4E Trigonometric identities | 4E 三角恒等式

📚 4E Trigonometric identities | 4E 三角恒等式

Trigonometric identities lie at the heart of IB Mathematics, providing powerful shortcuts for simplifying expressions, verifying equations, and modelling periodic phenomena. In topic 4E, students are expected not only to memorise the core identities but also to apply them flexibly in proofs and problem-solving. This article systematically presents the essential identities, their derivations, and practical strategies to help you gain confidence and accuracy.

三角恒等式是IB数学的核心内容，它们为化简表达式、验证等式以及建模周期现象提供了强有力的工具。在4E主题中，学生不仅要熟记基本恒等式，还需灵活运用它们进行证明和解题。本文系统梳理了关键的恒等式、推导过程以及实用技巧，帮助你建立信心并提高准确性。

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1. The Pythagorean Identities | 毕达哥拉斯恒等式

sin²θ + cos²θ = 1

This identity arises directly from the unit circle definition. It can be rearranged to express sin²θ or cos²θ in terms of the other. Dividing the equation by cos²θ or sin²θ yields two companion forms that are equally important.

这个恒等式直接源于单位圆的定义。它可以变形，用其中一个表示另一个的平方。将等式除以cos²θ或sin²θ会得到两个同样重要的相关形式。

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

These versions are particularly useful when an expression contains tangent and secant or cotangent and cosecant. Using them to replace a squared term often unlocks a path to simplification.

当表达式中含有正切与正割或余切与余割时，这些形式特别有用。用它们替换平方项往往能打开化简的突破口。

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2. Reciprocal and Quotient Identities | 倒数恒等式与商恒等式

The reciprocal relationships define cosecant, secant, and cotangent in terms of sine, cosine, and tangent. The quotient identities link tangent and cotangent directly to sine and cosine.

倒数关系用正弦、余弦和正切定义了余割、正割和余切。商恒等式则将正切和余切直接与正弦、余弦联系起来。

• sin θ = 1 / csc θ   cos θ = 1 / sec θ   tan θ = sin θ / cos θ

  sin θ = 1 / csc θ   cos θ = 1 / sec θ   tan θ = sin θ / cos θ

• csc θ = 1 / sin θ   sec θ = 1 / cos θ   cot θ = cos θ / sin θ

  csc θ = 1 / sin θ   sec θ = 1 / cos θ   cot θ = cos θ / sin θ

These are the working definitions you will return to again and again. When stuck, rewriting everything in terms of sine and cosine is often the most reliable first step.

这些是你将反复使用的工作定义。当思路卡住时，把一切都用正弦和余弦表示往往是可靠的第一步。

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3. Sum and Difference Formulas | 和角与差角公式

The sum and difference identities allow us to expand expressions like sin(α+β) into products of individual sines and cosines. They are essential for deriving double-angle and half-angle formulas.

和差公式能将sin(α+β)这类表达式展开为单个正弦、余弦的乘积形式。它们是推导倍角公式和半角公式的基础。

sin(α + β) = sin α cos β + cos α sin β

sin(α − β) = sin α cos β − cos α sin β

cos(α + β) = cos α cos β − sin α sin β

cos(α − β) = cos α cos β + sin α sin β

tan(α + β) = (tan α + tan β) / (1 − tan α tan β)

tan(α − β) = (tan α − tan β) / (1 + tan α tan β)

A common trick is to recognise that the cosine formulas reverse the sign pattern: cosine of a sum gives a minus, while cosine of a difference gives a plus. Practice with exact values helps lock these in.

一个常见的记忆点是余弦公式的符号模式相反：和的余弦中间是减号，差的余弦中间是加号。结合特殊角进行练习有助于牢固记忆。

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4. Double-Angle Formulas | 倍角公式

Setting α = β in the sum formulas produces the double-angle identities. These extremely versatile formulas appear in integration, differentiation, and equation solving.

在和角公式中令α = β即可得到倍角恒等式。这些用途极广的公式出现在积分、微分以及方程求解中。

sin 2θ = 2 sin θ cos θ

cos 2θ = cos²θ − sin²θ

cos 2θ = 2 cos²θ − 1

cos 2θ = 1 − 2 sin²θ

tan 2θ = (2 tan θ) / (1 − tan²θ)

The three forms of cos 2θ give you the power to express the square of sine or cosine linearly — a technique central to simplifying integrals like ∫cos²x dx.

cos 2θ的三种形式赋予你将正弦或余弦的平方线性化的能力——这一技巧对于简化∫cos²x dx之类的积分至关重要。

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5. Half-Angle Formulas | 半角公式

Half-angle identities express trigonometric functions of θ/2 in terms of cos θ. The ± sign depends on the quadrant in which θ/2 lies, and choosing the correct sign is a frequent examination point.

半角恒等式将θ/2的三角函数用cos θ表示。正负号取决于θ/2所在的象限，选择正确的符号是常见的考点。

sin(θ/2) = ±√[(1 − cos θ)/2]

cos(θ/2) = ±√[(1 + cos θ)/2]

tan(θ/2) = ±√[(1 − cos θ)/(1 + cos θ)] = sin θ/(1 + cos θ) = (1 − cos θ)/sin θ

Notice that the tangent half-angle formula has alternative rational forms that avoid the square root and the ± ambiguity, making them more convenient for many algebraic proofs.

请注意，正切半角公式有无需根号且无正负歧义的有理形式，这使得它们在许多代数证明中更为方便。

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6. Product-to-Sum and Sum-to-Product Formulas | 积化和差与和差化积公式

These identities convert products of sines and cosines into sums, and vice versa. They are indispensable in solving certain equations and in evaluating integrals of products.

这些恒等式能将正弦和余弦的乘积转化为和差形式，反之亦然。它们在求解某些方程和计算乘积积分时不可或缺。

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)]

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]

You can derive all product-to-sum formulas directly from the sum and difference formulas. The sum-to-product versions are particularly helpful when solving equations like sin 4x + sin 2x = 0.

所有积化和差公式都可以直接从和差公式推导出来。和差化积形式在求解像sin 4x + sin 2x = 0这样的方程时尤其有用。

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7. Cofunction and Negative-Angle Identities | 余函数恒等式与负角恒等式

Cofunction identities reveal the symmetry between sine and cosine, tangent and cotangent, secant and cosecant for complementary angles. Negative-angle identities describe which functions are odd and which are even.

余函数恒等式揭示了互余角的正弦与余弦、正切与余切、正割与余割之间的对称性。负角恒等式则描述了哪些函数是奇函数，哪些是偶函数。

• sin(π/2 − θ) = cos θ   cos(π/2 − θ) = sin θ   tan(π/2 − θ) = cot θ

  sin(π/2 − θ) = cos θ   cos(π/2 − θ) = sin θ   tan(π/2 − θ) = cot θ

• sin(−θ) = − sin θ   cos(−θ) = cos θ   tan(−θ) = − tan θ

  sin(−θ) = − sin θ   cos(−θ) = cos θ   tan(−θ) = − tan θ

Knowing that cosine is even and sine is odd helps simplify expressions with negative angles quickly, and it underpins the symmetry of the unit circle.

知道余弦是偶函数而正弦是奇函数，有助于快速化简包含负角的表达式，这也是单位圆对称性的基础。

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8. Strategies for Proving Identities | 证明恒等式的策略

Proving a trigonometric identity requires transforming one side of the equation step by step until it matches the other side. There is no single algorithm, but a structured approach dramatically increases your success rate.

证明三角恒等式需要逐步变换等号的一边，直到与另一边相同。虽然没有单一算法，但结构化的方法能大幅提高成功率。

Start by choosing the more complicated side and rewriting everything in terms of sine and cosine. Look for common denominators, factorisations, and opportunities to apply Pythagorean identities. If the denominator contains 1 ± cos θ or 1 ± sin θ, consider multiplying numerator and denominator by the conjugate. Avoid moving terms across the equals sign unless absolutely necessary; work on one side only.

从较复杂的一边入手，将一切用正弦和余弦表示。寻找公分母、因式分解以及应用毕达哥拉斯恒等式的机会。如果分母含有1 ± cos θ或1 ± sin θ，可考虑将分子和分母同时乘以共轭式。除非绝对必要，不要将项移过等号；只对一边进行变换。

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9. Simplifying Expressions | 化简表达式

Simplifying a trigonometric expression usually means reducing it to a form that contains as few functions as possible, ideally a single function or a simple fraction. Identities are the tools that make this possible.

化简三角表达式通常意味着将其化为包含尽可能少函数的形式，理想情况下为单一函数或简单的分式。恒等式正是实现这一目标的工具。

For example, to simplify (1 − cos 2x) / (sin 2x), replace 1 − cos 2x with 2 sin²x and sin 2x with 2 sin x cos x. The expression reduces to tan x. Practising such reductions sharpens algebraic intuition and prepares you for calculus applications.

例如，要化简(1 − cos 2x) / (sin 2x)，将1 − cos 2x替换为2 sin²x，将sin 2x替换为2 sin x cos x。表达式可化简为tan x。练习这类化简能提高代数直觉，为微积分应用做好准备。

(1 − cos 2x) / sin 2x = (2 sin²x) / (2 sin x cos x) = tan x

Always verify the domain of the original expression, as simplification might hide restrictions where denominators become zero.

务必验证原表达式的定义域，因为化简过程可能隐藏了分母为零的限制条件。

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10. Solving Trigonometric Equations | 解三角方程

Identities are the key to transforming an equation like 2 sin²x + 3 cos x = 0 into a quadratic in a single trigonometric function. Substitutions using Pythagorean and double-angle identities often convert a seemingly difficult equation into something manageable.

恒等式是将2 sin²x + 3 cos x = 0这类方程转化为单个三角函数的二次方程的关键。利用毕达哥拉斯恒等式和倍角恒等式进行替换，常常能将看似困难的方程变得易于处理。

Replace sin²x with 1 − cos²x, obtain a quadratic in cos x, solve, and then find all solutions in the required interval. Remember to check for extraneous solutions and to express answers using the general solution formula where appropriate: for sin x = k, x = nπ + (−1)ⁿ arcsin(k), and for cos x = k, x = 2nπ ± arccos(k), with n ∈ ℤ.

将sin²x替换为1 − cos²x，得到关于cos x的二次方程，求解后再在给定区间内找出所有解。注意检验增根，并在适当情况下使用通解公式：对于sin x = k，x = nπ + (−1)ⁿ arcsin(k)；对于cos x = k，x = 2nπ ± arccos(k)，其中n ∈ ℤ。

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11. Common Pitfalls and How to Avoid Them | 常见陷阱与规避方法

Misremembering a sign in a sum formula or applying the Pythagorean identity incorrectly can derail an entire proof. The most frequent errors include forgetting the ± in half-angle formulas, dividing by a trigonometric expression without checking whether it could be zero, and confusing the double-angle forms of cosine.

记错和角公式中的符号或错误地应用毕达哥拉斯恒等式，可能会使整个证明出错。最常见的错误包括：遗忘半角公式中的±号、在未检验三角表达式是否为零的情况下进行除法，以及混淆余弦的倍角形式。

To avoid these, always keep a concise formula sheet handy during practice until the identities become second nature. When using identities that involve square roots or fractions, immediately note the quadrant or domain conditions. Lastly, test your final result with a simple angle (like θ = 0 or π/4) as a quick sanity check.

为避免这些错误，在练习阶段可常备一份精简的公式表，直到恒等式变得像本能一样熟练。当使用含有根号或分式的恒等式时，立即标注象限或定义域条件。最后，用一个简单角度（如θ = 0或π/4）快速检验最终结果，作为理智检查。

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12. Summary and Practice Advice | 总结与练习建议

Mastering 4E Trigonometric identities is not about passive reading — it is about active manipulation. Work through proofs line by line, and practise with a mix of simplifying, proving, and equation-solving exercises. Pay special attention to linking Pythagorean identities with double-angle formulas, as this combination appears in many IB exam questions.

掌握4E三角恒等式靠的不是被动阅读，而是主动的代数操作。逐行进行证明练习，并混合练习化简、证明和解方程题型。特别关注将毕达哥拉斯恒等式与倍角公式结合使用，这种组合在大量IB考题中出现。

Create your own summary cards for the six families of identities covered here. Each day, write out the formulas from memory and test yourself with a timed ‘mini-proof’ to build both speed and reliability.

为本文涵盖的六类恒等式制作你自己的总结卡片。每天凭记忆写出公式，并用限时“小证明”自测，以同时提升速度和可靠性。

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