Using Trigonometric Identities | 运用三角恒等式

📚 Using Trigonometric Identities | 运用三角恒等式

Trigonometric identities are fundamental tools for simplifying expressions, solving equations, and proving relationships in A-Level Mathematics. Mastering them improves algebraic manipulation and prepares you for calculus, mechanics, and further pure mathematics. This article covers the Pythagorean identities, reciprocal and ratio identities, compound-angle and double-angle formulas, half-angle and power-reduction forms, the R-formula, and practical exam techniques.

三角恒等式是简化表达式、解方程和证明关系的核心工具，在 A-Level 数学中至关重要。熟练掌握它们能提升代数运算能力，并为微积分、力学和进阶纯数学打下基础。本文将涵盖勾股恒等式、倒数与比值恒等式、和差角公式、倍角公式、半角与降幂形式、R 公式以及实用的考试技巧。

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1. Fundamental Pythagorean Identities | 基本勾股恒等式

The three Pythagorean identities come from the unit circle and are the most frequently used identities in trigonometry. The first identity is sin²θ + cos²θ ≡ 1. Dividing through by cos²θ gives 1 + tan²θ ≡ sec²θ, and dividing by sin²θ gives 1 + cot²θ ≡ cosec²θ.

三个基本勾股恒等式源自单位圆，是三角学中使用最频繁的恒等式。第一个恒等式为 sin²θ + cos²θ ≡ 1。两边除以 cos²θ 可得 1 + tan²θ ≡ sec²θ，两边除以 sin²θ 可得 1 + cot²θ ≡ cosec²θ。

sin²θ + cos²θ ≡ 1

1 + tan²θ ≡ sec²θ

1 + cot²θ ≡ cosec²θ

These identities hold for all values of θ where the functions are defined. They are especially useful when you need to replace one trigonometric function with another, for example converting sin²θ into 1 - cos²θ or cos²θ into 1 - sin²θ.

这些恒等式对所有使函数有定义的 θ 都成立。当你需要用某个三角函数替换另一个时，它们特别有用，例如把 sin²θ 换成 1 - cos²θ，或把 cos²θ 换成 1 - sin²θ。

/25. This kind of conversion is common in exam questions that link tan, sec, sin, and cos.

例如，若 tanθ = 3/4，则 1 + tan²θ = 1 + 9/16 = 25/16，因此 sec²θ = 25/16，cos²θ = 16/25。这类转换在联系 tan、sec、sin 和 cos 的考题中很常见。

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2. Reciprocal and Ratio Identities | 倒数与比值恒等式

The reciprocal identities define sec, cosec, and cot in terms of cos, sin, and tan. They are the first step in many proofs because they allow you to express everything using sin and cos.

倒数恒等式用 cos、sin 和 tan 定义了 sec、cosec 和 cot。它们是许多证明的第一步，因为可以把所有函数都转化为 sin 和 cos。

secθ = 1/cosθ, cosecθ = 1/sinθ, cotθ = 1/tanθ = cosθ/sinθ

alid wherever the denominators are non-zero, so cosθ ≠ 0 for tan and sec, and sinθ ≠ 0 for cot and cosec.

比值恒等式为 tanθ = sinθ/cosθ，cotθ = cosθ/sinθ。只要分母不为零就成立，因此 tan 和 sec 要求 cosθ ≠ 0，cot 和 cosec 要求 sinθ ≠ 0。

In proof questions, a reliable strategy is to rewrite every term as sinθ or cosθ, combine fractions, and then simplify using sin²θ + cos²θ ≡ 1.

在证明题中，一个可靠策略是把每一项都改写成 sinθ 或 cosθ，合并分数，然后利用 sin²θ + cos²θ ≡ 1 化简。

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3. Simplifying Expressions with Identities | 用恒等式化简表达式

Simplification usually follows a clear order: convert all trigonometric functions into sin and cos, apply Pythagorean identities, factorise where possible, and combine fractions using a common denominator.

化简通常遵循明确顺序：把所有三角函数转化为 sin 和 cos，应用勾股恒等式，尽可能因式分解，并用公分母合并分数。

to get 2(sin²θ + cos²θ). Since sin²θ + cos²θ = 1, the expression simplifies to 2 × 1 = 2.

例如，化简 2sin²θ + 2cos²θ。提取 2 得 2(sin²θ + cos²θ)。由于 sin²θ + cos²θ = 1，表达式化简为 2 × 1 = 2。

inθ. Using 1 - cos²θ = sin²θ, the expression becomes sin²θ/sinθ = sinθ, provided sinθ ≠ 0.

另一个例子是 (1 - cos²θ)/sinθ。利用 1 - cos²θ = sin²θ，表达式变为 sin²θ/sinθ = sinθ，其中 sinθ ≠ 0。

0. These small conversions often turn a complex-looking expression into a single function.

类似地，tanθ cosθ = (sinθ/cosθ) × cosθ = sinθ，其中 cosθ ≠ 0。这些小转换常能把看似复杂的表达式变成单一函数。

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4. Proving Trigonometric Identities | 证明三角恒等式

To prove an identity, start from one side, usually the more complicated side, and transform it step by step until it matches the other side. Do not treat the identity as an equation and cross-multiply, because that assumes what you are trying to prove.

证明恒等式时，从一边开始，通常从较复杂的一边出发，逐步变形直到与另一边相同。不要把它当作方程来交叉相乘，因为那等于假设了你要证明的结论。

rything in terms of sin and cos, then simplify. For example, prove (1 + tan²θ)/(1 + cot²θ) ≡ n²θ. The left-hand side becomes sec²θ/cosec²θ = (1/cos²θ)/(1/sin²θ) = sin²θ/cos²θ = tan²θ.

一个有用方法是把所有式子改写成 sin 和 cos 再化简。例如证明 (1 + tan²θ)/(1 + cot²θ) ≡ tan²θ。左边变为 sec²θ/cosec²θ = (1/cos²θ)/(1/sin²θ) = sin²θ/cos²θ = tan²θ。

Always work vertically and show each step clearly. If one side contains fractions, combine them first. If both sides look complicated, simplify each side separately until they meet in the middle.

始终竖向书写并清楚展示每一步。如果某一边有分数，先合并。如果两边都很复杂，就分别化简直到中间相遇。

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5. Compound-Angle Identities | 和差角公式

tial for exact values such as sin75° and for solving equations involving sums or differences of angles.

和差角公式用于展开 A ± B 的 sin、cos 和 tan。它们对于求 sin75° 等精确值，以及解涉及角度和差的方程至关重要。

sin(A ± B) = sinA cosB ± cosA sinB

cos(A ± B) = cosA cosB ∓ sinA sinB

tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)

For example, sin75° = sin(45° + 30°) = sin45° cos30° + cos45° sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

例如，sin75° = sin(45° + 30°) = sin45° cos30° + cos45° sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4。

Be careful with signs: cos(A + B) has a minus sign in the middle, while cos(A - B) has a plus sign. The tan formula also has the opposite sign in the denominator.

注意符号：cos(A + B) 中间是减号，而 cos(A - B) 中间是加号。tan 公式分母中的符号也与分子相反。

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6. Double-Angle Identities | 倍角公式

B = θ. They are used to express trigonometric functions of 2θ in terms of functions of θ.

倍角公式由和差角公式令 A = B = θ 得到，用于把 2θ 的三角函数表示为 θ 的函数。

sin2θ = 2 sinθ cosθ

cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

tan2θ = 2tanθ/(1 - tan²θ)

The three forms of cos2θ are equivalent. Choose the form that matches the information given: if you know sinθ, use 1 - 2sin²θ; if you know cosθ, use 2cos²θ - 1.

cos2θ 的三种形式等价。选择与已知信息匹配的形式：如果已知 sinθ，用 1 - 2sin²θ；如果已知 cosθ，用 2cos²θ - 1。

θ = 1 - 2sin²θ = 1 - 2(9/25) = 1 - 18/25 = 7/25. This is much faster than finding θ first.

例如，若 sinθ = 3/5 且 θ 为锐角，则 cos2θ = 1 - 2sin²θ = 1 - 2(9/25) = 1 - 18/25 = 7/25。这比先求 θ 快得多。

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7. Half-Angle and Power-Reduction Forms | 半角与降幂形式

Rearranging the double-angle identities gives half-angle and power-reduction forms. These are especially useful in calculus when integrating even powers of sin and cos.

重新整理倍角公式可得到半角和降幂形式。它们在微积分中积分 sin 和 cos 的偶次幂时特别有用。

sin²θ = (1 - cos2θ)/2

cos²θ = (1 + cos2θ)/2

tan²θ = (1 - cos2θ)/(1 + cos2θ)

d cos(θ/2) = ±√((1 + cosθ)/2). The correct sign depends on the quadrant in which θ/2 lies.

对于半角，sin(θ/2) = ±√((1 - cosθ)/2)，cos(θ/2) = ±√((1 + cosθ)/2)。正负号取决于 θ/2 所在象限。

These forms also help in solving equations where you need to reduce a squared trigonometric term to a first-power term.

这些形式也有助于解方程，特别是需要把平方项降为一次项的时候。

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8. Solving Trigonometric Equations | 解三角方程

To solve a trigonometric equation, use identities to reduce it to a single trigonometric function, then solve the resulting equation within the given interval.

解三角方程时，先用恒等式将其化为单一三角函数，再在给定区间内解所得方程。

- 1 = 0 for 0° ≤ θ < 360°. Treat it as a quadratic in sinθ: (2sinθ - 1)(sinθ + 1) = 0.

例如，解 2sin²θ + sinθ - 1 = 0，0° ≤ θ < 360°。把它看成关于 sinθ 的二次方程：(2sinθ - 1)(sinθ + 1) = 0。

es θ = 30° and 150°, while sinθ = -1 gives θ = 270°. The full solution set is θ = 30°, 150

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