📚 Trigonometric Formulae: Core Identities and IB Applications | 三角公式：核心恒等式与IB应用
Trigonometric formulae form the backbone of advanced geometry, calculus, and mathematical modelling in the IB Mathematics curriculum. Whether you are taking Analysis & Approaches or Applications & Interpretation, mastering these identities enables you to simplify complex expressions, solve equations, and prove relationships. This revision guide walks you through every essential formula, from the basic reciprocal identities to the more advanced sum-to-product transformations, equipping you with clear explanations, exam tips, and practical examples to boost your confidence.
三角公式是IB数学课程中高级几何、微积分和数学建模的基石。无论你学习的是分析与方法还是应用与解释，掌握这些恒等式都能帮助你化简复杂表达式、求解方程并证明关系。本复习指南将带你逐一攻克从基本倒数恒等式到更高级的和差化积变换的每一个核心公式，提供清晰的解释、考试技巧和实用示例，助你增强信心。
1. Basic Reciprocal Identities | 基本倒数恒等式
The three fundamental trigonometric functions are sine, cosine, and tangent. Their reciprocal counterparts – cosecant, secant, and cotangent – are defined directly from them. These identities are the very first tools you need to rewrite expressions and solve equations in a more flexible form.
三个基本三角函数是正弦、余弦和正切。它们的倒数对应函数——余割、正割和余切——直接由它们定义。这些恒等式是你需要掌握的第一组工具，能让你以更灵活的形式重写表达式和求解方程。
csc θ = 1 / sin θ sec θ = 1 / cos θ cot θ = 1 / tan θ = cos θ / sin θ
treated as a foundational relationship. Remember that these functions are undefined whenever the denominator is zero, a key point for determining domain restrictions in IB problems.
恒等式 tan θ = sin θ / cos θ 也被视为一个基本关系。请记住，只要分母为零，这些函数就没有定义，这是确定IB问题中定义域限制的关键点。
2. Pythagorean Identities | 毕达哥拉斯恒等式
Derived directly from the unit circle definition, the Pythagorean identities are the most frequently used tools for transforming squared trigonometric terms. The primary identity relates the squares of sine and cosine.
毕达哥拉斯恒等式直接源自单位圆定义，是转换平方三角项时最常用的工具。主恒等式将正弦和余弦的平方关联起来。
sin²θ + cos²θ = 1
By dividing every term of this identity by cos²θ or by sin²θ, we obtain two additional forms that are extremely helpful when working with tangent and secant, or cotangent and cosecant.
将此恒等式的每一项除以 cos²θ 或 sin²θ，我们可以得到另外两种形式，在处理正切与正割，或余切与余割时非常有用。
1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ
These identities are essential for simplifying expressions like 1 − sin²θ into cos²θ, or for converting a trigonometric equation into a quadratic in tan θ. In IB exams, you will often use them to prove more complicated identities or to integrate trigonometric functions.
这些恒等式对于将 1 − sin²θ 化简为 cos²θ 或将三角方程转换为关于 tan θ 的二次方程至关重要。在IB考试中，你经常会用它们来证明更复杂的恒等式或对三角函数进行积分。
3. Sum and Difference Formulae | 和差角公式
When you need the sine, cosine, or tangent of an angle expressed as A+B or A−B, the compound angle formulae come into play. These essential expansions are provided in the IB formula booklet, but you must know how to apply them correctly.
当你需要计算用 A+B 或 A−B 表示的角的正弦、余弦或正切时，就要用到复合角公式。这些重要的展开式在IB公式手册中提供，但你必须懂得如何正确运用。
rules reveal that sin(A±B) mixes the product of sine and cosine of the individual angles.
正弦的和差规则表明 sin(A±B) 混合了各角正弦与余弦的乘积。
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
For cosine, the sign reverses in the product of sines, which is a common source of sign errors if not memorised carefully.
对于余弦，正弦乘积项的符号会反转，如果不仔细记忆，这是一个常见的符号错误来源。
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
The tangent addition formula can be derived from the sine and cosine results and is especially useful when dealing with gradients and angles of inclination in coordinate geometry.
正切的加法公式可由正弦和余弦的结果推导得出，在处理坐标几何中的斜率和倾角时特别有用。
tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
sin(45°+30°), but also underpin the entire calculus of trigonometric functions in later topics.
这些公式不仅能用于精确值计算，比如 sin 75° = sin(45°+30°)，还为后续章节中三角函数的整个微积分体系奠定了基础。
4. Double Angle Formulae | 倍角公式
iately yields the double angle identities. These are among the most heavily tested trigonometric tools in IB because they link an angle to its double, enabling equation solving and simplification.
在和角公式中令 A = B，立即得到倍角恒等式。这些是IB中考查最频繁的三角工具之一，因为它们将一个角与其双倍角联系起来，便于求解方程和化简。
sin 2θ = 2 sin θ cos θ
The cosine double angle formula is unique because it has three equivalent forms, giving you the flexibility to choose the one that best fits the context.
余弦的倍角公式独一无二，有三种等价形式，让你能灵活选择最适合上下文的一种。
cos 2θ = cos²θ − sin²θ
cos 2θ = 2 cos²θ − 1
cos 2θ = 1 − 2 sin²θ
The tangent double angle formula follows directly and is particularly useful for finding tan 2θ when tan θ is known, for instance in trigonometric substitution problems.
正切的倍角公式可直接得出，在已知 tan θ 求 tan 2θ 时特别有用，例如在三角代换问题中。
tan 2θ = 2 tan θ / (1 − tan²θ)
In many IB questions, these identities are used to express sin²θ or cos²θ in terms of cos 2θ, which is a powerful method for integrating squared trigonometric functions.
在许多IB题目中，这些恒等式被用来将 sin²θ 或 cos²θ 用 cos 2θ 表示，这是积分平方三角函数的有效方法。
5. Half-Angle Formulae | 半角公式
Rearranging the double angle identities for cos 2θ allows us to express the square of sine or cosine of a half-angle. These are indispensable when you need to find the exact value of a trigonometric function at an angle like 22.5° or to solve certain types of equations.
重新排列 cos 2θ 的倍角恒等式，我们可以表示半角的正弦或余弦的平方。当你需要求 22.5° 这类角的三角函数精确值或解某些类型的方程时，这些公式不可或缺。
sin(θ/2) = ±√((1 − cos θ) / 2)
cos(θ/2) = ±√((1 + cos θ) / 2)
nt in which θ/2 lies, so sketching a quick unit circle is always wise. The half-angle formula for tangent can be expressed in two convenient ways, both of which avoid the ambiguous sign.
正负号由 θ/2 所在的象限决定，因此快速画一个单位圆总是明智的。正切的半角公式可以用两种方便的形式表示，两者都避免了符号歧义。
tan(θ/2) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ
These forms are extremely practical in calculus for the Weierstrass substitution, and in IB they often appear in 'show that' proof questions where an expression without radicals is required.
这些形式在微积分的魏尔斯特拉斯代换中极为实用，在IB考试中它们常出现在要求不含根号的“证明”题中。
6. Product-to-Sum Formulae | 积化和差公式
Adding or subtracting the sum and difference sine and cosine formulae produces identities that convert products of trigonometric functions into sums. This is particularly helpful when integrating a product like sin 3x cos x, which cannot be integrated directly without conversion.
将和差角的正弦和余弦公式相加或相减，可以得到将三角函数乘积转换为和的恒等式。这在积分诸如 sin 3x cos x 的乘积时特别有用，因为不转换就无法直接积分。
sin A cos B = ½[sin(A + B) + sin(A − B)]
cos A sin B = ½[sin(A + B) − sin(A − B)]
cos A cos B = ½[cos(A + B) + cos(A − B)]
sin A sin B = ½[cos(A − B) − cos(A + B)]
Notice the symmetry and the careful placement of signs: the sum of two cosines produces a cosine sum, while the product of sines yields a difference of cosines. In IB Applications and Interpretation, these formulae can also model wave interference and oscillations.
注意其中的对称性和符号的仔细安排：两个余弦的和得到一个余弦和，而正弦的乘积得到余弦的差。在IB应用与解释课程中，这些公式还可以用于模拟波的干涉和振荡。
7. Sum-to-Product Formulae | 和差化积公式
The reverse process, converting a sum or difference of sines or cosines into a product, is just as critical. It enables the factoring of trigonometric expressions and is immensely powerful in solving equations where terms like sin 5θ + sin 3θ appear.
逆过程，即将正弦或余弦的和差转换为乘积，同样至关重要。它能对三角表达式进行因式分解，在求解含有 sin 5θ + sin 3θ 这类项的方程时威力巨大。
sin P + sin Q = 2 sin((P+Q)/2) cos((P−Q)/2)
sin P − sin Q = 2 cos((P+Q)/2) sin((P−Q)/2)
cos P + cos Q = 2 cos((P+Q)/2) cos((P−Q)/2)
cos P − cos Q = −2 sin((P+Q)/2) sin((P−Q)/2)
Memorising the factor of 2 and the half-angle arguments is essential. IB examiners frequently set questions where you must first use a sum-to-product transformation to create a factorised equation, then solve for the variable within a given interval.
记住因子 2 和半角参数至关重要。IB考官经常出题要求你首先使用和差化积变换生成因式分解的方程，然后在一个给定区间内求解变量。
8. Solving Trigonometric Equations Using Identities | 用恒等式解三角方程
comes manageable when you replace sin 2x with 2 sin x cos x, leading to a factorised form.
三角恒等式的真正威力在方程求解中得以显现。一个看似复杂的方程如 sin 2x = cos x，当你用 2 sin x cos x 替换 sin 2x 后，就变得易于处理，从而得到一个因式分解的形式。
from which you can find all solutions in the domain [0, 2π]. Always check the given domain and remember that extraneous solutions may arise if you square both sides at any step.
例如，从 2 sin x cos x − cos x = 0 提出公因式 cos x 得到 cos x (2 sin x − 1) = 0。求解得 cos x = 0 或 sin x = ½，由此可找出定义域 [0, 2π] 内的所有解。始终检查给定定义域，并记住如果任何一步两边平方，可能会产生增根。
sin²θ + sin θ − 1 = 0. Substituting u = sin θ turns this into a simple algebraic equation.
涉及多倍角的方程，如 cos 2θ = sin θ，可利用倍角形式 1 − 2 sin²θ = sin θ，重新整理成关于 sin θ 的二次方程：2 sin²θ + sin θ − 1 = 0。令 u = sin θ，便转化为一个简单的代数方程。
9. Proving Trigonometric Identities | 证明三角恒等式
IB proof questions require a clear, logical sequence of steps. The golden rule is to start with the more complicated side and manipulate it using known identities until it matches the simpler side.
IB的证明题要求清晰、逻辑严密的步骤。黄金法则是从较复杂的一侧入手，用已知恒等式进行变换，直到与较简单的一侧一致。
1 − cos 2θ = 2 sin²θ and 1 + cos 2θ = 2 cos²θ. The ratio simplifies immediately to tan²θ.
常见策略包括：将所有函数用正弦和余弦表示；通分；因式分解；以及利用毕达哥拉斯恒等式消去平方项。比如，要证明 (1 − cos 2θ) / (1 + cos 2θ) = tan²θ，可用 1 − cos 2θ = 2 sin²θ 和 1 + cos 2θ = 2 cos²θ 代换分子和分母，比值立即简化为 tan²θ。
Never move terms from one side to the other unless you are working on an equation. In a proof, keep the two sides separate and transform one into the other. Clearly state the identity used at each line to earn full marks in the IB examination.
除非是在方程中运算，否则切勿将一项从一侧移到另一侧。在证明中，保持两侧分开，将一侧变换为另一侧。在每一步清晰注明所使用的恒等式，以在IB考试中取得满分。
10. Applications in Geometry and Modelling | 几何与建模应用
Trigonometric formulae extend far beyond abstract equations. In geometry, the sine and cosine rules, combined with compound angle identities, allow you to find unknown sides and angles in non-right-angled triangles when given partial information.
三角公式的应用远不止抽象方程。在几何中，正弦定理和余弦定理与复合角恒等式相结合，使你能够在部分信息已知的情况下，求解非直角三角形的未知边和角。
sed as a single sine or cosine wave R sin(bt + d) using the sum formula in reverse. This harmonic form reveals the amplitude and phase shift directly, an exam favorite.
在建模周期现象，如潮汐高度、声波或旋转机械时，形如 f(t) = a sin(bt) + c cos(bt) 的函数可以利用和角公式逆向写成单一的正弦或余弦波 R sin(bt + d)。这种谐波形式直接展示了振幅和相位偏移，是考试常客。
product like sin 3x cos x is made possible by the product-to-sum formula. Without these formulae, calculus would be severely limited.
这些恒等式也是三角函数微积分的基础。例如，要积分 sin²x，需使用恒等式 sin²x = (1 − cos 2x)/2。积分乘积 sin 3x cos x 则依赖于积化和差公式。没有这些公式，微积分将严重受限。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One of the most frequent errors is mishandling the sign in the cosine sum formula or forgetting the factor of 2 in the double angle formula. Write key formulae at the top of your working paper to avoid mental blanks under pressure.
最常见的错误之一是弄错余弦和角公式中的符号，或忘记倍角公式中的因子 2。在草稿纸顶部写下关键公式，可避免压力下的大脑空白。
cos x by dividing both sides by cos x loses the cos x = 0 solution. Always factor instead.
另一个陷阱是在未经验证的情况下约去可能为零的三角项。例如，解 sin x cos x = cos x 时，两边同除以 cos x 会丢失 cos x = 0 的解。始终应先因式分解。
mmon deduction. Also, be mindful of radian mode unless specified otherwise, and always give final answers in the requested domain, often expressed in exact values involving π or surds.
使用半角或平方根恒等式时，要根据给定象限确定恰当的符号。在IB考试中，遗漏 ± 号是常见的扣分点。此外，除非另有说明，记得使用弧度模式，且最终答案应按照要求的定义域给出，通常用含 π 或根号的精确值表示。
12. Summary of Key Formulae | 关键公式总结
The table below collects the essential trigonometric identities for quick reference. Use it to check your memory before an exam, but remember that understanding how to apply them is far more important than rote learning.
下表汇集了核心三角恒等式，供快速查阅。请在考前用它检查记忆，但请记住，理解如何应用远比死记硬背更重要。
| Identity Category | 恒等式类别 | Key Formula | 关键公式 |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1 ; 1 + tan²θ = sec²θ ; 1 + cot²θ = csc²θ |
| Sum/Difference | sin(A±B) = sinA cosB ± cosA sinB ; cos(A±B) = cosA cosB ∓ sinA sinB |
| Double Angle | 2 sinθ cosθ ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ ; tan 2θ = 2 tanθ/(1−tan²θ) |
| Half-Angle | sin²(θ/2) = (1−cosθ)/2 ; cos²(θ/2) = (1+cosθ)/2 ; tan(θ/2) = sinθ/(1+cosθ) |
| Product-to-Sum | 2 sinA cosB = sin(A+B)+sin(A−B) ; 2 cosA cosB = cos(A+B)+cos(A−B) |
| Sum-to-Product | sin P+sin Q = 2 sin((P+Q)/2) cos((P−Q)/2) ; cos P+cos Q = 2 cos((P+Q)/2) cos((P−Q)/2) |
Regular practice with past paper questions will cement these identities in your mind. Challenge yourself to derive the less familiar ones from the core sum and difference formulae, which strengthens understanding and reduces dependency on the formula booklet.
定期练习历年真题将巩固这些恒等式。试着从核心和差角公式推导出不太熟悉的那些公式，这能加深理解，减少对公式手册的依赖。
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