📚 Using Trigonometric Identities | 三角函数恒等式的应用
In Edexcel A Level Mathematics, trigonometric identities are essential tools for simplifying expressions, proving results, finding exact values, and solving equations over a given interval. An identity is true for every value of the variable, whereas an equation is only true for particular values. This topic builds directly on the sine, cosine, and tangent functions you met in the first trigonometry module.
在 Edexcel A Level 数学中，三角恒等式是化简表达式、证明结论、求精确值以及在给定区间内解方程的核心工具。恒等式对变量的每一个值都成立，而方程只对特定值成立。本专题直接建立在你在第一个三角学模块中学过的正弦、余弦和正切函数之上。
1. Why Trigonometric Identities Matter | 为什么三角恒等式重要
Identities are not isolated formulas to memorise. They allow you to convert a complicated trigonometric expression into a simpler form, often turning a mixed equation into a quadratic or linear equation that can be solved using standard algebraic methods. In Edexcel exams, you will be asked to use identities both with and without a calculator, so fluency with the main forms is essential.
恒等式并不是需要死记硬背的孤立公式。它们能让你把复杂的三角表达式转换成更简单的形式，常常把一个混合方程转化为可以用标准代数方法求解的二次或线性方程。在 Edexcel 考试中，你会被要求在有计算器和无计算器的情况下使用恒等式，因此熟练掌握主要形式至关重要。
2. The Three Pythagorean Identities | 三个毕达哥拉斯恒等式
The most fundamental identity is sin² θ + cos² θ = 1. It comes from the unit circle definition of sine and cosine. Dividing every term by cos² θ gives tan² θ + 1 = sec² θ, and dividing by sin² θ gives 1 + cot² θ = cosec² θ. These three identities are the foundation for almost every proof and equation-solving task.
最基本的恒等式是 sin² θ + cos² θ = 1。它来自单位圆上正弦和余弦的定义。将每一项都除以 cos² θ 得到 tan² θ + 1 = sec² θ，除以 sin² θ 得到 1 + cot² θ = cosec² θ。这三个恒等式是几乎所有证明和解方程任务的基础。
| Identity | Origin | Common Use |
|---|---|---|
| sin² θ + cos² θ = 1 | Unit circle | Switching between sin and cos |
| tan² θ + 1 = sec² θ | Divide by cos² θ | Linking tan and sec |
| 1 + cot² θ = cosec² θ | Divide by sin² θ | Linking cot and cosec |
Always check whether the identity needs to be rearranged. For example, sin² θ = 1 − cos² θ is very useful when you want to convert a squared sine term into a cosine term.
一定要检查恒等式是否需要变形。例如，sin² θ = 1 − cos² θ 在你希望把正弦平方项转换为余弦项时非常有用。
3. Compound Angle Identities: Sine and Cosine | 和角公式：正弦与余弦
For angles A and B, the compound angle identities are:
对于角 A 和 B，和角公式如下：
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
Notice that for sine, the sign on the right-hand side matches the sign inside the brackets: plus with plus, minus with minus. For cosine, the sign is reversed: cos(A + B) has a minus, and cos(A − B) has a plus. This sign reversal is one of the most common sources of error.
注意，对于正弦，右边的符号与括号内的符号一致：加号对应加号，减号对应减号。对于余弦，符号则相反：cos(A + B) 中为减号，cos(A − B) 中为加号。这种符号反转是最常见的错误来源之一。
4. Compound Angle Identities: Tangent | 和角公式：正切
The tangent version follows from dividing the sine and cosine compound angle results:
正切的和角公式由正弦和余弦的和角公式相除得到：
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
Here the numerator sign matches A ± B, but the denominator uses the opposite sign. If 1 ∓ tan A tan B = 0, the tangent is undefined, which corresponds to A ± B being an odd multiple of 90° or π/2 radians.
这里分子的符号与 A ± B 一致，但分母使用相反的符号。如果 1 ∓ tan A tan B = 0，正切无定义，这对应于 A ± B 为 90° 或 π/2 弧度的奇数倍。
5. Double Angle Identities | 二倍角公式
Double angle formulas are special cases of the compound angle identities with A = B. They are heavily used in solving trigonometric equations, differentiating, and integrating.
二倍角公式是 A = B 时和角公式的特殊情况。它们在解三角方程、求导和积分中被大量使用。
sin 2A = 2 sin A cos A
cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A
tan 2A = 2 tan A/(1 − tan² A)
The three forms of cos 2A are equivalent, but choosing the right form is often the key step in a question. The first form links directly to cos² A and sin² A; the second and third forms allow you to eliminate one of these squares.
cos 2A 的三种形式是等价的，但选择正确的形式往往是解题的关键一步。第一种形式直接联系 cos² A 和 sin² A；第二和第三种形式可以消去其中一个平方项。
6. Choosing the Right Form of cos 2A | 选择合适的 cos 2A 形式
When an equation contains both cos 2θ and sin² θ, use cos 2θ = 1 − 2 sin² θ. This substitution removes the double angle and leaves a quadratic or linear equation in sin θ. Similarly, if the equation contains cos 2θ and cos² θ, use cos 2θ = 2 cos² θ − 1.
当方程中同时含有 cos 2θ 和 sin² θ 时，使用 cos 2θ = 1 − 2 sin² θ。这种替换消去了二倍角，留下关于 sin θ 的二次或线性方程。类似地，如果方程中含有 cos 2θ 和 cos² θ，则使用 cos 2θ = 2 cos² θ − 1。
This technique creates a hidden quadratic. Many A Level questions are designed so that after substitution you get something like 2 cos² θ + 3 cos θ − 2 = 0, which factorises easily. Recognising this pattern saves time and reduces mistakes.
这种技巧会产生一个隐藏的二次方程。许多 A Level 题目在代入后都会得到类似 2 cos² θ + 3 cos θ − 2 = 0 的式子，该式很容易因式分解。识别这个规律可以节省时间并减少错误。
7. Solving Equations Using Identities | 用恒等式解方程
Let’s solve 2 sin² θ + 3 cos θ = 0 for 0° ≤ θ < 360°. Begin by replacing sin² θ with 1 − cos² θ:
我们来解方程 2 sin² θ + 3 cos θ = 0，其中 0° ≤ θ < 360°。首先将 sin² θ 替换为 1 − cos² θ：
2(1 − cos² θ) + 3 cos θ = 0
This simplifies to 2 − 2 cos² θ + 3 cos θ = 0, or equivalently 2 cos² θ − 3 cos θ − 2 = 0. Let c = cos θ:
化简得 2 − 2 cos² θ + 3 cos θ = 0，等价于 2 cos² θ − 3 cos θ − 2 = 0。设 c = cos θ：
(2c + 1)(c − 2) = 0
Hence cos θ = −1/2 or cos θ = 2. Since cos θ can never be greater than 1, only cos θ = −1/2 is valid. In the interval 0° ≤ θ < 360°, the solutions are θ = 120° and θ = 240°.
因此 cos θ = −1/2 或 cos θ = 2。由于 cos θ 不可能大于 1，只有 cos θ = −1/2 有效。在区间 0° ≤ θ < 360° 内，解为 θ = 120° 和 θ = 240°。
8. Proving Trigonometric Identities | 证明三角恒等式
To prove an identity, start from one side, usually the more complicated side, and rewrite it using known identities. A reliable strategy is to express everything in terms of sin θ and cos θ, then simplify or factorise.
证明恒等式时，从一边开始，通常是较复杂的一边，用已知恒等式进行改写。一个可靠的策略是将所有项都写成 sin θ 和 cos θ，然后化简或因式分解。
Example: prove (sin θ + cos θ)² = 1 + sin 2θ.
例如：证明 (sin θ + cos θ)² = 1 + sin 2θ。
LHS = sin² θ + 2 sin θ cos θ + cos² θ = (sin² θ + cos² θ) + 2 sin θ cos θ = 1 + sin 2θ
The key step is recognising that 2 sin θ cos θ = sin 2θ. Always show every step clearly; marks are awarded for correct use of identities and logical progression.
关键一步是识别出 2 sin θ cos θ = sin 2θ。一定要清晰地展示每一步；评分标准会奖励正确使用恒等式和逻辑清晰的推导过程。
9. Harmonic Form: R sin(θ ± α) and R cos(θ ± α) | 辅助角形式：R sin(θ ± α) 与 R cos(θ ± α)
Edexcel also expects you to express expressions of the form a sin θ + b cos θ as R sin(θ ± α) or R cos(θ ± α). Here R = √(a² + b²) and α is an angle chosen so that the expansion matches the original coefficients.
Edexcel 还要求你将 a sin θ + b cos θ 形式的表达式写成 R sin(θ ± α) 或 R cos(θ ± α) 的形式。其中 R = √(a² + b²)，α 的选取要使展开式与原系数匹配。
Example: express 3 sin θ + 4 cos θ as R sin(θ + α). First, R = √(3² + 4²) = 5. Then compare 5 sin(θ + α) = 5 sin θ cos α + 5 cos θ sin α, so cos α = 3/5 and sin α = 4/5. Therefore α ≈ 53.13°.
例如：将 3 sin θ + 4 cos θ 写成 R sin(θ + α)。首先，R = √(3² + 4²) = 5。然后比较 5 sin(θ + α) = 5 sin θ cos α + 5 cos θ sin α，得到 cos α = 3/5，sin α = 4/5。因此 α ≈ 53.13°。
This form is particularly useful for solving equations such as 3 sin θ + 4 cos θ = 2, because it reduces the left-hand side to a single sine term. It also reveals the maximum and minimum values of the expression: ±R.
这种形式对于解诸如 3 sin θ + 4 cos θ = 2 的方程特别有用，因为它把左边化为一个单一的正弦项。它还能揭示表达式的最大值和最小值：±R。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Be careful with the following exam pitfalls:
请留意以下考试中的常见陷阱：
-
Forgetting that cos(A + B) = cos A cos B − sin A sin B has a minus sign
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply