# 恒等式 tan θ = sin θ / cos θ

> 爱德思国际GCSE进阶纯数学 · 4PM1
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-further-pure-math-s10-the-identity-tan-sin-cos/

本指南将指导你在爱德思IGCSE进阶纯数学（4PM1）考试中应用$\tan \theta = \frac{\sin \theta}{\cos \theta}$恒等式。你将学习使用这个印在考试公式表上的公式化简三角表达式、求解基础方程。

**先修:** [直角三角形中正弦、余弦、正切的基础定义](https://www.owlsprep.com/zh/study/edexcel-igcse-further-pure-math-s10-trigonometric-ratios/); [30°、45°、60°的三角精确值](https://www.owlsprep.com/zh/study/edexcel-igcse-further-pure-math-s10-exact-trig-values/)

## 学习目标

- 牢记恒等式 tan θ = sin θ / cos θ，注意该恒等式已在4PM1公式表中给出
- 运用正切恒等式化简三角表达式
- 通过恒等式改写正切项，求解基础三角方程

## 什么是tan θ = sin θ / cos θ恒等式？

**tan θ恒等式** — 这是一个三角恒等式，表明对于所有满足$\cos \theta \neq 0$的θ值，均有$\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$。该恒等式已在爱德思4PM1公式表中给出。

*例:* For $\theta = 45^\circ$, $\sin 45 = \frac{\sqrt{2}}{2}$, $\cos 45 = \frac{\sqrt{2}}{2}$, so $\tan 45 = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$, which matches the known exact value.

由于除以零无定义，该恒等式对所有cos θ不为零的角度θ均成立。当cos θ = 0（例如θ = 90°、270°）时，tan θ无定义，因此这些情况下该恒等式不适用。

> **info**
>
> 你无需为考试背诵该恒等式，因为它已经印在官方公式表上。但你需要能立刻识别出它，并清楚它的适用场景。

**例题:** 使用三角精确值验证该恒等式在θ = 30°时成立。

1. 步骤1：写出sin 30°、cos 30°和tan 30°的精确值。
2. $$\sin 30^\circ = \frac{1}{2}, \cos 30^\circ = \frac{\sqrt{3}}{2}, \tan 30^\circ = \frac{1}{\sqrt{3}}$$
3. 步骤2：计算θ = 30°时sin θ / cos θ的数值。
4. $$\frac{\sin 30^\circ}{\cos 30^\circ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}$$
5. 步骤3：将结果与tan 30°对比：二者数值相等，因此该恒等式在θ = 30°时成立。

> **考试提示:** 使用该恒等式前务必确认cos θ ≠ 0，尤其是当你处理接近90°或270°的角度时。

## 使用恒等式化简三角表达式

该恒等式最常见的用途之一就是化简同时包含tan、sin和cos项的表达式。你可以将tan θ替换为sin θ / cos θ，或者将sin θ与cos θ的比值改写为tan θ，以此减少表达式中的项数。

**例题:** 将表达式$\tan \theta \times \cos \theta$尽可能化简。

1. 步骤1：使用恒等式将tan θ替换为sin θ / cos θ。
2. $$\tan \theta \times \cos \theta = \frac{\sin \theta}{\cos \theta} \times \cos \theta$$
3. 步骤2：在cos θ ≠ 0的前提下，约去分子和分母中公共的cos θ项。
4. $$\frac{\sin \theta}{\cancel{\cos \theta}} \times \cancel{\cos \theta} = \sin \theta$$
5. 步骤3：化简后的表达式为sin θ，对所有满足cos θ ≠ 0的θ均成立。

**概念自测**

测试你对化简知识点的理解

1. What is the simplified form of $\frac{\sin \theta}{\tan \theta}$?

   - cos θ
   - sin θ
   - tan θ
   - 1

   *答案:* cos θ

   *解析:* Correct: substitute tan θ = sin θ / cos θ, so the fraction becomes sin θ divided by (sin θ / cos θ) = cos θ.

> **考试提示:** 如果一个表达式中有多个正切项，将所有tan替换为sin/cos通常可以约去公因子，大幅简化表达式。

*计算器:* allowed

## 使用恒等式求解三角方程

你可以使用该恒等式求解同时包含tan、sin和cos项的方程，方法是将所有项改写为sin和cos的形式，或者重新整理项，将其组合为可替换为tan的sin/cos比值。

**例题:** 在0° ≤ θ < 360°且cos θ ≠ 0的条件下，求解方程$2 \sin \theta = \tan \theta$。

1. 步骤1：将方程中的tan θ替换为sin θ / cos θ。
2. $$2 \sin \theta = \frac{\sin \theta}{\cos \theta}$$
3. 步骤2：将所有项移到等号一侧，提取公因子sin θ。
4. $$2 \sin \theta - \frac{\sin \theta}{\cos \theta} = 0 \implies \sin \theta \left(2 - \frac{1}{\cos \theta}\right) = 0$$
5. 步骤3：令每个因子等于零，求解θ。
6. 情况1：sin θ = 0。在给定范围内的解为θ = 0°、180°。
7. 情况2：$2 - 1/\cos \theta = 0$，推导得$1/\cos \theta = 2$，即$\cos \theta = 1/2$。该取值范围内的解为$\theta = 60^\circ$、$300^\circ$。
8. 步骤4：验证所有解都不会使$\cos \theta = 0$，所有解均有效。最终解为：$0^\circ$、$60^\circ$、$180^\circ$、$300^\circ$。

> **考试提示:** 除非你能百分百确定sin θ或cos θ不可能为零，否则绝对不要将方程两边同时除以sin θ或cos θ，否则会丢失有效解。请始终采用因式分解的方式，如示例所示。

*计算器:* allowed

## 常见错误

- **错误做法:** Using the identity when cos θ = 0
  - 原因: Division by zero is undefined, so tan θ is undefined at these angles, making the identity invalid
  - 正确做法: Always check that cos θ ≠ 0 for your angle values before applying the identity
- **错误做法:** Dividing both sides of an equation by sin θ or cos θ to simplify
  - 原因: This removes any solutions where the divided term equals zero, leading to incomplete answers
  - 正确做法: Rearrange all terms to one side and factor out the common trigonometric term instead of dividing
- **错误做法:** Forgetting the identity is only valid for real angles where tan θ is defined
  - 原因: Applying it to angles like 90° will lead to mathematical errors
  - 正确做法: Exclude angles where cos θ = 0 (90°, 270° etc.) from your solution set when using this identity
- **错误做法:** Rewriting cos θ / sin θ as tan θ
  - 原因: The identity states tan θ = sin θ / cos θ, not the reverse. Cotangent is out of scope for 4PM1 so you should not use that term either
  - 正确做法: Only replace a fraction of sin θ over cos θ with tan θ, leave cos θ / sin θ as is unless you can simplify it another way
- **错误做法:** Memorizing the identity unnecessarily
  - 原因: It is provided on the formula sheet, so you waste memory space that could be used for non-provided formulae
  - 正确做法: Focus on learning to apply the identity correctly rather than memorizing it

## 速查表

| Task | Action | Validity Note |
| --- | --- | --- |
| Verify the identity | Substitute known sin/cos values and compare to tan | Only for θ where cos θ ≠ 0 |
| Simplify expressions with tan | Replace tan θ with sin θ / cos θ, cancel common factors | Exclude θ where cos θ = 0 |
| Solve equations with mixed tan/sin/cos | Substitute tan, rearrange and factor, solve for θ | Check no solutions have cos θ = 0 |

## 下一步

Now that you have mastered applying the $\tan \theta = \frac{\sin \theta}{\cos \theta}$ identity, you are ready to build on this knowledge with other core trigonometric identities and problem-solving techniques for your Edexcel IGCSE Further Pure Math exam. Next, you will learn the Pythagorean identity $\sin^2 \theta + \cos^2 \theta \equiv 1$, which is often used alongside the tan identity to simplify more complex expressions. Following that, you will cover trigonometric addition formulae and learn to solve a wider range of trigonometric equations, both of which regularly appear in 4PM1 exam papers. Make sure you practice applying this identity to mixed problems to reinforce your understanding before moving to more advanced topics.

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来自 [OwlsPrep](https://www.owlsprep.com) —— A-Level / IB / AP / IGCSE 免费学习指南，依据官方考纲编写。原页面：https://www.owlsprep.com/zh/study/edexcel-igcse-further-pure-math-s10-the-identity-tan-sin-cos/