📚 Exercise 22A: Trigonometric Graphs & Identities | 练习22A：三角函数图像与恒等式
This article provides a detailed walkthrough for Exercise 22A, a typical IB Mathematics: Analysis and Approaches (SL/HL) problem set focusing on trigonometric graphs, the unit circle, and core identities. We will break down key concepts and model solutions to help you master the skills required for Paper 1 and Paper 2 questions. Each example is explained with step-by-step reasoning, followed by the same explanation in Chinese. By the end, you will be able to sketch transformed sine and cosine curves, prove fundamental identities, and solve trigonometric equations efficiently.
本文为练习22A提供详尽解析，这是IB数学分析与方法（SL/HL）中典型的习题集，重点涵盖三角函数图像、单位圆及核心恒等式。我们将分解关键概念并给出范例解答，帮助您掌握卷一和卷二题目所需的技能。每个例题均以逐步推理的方式呈现，并附上等同的中文解释。阅读结束后，您将能够熟练绘制经变换的正弦和余弦曲线、证明基本恒等式，并高效求解三角方程。
1. Radian Measure and the Unit Circle | 弧度制与单位圆
Radian measure is the foundation of all calculus-based trigonometry. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. This means a full revolution (360°) equals 2π radians. In Exercise 22A, you must often convert between degrees and radians. The unit circle is a circle of radius 1 centred at the origin. Its equation is x² + y² = 1, and because any point (cos θ, sin θ) lies on the circle, we immediately obtain the identity sin²θ + cos²θ = 1. Understanding the unit circle allows you to recognise the signs of sine, cosine, and tangent in each quadrant effortlessly.
弧度制是所有涉及微积分的三角学的基础。一弧度是指圆心角所对的弧长等于半径时所对应的圆心角。这意味着一整圈（360°）等于2π弧度。在练习22A中，您经常需要在度与弧度之间进行转换。单位圆是圆心在原点、半径为1的圆。其方程为x² + y² = 1，又因任意点(cos θ, sin θ)均落在该圆上，我们立刻得到恒等式sin²θ + cos²θ = 1。理解单位圆能让你毫不费力地判断正弦、余弦和正切在各个象限中的符号。
2. Special Angles and Exact Values | 特殊角及其精确值
IB examiners expect you to recall exact values for sine, cosine, and tangent at 0, π/6, π/4, π/3, π/2, and their multiples. These values derive from the 30-60-90 and 45-45-90 triangles. For instance, sin(π/6) = 1/2, tan(π/4) = 1, and cos(π/3) = 1/2. Exercise 22A often tests whether you can apply these without a calculator. Using the unit circle, you can extend these values to angles in radians such as 5π/6 or 7π/4. Always simplify fractions and rationalise denominators when needed.
IB考官希望你能记住0、π/6、π/4、π/3、π/2及其整数倍角的正弦、余弦和正切的精确值。这些值源自30-60-90和45-45-90三角形。例如，sin(π/6) = 1/2，tan(π/4) = 1，cos(π/3) = 1/2。练习22A经常检验你能否在不使用计算器的情况下直接应用这些值。借助单位圆，你能将这些值推广至弧度制下的角，如5π/6或7π/4。需要时务必化简分数并有理化分母。
3. Graphs of y = sin x, y = cos x, and y = tan x | y = sin x、y = cos x 与 y = tan x 的图像
The base sine curve has amplitude 1, period 2π, and passes through the origin. The cosine curve is a horizontal translation of the sine curve: cos x = sin(x + π/2). The tangent function has period π and vertical asymptotes at x = π/2 + kπ. These parent graphs are the starting point for all transformations in Exercise 22A. Always label key points such as intercepts, maxima, minima, and asymptotes clearly. For tan x, note that its range is R, but its domain excludes the asymptote values.
基本正弦曲线的振幅为1，周期为2π，且经过原点。余弦曲线可视为正弦曲线向左平移π/2：cos x = sin(x + π/2)。正切函数的周期为π，并在x = π/2 + kπ处存在铅直渐近线。这些母图是练习22A中所有图像变换的起点。请务必清晰地标出截距、最大值、最小值和渐近线等关键点。对于tan x，注意其值域为全体实数，但定义域不包含渐近线处的值。
4. Transformations: Amplitude, Period, and Phase Shift | 变换：振幅、周期与相移
A general sine function can be written as y = a sin(bx + c) + d. Here |a| is the amplitude, 2π / |b| is the period, -c/b is the phase shift, and d is the vertical shift. In Exercise 22A you are asked to sketch curves such as y = 3 sin(2x – π/4) + 1. Start by identifying the sequence of transformations: horizontal compression by factor 1/2, phase shift to the right by π/8, vertical stretch by factor 3, and vertical translation up by 1. Plot the five key points of the transformed cycle to ensure accuracy.
一般的正弦函数可写为 y = a sin(bx + c) + d 的形式。其中 |a| 是振幅，2π / |b| 是周期，-c/b 是相移，d 是垂直平移。练习22A要求你绘制如 y = 3 sin(2x – π/4) + 1 之类的曲线。首先识别变换的顺序：水平方向压缩为原来的1/2，向右平移π/8，垂直方向拉伸为原来的3倍，再向上平移1。标出变换后一个周期内的五个关键点，以确保图形准确。
5. Core Trigonometric Identities | 核心三角恒等式
The two foundational identities are sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ. From these, we derive others like 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. Exercise 22A typically includes proving or using identities to simplify expressions. For example, to prove (sin θ + cos θ)² = 1 + sin 2θ, expand the left side as sin²θ + 2 sin θ cos θ + cos²θ, then apply the Pythagorean identity and the double-angle formula sin 2θ = 2 sin θ cos θ. The key is to work on one side until it matches the other.
两个最基本的恒等式是 sin²θ + cos²θ = 1 和 tan θ = sin θ / cos θ。由此可推导出 1 + tan²θ = sec²θ 和 1 + cot²θ = csc²θ 等恒等式。练习22A通常要求证明或运用恒等式来化简表达式。例如，要证明 (sin θ + cos θ)² = 1 + sin 2θ，可展开左边得到 sin²θ + 2 sin θ cos θ + cos²θ，然后应用勾股恒等式及倍角公式 sin 2θ = 2 sin θ cos θ。关键是从一边入手，逐步变形至与另一边一致。
6. Solving Basic Trigonometric Equations | 求解基本三角方程
To solve an equation like sin x = 1/2 for 0 ≤ x < 2π, recall that sin x is positive in the first and second quadrants. The reference angle is π/6. Thus the solutions are x = π/6 and x = π - π/6 = 5π/6. When the coefficient of x is not 1, such as in cos(2x) = √3/2, first solve for the angle inside the bracket: 2x = π/6 + 2kπ or 2x = 11π/6 + 2kπ. Then divide and consider the given domain. Always check that all solutions fall within the required interval.
要解如 sin x = 1/2 且 0 ≤ x < 2π 的方程，应想到 sin x 在第一象限和第二象限为正。参考角为 π/6。因此解为 x = π/6 及 x = π - π/6 = 5π/6。当 x 的系数不为1时，例如 cos(2x) = √3/2，先解出括号内的角：2x = π/6 + 2kπ 或 2x = 11π/6 + 2kπ。然后除以系数，并结合给定区间进行取舍。务必检查所有解是否都在指定范围内。
7. Worked Example: Proving an Identity | 范例：证明恒等式
Question: Prove that (1 – cos 2θ) / (sin 2θ) = tan θ.
Solution (English): Start with the left side. Use the double-angle identities cos 2θ = 1 – 2 sin²θ and sin 2θ = 2 sin θ cos θ. Then numerator becomes 1 – (1 – 2 sin²θ) = 2 sin²θ. The fraction turns into (2 sin²θ) / (2 sin θ cos θ) = sin θ / cos θ = tan θ, which equals the right side. Therefore the identity is proved. This is a classic Exercise 22A problem, often used to strengthen algebraic manipulation with trigonometric forms.
问题：证明 (1 – cos 2θ) / (sin 2θ) = tan θ。
解答（中文）：从左边入手。利用倍角公式 cos 2θ = 1 – 2 sin²θ 和 sin 2θ = 2 sin θ cos θ。分子变为 1 – (1 – 2 sin²θ) = 2 sin²θ。整个分式变成 (2 sin²θ) / (2 sin θ cos θ) = sin θ / cos θ = tan θ，与右边相等。因此恒等式得证。这是一道经典的练习22A题目，常用于强化三角表达式中的代数操作。
8. Worked Example: Equation with Transformations | 范例：含变换的方程
Question: Solve 2 cos²x + 3 sin x = 0 for 0 ≤ x ≤ 2π.
Solution (English): Use the identity cos²x = 1 – sin²x to rewrite the equation as 2(1 – sin²x) + 3 sin x = 0. Simplify to -2 sin²x + 3 sin x + 2 = 0, or multiply by -1: 2 sin²x – 3 sin x – 2 = 0. Let u = sin x. Then 2u² – 3u – 2 = 0. Factorising gives (2u + 1)(u – 2) = 0, so u = -1/2 or u = 2. Since sin x = 2 has no real solution, we take sin x = -1/2. Sine is negative in quadrants III and IV. The reference angle is π/6. Solutions in the given interval are x = 7π/6 and x = 11π/6.
问题：求解 2 cos²x + 3 sin x = 0，其中 0 ≤ x ≤ 2π。
解答（中文）：利用恒等式 cos²x = 1 – sin²x 将方程改写为 2(1 – sin²x) + 3 sin x = 0。化简得 -2 sin²x + 3 sin x + 2 = 0，或乘以 -1：2 sin²x – 3 sin x – 2 = 0。设 u = sin x，得到 2u² – 3u – 2 = 0。因式分解为 (2u + 1)(u – 2) = 0，故 u = -1/2 或 u = 2。sin x = 2 无实数解，因此取 sin x = -1/2。正弦值为负出现在第三、四象限。参考角为 π/6。给定区间内的解为 x = 7π/6 和 x = 11π/6。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Many students lose marks by forgetting to check the domain or by giving answers in the wrong quadrant. In Exercise 22A, always verify that your solutions satisfy the original equation, especially when squaring both sides has been used. Another frequent error is mishandling the period when dealing with compound angles: for sin(3x), the period is 2π/3, so you must list all solutions up to the given boundary. Additionally, when proving identities, do not move terms across the ‘equals’ sign; work exclusively on one side. And for graph sketching, mixing up the phase shift direction is common: y = sin(x + c) shifts to the left by c, not to the right.
许多学生因忘记检查定义域或给出错误象限的解而失分。在练习22A中，务必验证你的解是否满足原方程，尤其是当使用了方程两边平方的操作时。另一个常见错误是处理复合角时弄错周期：对于 sin(3x)，周期为 2π/3，因此必须在给定区间内列出所有解。此外，证明恒等式时，不要将项移过等号；只对一边进行操作。对于图像绘制，混淆相移方向也很普遍：y = sin(x + c) 是向左平移 c 个单位，而非向右。
10. Strategy for Tackling Exercise 22A Efficiently | 高效攻克练习22A的策略
Begin with a quick review of exact values and graph shapes. When given a problem, decide immediately whether it is an identity, equation, or graphing task. For identities, underline the side that appears more complex and transform it using substitution. For equations, isolate the trigonometric function and use the unit circle. For graphs, mark the amplitude, period, and phase shift before plotting any points. Time management is key: if a proof is taking too long, check whether a simpler identity can be applied. Practise with past-paper questions that mimic the structure of Exercise 22A, and always self-correct using the mark scheme to internalise the required level of detail.
首先快速复习精确值和图像形状。拿到题目后，立即判断它是恒等式、方程还是作图题。对于恒等式，勾画出看起来更复杂的一边，并运用代换进行变形。对于方程，先分离出三角函数部分，然后利用单位圆求解。对于图像，先标出振幅、周期和相移，再开始描点。时间管理至关重要：若一项证明耗时过长，检查一下是否有更简单的恒等式可用。多用与练习22A结构相似的往年真题进行练习，并始终对照评分方案自我批改，以将答题所需的详细程度内化于心。
11. Extensions: Beyond the Standard Set | 拓展：超越标准题集
Stronger students can explore how Exercise 22A concepts connect to calculus. For instance, differentiating y = sin(2x) using the chain rule requires recognising the inner function 2x, directly linked to period change. Graphs of trigonometric functions are the basis for modelling periodic behaviour in physics, such as simple harmonic motion. The identity sin²θ = (1 – cos 2θ)/2 is a power-reducing formula essential for integration in HL. Understanding the symmetrical properties from the unit circle also lays the groundwork for complex numbers and Euler’s formula e^(iθ) = cos θ + i sin θ. A deep grasp now will pay dividends across the IB syllabus.
学有余力的同学可以探索练习22A的概念如何与微积分相关联。例如，用链式法则对 y = sin(2x) 求导需要识别内层函数2x，这与周期变化直接相关。三角函数图像是物理中简谐运动等周期现象建模的基础。恒等式 sin²θ = (1 – cos 2θ)/2 是降幂公式，对HL阶段的积分至关重要。理解单位圆所呈现的对称性质也为复数及欧拉公式 e^(iθ) = cos θ + i sin θ 打下了基础。此时深入掌握，日后在IB全课程中将收益颇丰。
12. Final Summary and Practice Tips | 总结与练习提示
Exercise 22A is designed to build fluency in handling trigonometric functions without a calculator. Revise radian measures, exact values, and graph transformations daily in short bursts. Create a one-page summary sheet containing the unit circle, key identities, and transformation rules. Attempt the exercise under timed conditions, then review every mistake explicitly. Remember that the marking scheme rewards clear working, so always show the substitution steps when solving equations and label your graphs fully. With consistent practice, the techniques demonstrated here will become second nature, raising both your confidence and your final grade.
练习22A旨在培养在不使用计算器的情况下熟练处理三角函数的能力。每天利用短时间复习弧度制、精确值及图像变换。制作一张包含单位圆、关键恒等式和变换规则的单页总结。在限时条件下完成练习，然后明确地回顾每一处错误。请记住，评分方案会给清晰的解题步骤以奖励，因此在解方程时务必写出代换步骤，并完整标注图像。通过持续练习，本文展示的技巧将成为你的第二天性，同时提升你的信心和最终成绩。
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