\tan$ $+$；QIV $\cos$ $+$。("All Students Take Calculus"。)
Pythagorean identity from the unit circle?单位圆推出的勾股恒等式？
$$\sin^{2}\theta + \cos^{2}\theta = 1$$ (equation of the unit circle in trig form)（单位圆方程的三角形式）
Reference angle in each quadrant?各象限参考角？
QI: $\theta$. QII: $\pi - \theta$. QIII: $\theta - \pi$. QIV: $2\pi - \theta$.QI: $\theta$；QII: $\pi - \theta$；QIII: $\theta - \pi$；QIV: $2\pi - \theta$。
$\sin, \cos, \tan$ at $\pi/3$?$\pi/3$ 处 $\sin, \cos, \tan$？
$$\tfrac{\sqrt{3}}{2}, \;\; \tfrac{1}{2}, \;\; \sqrt{3}$$
$\sin, \cos, \tan$ at $\pi/4$?$\pi/4$ 处 $\sin, \cos, \tan$？
$$\tfrac{\sqrt{2}}{2}, \;\; \tfrac{\sqrt{2}}{2}, \;\; 1$$
Reciprocal definitions?倒数定义？
$$\csc\theta = \tfrac{1}{\sin\theta}, \;\; \sec\theta = \tfrac{1}{\cos\theta}, \;\; \cot\theta = \tfrac{1}{\tan\theta}$$
Period of $\sin x, \cos x, \tan x$?$\sin x, \cos x, \tan x$ 的周期？
$$2\pi, \;\; 2\pi, \;\; \pi$$
Symmetry of $\sin x, \cos x, \tan x$?$\sin x, \cos x, \tan x$ 的对称性？
$\sin$ odd, $\cos$ even, $\tan$ odd.$\sin$ 奇、$\cos$ 偶、$\tan$ 奇。
Asymptotes of $\tan x$?$\tan x$ 的渐近线？
$$x = \tfrac{\pi}{2} + k\pi, \;\; k \in \mathbb{Z}$$ (zeros of $\cos x$)（$\cos x$ 的零点）
Four-parameter sinusoidal form?四参数正弦形式？
$$y = A \sin\!\big(B(x - C)\big) + D$$
Amplitude, period, phase, midline?振幅、周期、相位、中线？
$|A|$, $2\pi/|B|$ (sin/cos) or $\pi/|B|$ (tan), $C$, $y = D$.$|A|$；$2\pi/|B|$（sin/cos）或 $\pi/|B|$（tan）；$C$；$y = D$。
$\sin x \leftrightarrow \cos x$ shift?$\sin x$ 与 $\cos x$ 的相位关系？
$$\cos x = \sin\!\big(x + \tfrac{\pi}{2}\big)$$ (cosine = sine shifted left by $\pi/2$)（余弦 = 正弦左移 $\pi/2$）
Mixed Practice综合练习

Practice Quiz综合测验

Convert $\theta = -\pi/3$ to degrees.把 $\theta = -\pi/3$ 化为角度。
Q1
$-90^{\circ}$
$-60^{\circ}$
$-30^{\circ}$
$-120^{\circ}$
Multiply by $180/\pi$: $(-\pi/3) \cdot 180/\pi = -60^{\circ}$.乘 $180/\pi$：$(-\pi/3) \cdot 180/\pi = -60^{\circ}$。
$\pi = 180^{\circ}$, so $\pi/3 = 60^{\circ}$. Negative angles keep the magnitude.$\pi = 180^{\circ}$，故 $\pi/3 = 60^{\circ}$。负角度保持绝对值不变。
Find the exact value of $\sin(5\pi/4)$.求 $\sin(5\pi/4)$ 的精确值。
Q2
$\sqrt{2}/2$
$\sqrt{3}/2$
$-\sqrt{2}/2$
$-1/2$
$5\pi/4 = 225^{\circ}$, QIII. Reference $= 5\pi/4 - \pi = \pi/4$. $\sin(\pi/4) = \sqrt{2}/2$. QIII makes sine negative: $\sin(5\pi/4) = -\sqrt{2}/2$.$5\pi/4 = 225^{\circ}$，QIII。参考 $= 5\pi/4 - \pi = \pi/4$。$\sin(\pi/4) = \sqrt{2}/2$。QIII 中正弦为负：$\sin(5\pi/4) = -\sqrt{2}/2$。
Quadrant first (QIII, $\sin -$), then reference-angle value ($\sqrt{2}/2$), then attach the sign.先判象限（QIII，$\sin -$），再算参考角值（$\sqrt{2}/2$），最后加符号。
If $\sin\theta = -3/5$ and $\theta$ is in QIII, find $\cos\theta$ as an exact fraction.若 $\sin\theta = -3/5$ 且 $\theta$ 在 QIII，求 $\cos\theta$ 的精确分数。
Q3
$-4/5$
$4/5$
$-3/4$
$3/4$
$\cos^{2}\theta = 1 - 9/25 = 16/25$, so $|\cos\theta| = 4/5$. QIII makes $\cos$ negative, so $\cos\theta = -4/5$.$\cos^{2}\theta = 1 - 9/25 = 16/25$，故 $|\cos\theta| = 4/5$。QIII 中 $\cos$ 为负，故 $\cos\theta = -4/5$。
Pythagorean identity gives the magnitude; the quadrant gives the sign.勾股恒等式给绝对值，象限给符号。
Find the exact value of $\tan(4\pi/3)$.求 $\tan(4\pi/3)$ 的精确值。
Q4
$-\sqrt{3}$
$-\sqrt{3}/3$
$\sqrt{3}$
$1$
$4\pi/3 = 240^{\circ}$, QIII. Reference $= 4\pi/3 - \pi = \pi/3$. $\tan(\pi/3) = \sqrt{3}$. QIII makes tan positive (ASTC), so $\tan(4\pi/3) = \sqrt{3}$.$4\pi/3 = 240^{\circ}$，QIII。参考 $= 4\pi/3 - \pi = \pi/3$。$\tan(\pi/3) = \sqrt{3}$。QIII 中 tan 为正（ASTC），故 $\tan(4\pi/3) = \sqrt{3}$。
Tangent is positive in QI and QIII. Reference angle $\pi/3$ gives magnitude $\sqrt{3}$.正切在 QI、QIII 为正。参考角 $\pi/3$ 给绝对值 $\sqrt{3}$。
Find the exact value of $\csc(7\pi/4)$.求 $\csc(7\pi/4)$ 的精确值。
Q5
$\sqrt{2}$
$2$
$-2$
$-\sqrt{2}$
$7\pi/4 = 315^{\circ}$, QIV. Reference $\pi/4$, so $|\sin| = \sqrt{2}/2$ and $\sin(7\pi/4) = -\sqrt{2}/2$ (QIV, $\sin -$). Then $\csc = 1/\sin = -2/\sqrt{2} = -\sqrt{2}$.$7\pi/4 = 315^{\circ}$，QIV。参考 $\pi/4$，$|\sin| = \sqrt{2}/2$，$\sin(7\pi/4) = -\sqrt{2}/2$（QIV，$\sin -$）。故 $\csc = 1/\sin = -2/\sqrt{2} = -\sqrt{2}$。
Compute sine first, then take the reciprocal and rationalise. QIV makes sine (and hence csc) negative.先求 sin 再取倒数并有理化。QIV 中 sin（与 csc）为负。
What is the amplitude of $y = -5 \sin(2x + \pi) + 3$?$y = -5 \sin(2x + \pi) + 3$ 的振幅为？
Q6
$-5$
$5$
$3$
$2$
Amplitude is $|A|$, never negative. Here $A = -5$, so amplitude $= 5$. The negative sign flips the curve vertically but does not change amplitude.振幅为 $|A|$，永不为负。此处 $A = -5$，故振幅 $= 5$。负号竖直反射曲线，但不改变振幅。
Amplitude $= |A|$. The negative makes the curve a reflection of the basic sine.振幅 $= |A|$。负号使曲线相对基本正弦做竖直反射。
What is the period of $y = \tan(\pi x / 4)$?$y = \tan(\pi x / 4)$ 的周期为？
Q7
$\pi$
$2$
$4$
$8$
Tangent period $= \pi/|B|$. Here $B = \pi/4$, so period $= \pi / (\pi/4) = 4$.正切周期 $= \pi/|B|$。此处 $B = \pi/4$，故周期 $= \pi / (\pi/4) = 4$。
Tangent uses $\pi/|B|$, not $2\pi/|B|$. Plug $B = \pi/4$ directly.正切用 $\pi/|B|$，非 $2\pi/|B|$。代入 $B = \pi/4$。
A Ferris wheel of radius $20$ m has its centre $25$ m above the ground and completes one revolution every $60$ seconds. A rider's height $h(t)$ (in metres) above the ground as a function of time $t$ (in seconds, starting at the lowest point) is which of the following? 🇺🇸 HSF-TF.B.5某摩天轮半径 $20$ m，圆心距地 $25$ m，每 $60$ 秒转一圈。乘客从最低点开始计时，距地高度 $h(t)$（米，$t$ 为秒）应取下列哪一式？🇺🇸 HSF-TF.B.5
Q8
$h(t) = 20 \sin(\pi t / 30) + 25$
$h(t) = -20 \cos(\pi t / 30) + 25$
$h(t) = 25 \cos(\pi t / 30) + 20$
$h(t) = 20 \cos(\pi t / 60) + 25$
Midline $D = 25$ (centre height). Amplitude $|A| = 20$ (radius). Period $60$ s gives $B = 2\pi/60 = \pi/30$. At $t = 0$, $h = 25 - 20 = 5$ m (lowest). $-\cos(0) = -1$ delivers that, so $h(t) = -20 \cos(\pi t / 30) + 25$.中线 $D = 25$（圆心高）；振幅 $|A| = 20$（半径）；周期 $60$ s 给 $B = 2\pi/60 = \pi/30$。$t = 0$ 时 $h = 25 - 20 = 5$ m（最低）。$-\cos(0) = -1$ 恰能做到，故 $h(t) = -20 \cos(\pi t / 30) + 25$。
Identify centre, radius, period, and start-position in turn. The "starts at lowest point" condition pins down the sign on the cosine.依次定圆心、半径、周期、起点。"从最低点开始"决定 cos 前符号。
Two coterminal angles for $\theta = 11\pi/6$ are: 🇨🇦 AB Math 30-1 ind. 1.7$\theta = 11\pi/6$ 的两个共终边角为：🇨🇦 AB Math 30-1 指标 1.7
Q9
$-\pi/6$ and与 $23\pi/6$
$\pi/6$ and与 $-23\pi/6$
$5\pi/6$ and与 $-7\pi/6$
$11\pi/3$ and与 $-11\pi/3$
Coterminal angles differ by integer multiples of $2\pi$. $11\pi/6 - 2\pi = -\pi/6$, and $11\pi/6 + 2\pi = 11\pi/6 + 12\pi/6 = 23\pi/6$. Both share the QIV terminal side.共终边角相差 $2\pi$ 的整数倍。$11\pi/6 - 2\pi = -\pi/6$；$11\pi/6 + 2\pi = 23\pi/6$。两者终边同在 QIV。
Add or subtract $2\pi = 12\pi/6$ to land coterminal candidates.加减 $2\pi = 12\pi/6$ 即得共终边候选。
A graph has midline $y = -2$, amplitude $4$, period $\pi$, and a maximum at $x = \pi/4$. Write the equation in the form $y = A \cos(B(x - C)) + D$. 🇨🇦 AB Math 30-1 ind. 4.9某图象中线 $y = -2$、振幅 $4$、周期 $\pi$、$x = \pi/4$ 处取最大。以 $y = A \cos(B(x - C)) + D$ 写出方程。🇨🇦 AB Math 30-1 指标 4.9
Q10
$y = 4 \cos(\pi(x - \pi/4)) - 2$
$y = 4 \cos(2(x - \pi/4)) - 2$
$y = -4 \cos(2(x - \pi/4)) - 2$
$y = 4 \cos(2x - \pi/4) - 2$
$A = 4, D = -2$ from amplitude and midline. $B = 2\pi/\pi = 2$ from period. Cosine reaches its max where its argument is zero, so $B(x - C) = 0$ at $x = \pi/4$ gives $C = \pi/4$. Result: $y = 4 \cos(2(x - \pi/4)) - 2$. (Option (d) looks similar but forgets to factor $B$ , $\cos(2x - \pi/4) = \cos(2(x - \pi/8))$, wrong phase.)由振幅与中线 $A = 4, D = -2$；由周期 $B = 2\pi/\pi = 2$。余弦在自变量为零处取最大，故 $B(x - C) = 0$ 在 $x = \pi/4$ 给 $C = \pi/4$。结果 $y = 4 \cos(2(x - \pi/4)) - 2$。（选项 (d) 形式相近但未把 $B$ 提出：$\cos(2x - \pi/4) = \cos(2(x - \pi/8))$，相位错。）
Always factor $B$ out before reading $C$. Cosine maxes at argument zero; sine starts at midline going up.读 $C$ 前先把 $B$ 提出。余弦在自变量为零处取最大；正弦在中线处向上启动。
Self-check自查

Readiness Checklist准备就绪清单

Tick each item when you can do it cold, without notes, on a first attempt.能在无笔记、首次尝试下完成，再勾选每一项。

0 / 12 mastered已掌握 0 / 12
Next Steps后续单元

What This Feeds Into本单元的去向

The unit circle is the hinge between geometry-style trig (Right-Triangle Trigonometry) and analysis-style trig (Trigonometric Identities onward). Every later use of sine, cosine, and tangent , in calculus derivatives, in Fourier-style signal modelling, in IB Math AA HL's complex-number polar form , assumes you have the radian-mode, all-real-angle picture solid. The cross-references below point at units already shipped in this repo.单位圆是几何式三角学（直角三角形三角学）与分析式三角学（三角恒等式起）之间的枢纽。后续所有正弦、余弦、正切的使用 , 微积分导数、傅里叶式信号建模、IB Math AA HL 复数极形式 , 都默认你已牢牢掌握"弧度制 + 全体实数角"的图景。下方链接指向本仓库已有的相关单元。

Within High School Math.在 HS Math 内部。

The Right-Triangle Trigonometry guide supplies the SOH CAH TOA values that re-appear at the QI special angles here. Trigonometric Identities and Equations builds on the Pythagorean identity from §2 and the reciprocal definitions from §3 to develop sum-difference and double-angle identities and the trig-equation toolkit. Function Transformations and Composition uses the $y = A \sin(B(x - C)) + D$ pattern from §7 as a base case for transformations of arbitrary $y = f(x)$. The Vectors guide decomposes a vector into components via $\cos\theta$ and $\sin\theta$ , the same unit-circle reading you mastered here.直角三角形三角学指南提供的 SOH CAH TOA 值，本单元在 QI 特殊角处重现。三角恒等式与方程指南基于本 §2 勾股恒等式与 §3 倒数定义，发展和差与倍角恒等式以及三角方程工具组。函数变换与复合指南以本 §7 的 $y = A \sin(B(x - C)) + D$ 模式为基础，扩展到任意 $y = f(x)$ 的变换。向量指南用 $\cos\theta$、$\sin\theta$ 分解分量 , 正是本单元的单位圆读法。

Across the AP and IB feeders in this repo.本仓库中的 AP 与 IB 衔接单元。

IB Math HL B4 · Trigonometric Functions (sin, cos, tan on the real line; transformations $A \sin(B(x - C)) + D$ at HL depth)IB Math HL B4 · 三角函数（sin、cos、tan 在实数上；HL 级变换 $A \sin(B(x - C)) + D$） IB Math HL C2 · Trigonometry Applications (radian-mode applications)IB Math HL C2 · 三角学应用（弧度制应用） AP Calculus Unit 2 · Differentiation (derivatives of $\sin x, \cos x$ assume radians + unit-circle fluency)AP Calculus Unit 2 · 微分（$\sin x, \cos x$ 的导数默认弧度 + 单位圆熟练） AP Physics Unit 6 · Simple Harmonic Motion ($x(t) = A \cos(\omega t + \phi)$ is the §7 form with $\omega = B$)AP Physics Unit 6 · 简谐运动（$x(t) = A \cos(\omega t + \phi)$ 即本 §7 形式，$\omega = B$）

If you are aiming for AP Pre-Calc, the radian + unit-circle + transformations stack here is Unit 3 of the AP Pre-Calc course in concentrated form. If you are aiming for AP Calculus, fluent recall of $\sin, \cos$ at multiples of $\pi/6$ and $\pi/4$ in radians is required from the first day. For IB Math AA HL, Topic B4 picks up directly from here with deeper identity work and the function $f(x) = a \sin(b(x - c)) + d$ as a modelling tool.备考 AP Pre-Calc：本单元"弧度 + 单位圆 + 变换"组合即 AP Pre-Calc Unit 3 的浓缩。备考 AP Calculus：第一天就要求熟练默写 $\pi/6, \pi/4$ 倍数处的 $\sin, \cos$（弧度制）。备考 IB Math AA HL：Topic B4 正是从本处接续，深入恒等式工作并把 $f(x) = a \sin(b(x - c)) + d$ 作为建模工具。

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High School Math · Unit-Circle Trigonometry and Trigonometric Functions · v1高中数学 · 单位圆三角学与三角函数 · v1