Exercise 22D: Solving Trigonometric Equations | 练习22D：解三角方程

📚 Exercise 22D: Solving Trigonometric Equations | 练习22D：解三角方程

These exercises are designed to build fluency in solving equations that involve sine, cosine, tangent, and their reciprocal functions. Whether you are working within a specified interval or finding the general solution, the techniques in Exercise 22D require a solid command of trigonometric identities, the unit circle, and algebraic manipulation. The following revision guide walks you through the key concepts and provides a structured approach to each problem type you will encounter, helping you avoid common pitfalls and gain confidence for your IB Mathematics exams.

这组练习旨在培养你解包含正弦、余弦、正切及其倒数函数的方程的能力。无论你是在指定区间内求解还是寻找通解，练习22D中的技巧都需要你牢牢掌握三角恒等式、单位圆和代数处理。下面的复习指南将带你梳理关键概念，并为每类问题提供系统的解题思路，帮助你避开常见陷阱，在IB数学考试中建立信心。

1. Understanding the Purpose of Exercise 22D | 理解练习22D的目的

This exercise set typically consolidates earlier work on trigonometric functions and identities by applying them to equations. It tests your ability to recognise equation types, choose appropriate identities, and work with angular measures in both degrees and radians. You are expected to produce exact answers using special angles and to extend your reasoning to an infinite set of solutions when required.

这套练习通常通过将方程应用于之前学过的三角函数和恒等式来进行巩固。它考查你识别方程类型、选择合适的恒等式以及同时使用角度制和弧度制运算的能力。你需要利用特殊角给出精确答案，并在需要时将推理扩展到无穷多组解。

Many problems in Exercise 22D mirror Paper 1 and Paper 2 questions, particularly those that ask you to “solve for x in [0, 2π]” or to find the general solution. The drill is not only about getting the right answer but also about presenting your working clearly, which is a core skill assessed in IB Mathematics.

练习22D中的许多题目与试卷一和试卷二的考题相似，尤其是那些要求“在[0, 2π]内求解x”或求通解的题目。这项训练不仅是为了得出正确答案，还在于清晰地呈现解题过程，这也是IB数学评估的核心技能之一。

2. Essential Trigonometric Identities to Review | 必须回顾的三角恒等式

Before tackling any equation, recall the fundamental identities. The Pythagorean identities, double‑angle formulas, and reciprocal relations are your main tools. For Exercise 22D you should have these at your fingertips:

在着手解任何方程之前，先回想基本恒等式。勾股恒等式、倍角公式和倒数关系是你的主要工具。对于练习22D，你应该熟练掌握以下内容：

sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
1 + cot²θ = csc²θ
sin(2θ) = 2 sinθ cosθ
cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ
tanθ = sinθ / cosθ
cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ

Sometimes the equation will be given in terms of a reciprocal function, such as csc x = 2. Convert it straight away to 1/sin x = 2, which gives sin x = 1/2. This simple trick prevents confusion and allows you to use the same solving strategy for all sine and cosine equations.

有时方程会以倒数函数的形式给出，如 csc x = 2。应立即将其转化为 1/sin x = 2，即 sin x = 1/2。这个简单的技巧可以避免混淆，并让你对所有正弦和余弦方程使用相同的求解策略。

3. Solving Linear Trigonometric Equations | 解线性三角方程

A linear trigonometric equation looks like a sin x + b = c or cos x = k. The first step is always to isolate the trigonometric function. For instance, to solve 2 sin x + 1 = 0, you would rearrange to sin x = -1/2. Once isolated, use the unit circle or special angle values to locate the principal solutions.

线性三角方程的形式如 a sin x + b = c 或 cos x = k。第一步始终是分离出三角函数。例如，要解 2 sin x + 1 = 0，你会将式子变形为 sin x = -1/2。完成分离后，利用单位圆或特殊角值来确定主值解。

In an interval such as [0, 2π], you then find all angles whose reference angle is π/6 and whose sine is negative. The solutions are in the third and fourth quadrants: π + π/6 = 7π/6 and 2π – π/6 = 11π/6. Always sketch a quick unit‑circle diagram to avoid missing a quadrant.

在诸如[0, 2π]的区间内，你需要找出所有参考角为π/6且正弦为负的角。这些解在第三和第四象限：π + π/6 = 7π/6 以及 2π – π/6 = 11π/6。始终快速画一个单位圆示意图，以免漏掉某个象限。

4. Using the Unit Circle to Find All Solutions | 利用单位圆求全部解

The unit circle is the fastest way to generate all angles that satisfy an equation in a given interval. For sin θ = 1/2, you immediately recognise that the principal angle is π/6 in Quadrant I, and the supplementary solution is π – π/6 = 5π/6 in Quadrant II. For cosine equations, the second solution lies in the fourth quadrant, giving θ = 2π – reference angle.

单位圆是生成给定区间内所有满足方程的角的最快方式。对于 sin θ = 1/2，你能立即认出第一象限的主角是π/6，第二象限的补角解是π – π/6 = 5π/6。对于余弦方程，第二个解位于第四象限，即 θ = 2π – 参考角。

When the interval is given in degrees, such as 0° ≤ x ≤ 360°, you follow the same logic: reference angle 30° gives solutions 30° and 150° for sine, or 30° and 330° for cosine. Practice converting between degrees and radians quickly, as IB questions often mix the two.

当区间以度为单位时，如0° ≤ x ≤ 360°，遵循同样的逻辑：参考角30°对于正弦给出解30°和150°，对于余弦则给出30°和330°。要练习快速进行度与弧度的转换，因为IB考试题常常混合使用两者。

5. Writing General Solutions Using π-notation | 用π记号书写通解

IB exam questions sometimes ask for the general solution, which expresses all possible angles over the real numbers. For a sine equation sin x = k, the general solution is written as x = nπ + (-1)ⁿα, where α is the principal angle and n ∈ ℤ. For cosine, the form is x = 2nπ ± α. For tangent, because the period is π, the general solution is simply x = nπ + α.

IB考试题有时会要求写出通解，即表达实数范围内的所有可能角度。对于正弦方程 sin x = k，通解写作 x = nπ + (-1)ⁿα，其中α为主角，n ∈ ℤ。对于余弦，形式为 x = 2nπ ± α。对于正切，由于其周期为π，通解仅为 x = nπ + α。

Using these compact forms is much cleaner than listing every solution. For instance, the general solution of sin x = 1/2 is x = nπ + (-1)ⁿ(π/6). Make sure you define n as an integer. This notation is especially efficient when you need to combine solutions or work with double angles.

使用这些紧凑形式比列出每个解要整洁得多。例如，sin x = 1/2的通解是 x = nπ + (-1)ⁿ(π/6)。务必注明n是整数。当你需要合并解或处理倍角时，这种记法尤为高效。

6. Tackling Quadratic Trigonometric Equations | 解决二次三角方程

Some equations appear as quadratics in sin x, cos x, or tan x, for example 2 sin²x – sin x – 1 = 0. Treat the trigonometric function as a variable, say u = sin x, and factor or use the quadratic formula. You get u = 1 or u = -1/2. Then solve sin x = 1 and sin x = -1/2 separately within the interval.

有些方程表现出关于 sin x、cos x 或 tan x 的二次形式，例如 2 sin²x – sin x – 1 = 0。可以将三角函数视为变量，如设 u = sin x，然后因式分解或使用求根公式。你得到 u = 1 或 u = -1/2。再在区间内分别求解 sin x = 1 和 sin x = -1/2。

Always check whether the value obtained is within the range [-1, 1] for sine and cosine. If a quadratic yields sin x = 2, discard it immediately, as it leads to no real solution. This checking step catches many careless errors before you waste time searching for non‑existent angles.

始终检查所求值是否在正弦和余弦的[-1, 1]范围内。如果由二次方程得到 sin x = 2，立即舍去，因为它无实数解。这个检查步骤可在你浪费时间寻找不存在的角度前找出许多粗心错误。

7. Handling Equations with Double or Multiple Angles | 处理倍角或多倍角方程

When the argument is 2x, 3x, or x/2, first solve for the multiple angle as usual, then divide by the coefficient at the end. For instance, to solve cos(2x) = 1/2 on [0, 2π], let u = 2x, find all u in [0, 4π] that satisfy cos u = 1/2, and finally divide each u by 2. The expanded interval is critical: if x is in [0, 2π], then 2x is in [0, 4π].

当角度是2x、3x或x/2时，首先照常解出多倍角，最后再除以系数。例如，要在[0, 2π]上解 cos(2x) = 1/2，可设 u = 2x，求出[0, 4π]上所有满足 cos u = 1/2 的 u，最后再将每个u除以2。扩展区间至关重要：如果x在[0, 2π]内，那么2x就在[0, 4π]内。

This method avoids the common mistake of only giving one or two solutions. Write out the general solution for u, then set n = 0, 1, 2, … to generate values within the expanded interval, and only then divide by the coefficient. Doing so guarantees you collect every valid value of x.

该方法可避免只给出一个或两个解的常见错误。先写出u的通解，然后令 n = 0, 1, 2, … 以生成扩展区间内的值，最后才除以系数。这样做能确保你收集到每一个有效的x值。

8. Using Identities to Simplify Equations | 利用恒等式化简方程

Sometimes an equation looks messy because it mixes different functions. For example, an equation containing both sin²x and cos x can be transformed into a quadratic in cos x by substituting sin²x = 1 – cos²x. Similarly, an equation like sin x = tan x is best rewritten as sin x = sin x / cos x, then rearranged and factored.

有时方程因混合了不同的函数而显得杂乱。例如，一个同时含有 sin²x 和 cos x 的方程可以通过代入 sin²x = 1 – cos²x 转化为关于 cos x 的二次方程。类似地，像 sin x = tan x 这样的方程最好改写成 sin x = sin x / cos x，然后移项并因式分解。

When you use an identity, be mindful of domain restrictions. Dividing both sides by sin x loses sin x = 0 solutions; instead, bring all terms to one side and factor. A disciplined approach is to avoid dividing by a variable expression unless you are certain it is never zero.

使用恒等式时，要注意定义域限制。两边同除以 sin x 会丢失 sin x = 0 的解；正确的做法是将所有项移到一边然后因式分解。严谨的做法是除非你确信一个含变量的表达式永不为零，否则不要用它去除等式两边。

9. Checking for Extraneous Solutions and Lost Roots | 检查增根与失根

Every operation you perform, especially squaring both sides or using an identity that changes the domain, can introduce extraneous solutions or remove valid ones. After solving, substitute your answers back into the original equation. In Exercise 22D, problems that involve squaring might give a value that satisfies the squared version but not the original sign condition.

你进行的每一步运算，尤其是平方两边或使用改变定义域的恒等式，都可能引入增根或丢失有效根。求解完毕后，务必将答案代回原方程。在练习22D中，涉及平方的题目可能会给出一个满足平方后式子但不满足原符号条件的结果。

For example, squaring sin x = cos x + 1 can yield an extra solution where sin x is negative, though the original equation required a specific sign. Always interpret your final values in light of the original equation’s structure. This habit is what distinguishes a careful solver from a rushed one.

例如，将 sin x = cos x + 1 两边平方可能产生一个 sin x 为负的额外解，但原方程要求特定的符号。始终根据原方程的结构来解释你得到的最终值。这个习惯正是严谨解题者与粗心解题者的区别所在。

10. Working Efficiently Under Timed Conditions | 在限时条件下高效解题

In an IB exam, time pressure can lead to simple mistakes. Set up a routine: first, identify the type of equation; second, isolate the trigonometric expression; third, sketch the unit circle and mark the quadrants; fourth, list solutions in order. For quadratic forms, factor mentally or with a quick sketch. Avoid long decimal approximations unless the question explicitly asks for them.

在IB考试中，时间压力可能导致简单错误。建立一套程序：首先，识别方程类型；其次，分离出三角函数表达式；第三，画出单位圆并标记象限；第四，按顺序列出解。对于二次形式，用心算或简单草图进行因式分解。除非题目明确要求，否则避免使用冗长的小数近似值。

Practice Exercise 22D problems with a timer, and write your solutions as you would in an exam booklet. Clear, logical steps not only earn method marks but also allow you to spot errors when checking. Remember, the IB values reasoning as much as the final answer.

用计时器练习练习22D的题目，并像在考试答题本中那样书写解题过程。清晰、有逻辑的步骤不仅能赢得方法分，还能让你在检查时发现错误。请记住，IB重视推理过程与最终答案同等重要。

11. Common Mistakes to Avoid | 需要避免的常见错误

One frequent error is forgetting that the calculator gives only the principal value for inverse trigonometric functions. If you ask for arcsin(0.5), your calculator displays π/6, but you still need the supplementary angle. Another error is misapplying the general solution formula: the (-1)ⁿ factor belongs only to sine, not cosine or tangent.

一个常见错误是忘记计算器只给出反三角函数的主值。如果你求 arcsin(0.5)，计算器显示π/6，但你仍需补角。另一个错误是误用通解公式：(-1)ⁿ因子只适用于正弦，不适用于余弦或正切。

Solutions are often lost when students divide by sin x or cos x too early. Instead of dividing, factor out the function. For example, tan x = sin x should become sin x / cos x = sin x, then sin x (1/cos x – 1) = 0. This keeps sin x = 0 as a viable solution set.

当学生过早除以 sin x 或 cos x 时，常常会丢失解。不要除法，而应将函数提取出来。例如，tan x = sin x 应变为 sin x / cos x = sin x，然后 sin x (1/cos x – 1) = 0。这样就保留了 sin x = 0 这组有效解。

12. How to Review and Master Exercise 22D | 如何复习并掌握练习22D

After completing the exercise, group the problems by type: linear, quadratic, multiple-angle, and identity‑based. Redo a representative question from each group without looking at your notes. Pay close attention to problems that ask for solutions in a specific domain, as these are the ones most likely to appear on the final examination.

完成练习后，将题目按类型分组：线性、二次、多倍角以及基于恒等式的方程。从每组中挑出一道有代表性的题目，尝试不看笔记重做一遍。要特别关注那些要求特定定义域内解的题目，因为它们最有可能出现在期末考试中。

Write summary cards with the general solution forms and the most‑used identities. Reviewing these cards the night before a test can keep the patterns fresh in your mind. With consistent practice, the techniques in Exercise 22D will become automatic, freeing up working memory for more complex problem‑solving on the day of the exam.

制作总结卡片，写上通解形式和最常用恒等式。在考试前一晚复习这些卡片可以使这些模式在脑海中保持清晰。通过持续练习，练习22D中的技巧将变成自动反应，从而在考试当天为更复杂的问题解决腾出工作记忆。

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Exercise 22D: Solving Trigonometric Equations | 练习22D：解三角方程