# 三角比及其图像

> 爱德思国际GCSE进阶纯数学 · 4PM1 2016年考纲（高阶）
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-further-pure-math-s10-trigonometric-ratios-and-their-graphs/

本指南涵盖核心三角比性质、sin/cos/tan的图像形状、30/45/60°精确值，以及任意大小角度的CAST图使用方法，完全匹配爱德思4PM1考纲要求。

**先修:** [基础直角三角学](https://www.owlsprep.com/zh/study/edexcel-igcse-further-pure-math-s10-right-angled-trigonometry/); [角度制与弧度制的角度度量](https://www.owlsprep.com/zh/study/edexcel-igcse-further-pure-math-s10-angle-measurement-degrees-radians/)

## 学习目标

- 定义任意大小角度（角度制或弧度制）的sin、cos、tan，并绘制它们的图像，保证周期、振幅和截距正确
- 熟记30°、45°、60°及其对应弧度值的三角精确值
- 运用CAST图和图像对称性求相关角度（例如120°、300°）的三角比

## 任意大小角度的三角比

**任意角度的三角比** — 对于从x轴正方向逆时针测量的角度θ，若其终边上有一点$(x,y)$，到原点的距离为$r$，则：$\sin\theta = \frac{y}{r}$，$\cos\theta = \frac{x}{r}$，$\tan\theta = \frac{y}{x}$

该定义适用于大于90°的角、负角，以及角度制和弧度制下的所有角，摆脱了直角三角形的限制。每个三角比的符号取决于角度终边所在的象限。

> **mnemonic**
>
> CAST规则：从第四象限开始逆时针移动：**C**os（余弦）为正、**A**ll（全部）为正、**S**in（正弦）为正、**T**an（正切）为正。该规则可帮你快速判断每个象限内哪些三角比的符号为正。

**例题:** 判断$\sin 210^\circ$、$\cos 330^\circ$和$\tan 150^\circ$的符号。

1. 确定每个角度所在的象限：210°位于第三象限（180°–270°），330°位于第四象限（270°–360°），150°位于第二象限（90°–180°）
2. 应用CAST规则：第三象限只有正切为正，因此$\sin 210^\circ$为负。第四象限只有余弦为正，因此$\cos 330^\circ$为正。第二象限只有正弦为正，因此$\tan 150^\circ$为负。

**概念自测**

1. What is the sign of $\cos 225^\circ$?

   - Positive
   - Negative
   - Zero

   *解析:* 225° falls in Q3, where only tan ratios are positive, so cos 225° is negative.

*计算器:* allowed

## 30°、45°、60°的三角精确值

你必须熟记这些常见角度的根式精确值，4PM1公式表中不会给出这些数值。必要时你可以通过30-60-90和45-45-90直角三角形推导它们。

| Angle (°) | Angle (rad) | $\sin\theta$ | $\cos\theta$ | $\tan\theta$ |
| --- | --- | --- | --- | --- |
| 30 | $\frac{\pi}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| 45 | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
| 60 | $\frac{\pi}{3}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |

**例题:** 求$\cos 30^\circ + \sin 45^\circ$的精确值，结果保留根式形式。

1. 回忆精确值：$\cos 30^\circ = \frac{\sqrt{3}}{2}$，$\sin 45^\circ = \frac{\sqrt{2}}{2}$
2. 将两分数相加：$\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} = \frac{\sqrt{3} + \sqrt{2}}{2}$
3. 该结果无法进一步化简，因此最终答案为$\frac{\sqrt{3} + \sqrt{2}}{2}$

> **tip**
>
> 除非题目明确要求给出小数近似值，否则请始终将结果保留为根式精确形式，避免丢失精度得分。

*计算器:* allowed

## 利用对称性和CAST图求相关角的三角比

你可以利用图像对称性或CAST图，将0–90°范围外的角度关联到其锐角参考角（终边与x轴之间的最小夹角），从而求出对应的三角值。

- Q2 (90° < θ < 180°): Reference angle = $180^\circ - \theta$, sin positive, cos and tan negative
- Q3 (180° < θ < 270°): Reference angle = $\theta - 180^\circ$, tan positive, sin and cos negative
- Q4 (270° < θ < 360°): Reference angle = $360^\circ - \theta$, cos positive, sin and tan negative

**例题:** 求$\sin 120^\circ$和$\tan 300^\circ$的精确值。

1. 对于$\sin 120^\circ$：120°位于第二象限，参考角 = $180 - 120 = 60^\circ$。正弦在第二象限为正，因此$\sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2}$
2. 对于$\tan 300^\circ$：300°位于第四象限，参考角 = $360 - 300 = 60^\circ$。正切在第四象限为负，因此$\tan 300^\circ = -\tan 60^\circ = -\sqrt{3}$

**概念自测**

1. What is the exact value of $\cos 210^\circ$?

   - $\frac{\sqrt{3}}{2}$
   - $-\frac{\sqrt{3}}{2}$
   - $\frac{1}{2}$
   - $-\frac{1}{2}$

   *解析:* 210° is in Q3, reference angle = 30°, cos is negative in Q3, so $\cos 210^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$

*计算器:* allowed

## sin x、cos x、tan x的图像

你必须能够准确绘制这三个三角函数的图像，保证周期、振幅、截距正确，并根据要求将坐标轴标注为角度或弧度。

| Function | Period | Amplitude | Range | Key intercepts (0 ≤ x ≤ 2π) |
| --- | --- | --- | --- | --- |
| $y = \sin x$ | $2\pi$ (360°) | 1 | $-1 ≤ y ≤ 1$ | $(0,0), (\pi,0), (2\pi,0)$ |
| $y = \cos x$ | $2\pi$ (360°) | 1 | $-1 ≤ y ≤ 1$ | $(\frac{\pi}{2},0), (\frac{3\pi}{2},0)$ |
| $y = \tan x$ | $\pi$ (180°) | Undefined | All real numbers | $(0,0), (\pi,0), (2\pi,0)$ |

**例题:** 绘制$0^\circ ≤ x ≤ 360^\circ$范围内$y = \sin x$的图像，标注所有截距、最大值点和最小值点。

1. 绘制x轴，标注为`x（角度）`，范围从0到360，y轴范围从-1到1
2. 在点$(0,0)$、$(180,0)$、$(360,0)$处标注截距
3. 在点$(90, 1)$处标注最大值点，在点$(270, -1)$处标注最小值点
4. 绘制一条连接这些点的平滑周期性波形，使其符合标准正弦曲线的形状

> **warning**
>
> 正切函数 tan x 的图像在 $90^\circ, 270^\circ, \frac{\pi}{2}, \frac{3\pi}{2}$ 等处存在垂直渐近线。请勿跨这些渐近线连接正切图像的两支，否则会被扣分。

*计算器:* allowed

## 常见错误

- **错误做法:** Using a calculator to find decimal values when exact surd answers are required
  - 原因: Exact value questions test recall of core knowledge, decimal approximations lose accuracy marks
  - 正确做法: Use memorised exact 30/45/60° values and CAST symmetry for exact value questions, unless told otherwise
- **错误做法:** Stating the period of tan x is 360° / 2π
  - 原因: Tan x repeats every 180° / π, unlike sin and cos which repeat every 360° / 2π
  - 正确做法: Memorise: period of sin/cos = 360° (2π), period of tan = 180° (π)
- **错误做法:** Assigning the wrong sign to trig ratios in non-acute quadrants
  - 原因: Forgetting CAST rules leads to sign errors that lose all marks for a question
  - 正确做法: First identify the angle's quadrant, use CAST to confirm the sign before calculating the ratio from the reference angle
- **错误做法:** Sketching tan x as a continuous curve across asymptotes
  - 原因: Tan x is undefined at odd multiples of 90° / π/2, so it has breaks at these points
  - 正确做法: Draw separate disconnected branches of tan x on either side of each vertical asymptote
- **错误做法:** Forgetting to label graph axes as degrees or radians
  - 原因: The same x value has different meaning in degrees vs radians, examiners require clear labelling for full marks
  - 正确做法: Always add an explicit label to the x-axis (e.g. `x (degrees)` or `x (radians)` when sketching trig graphs

## 速查表

| Concept | Key Details |
| --- | --- |
| Exact values (30/45/60°) | sin30=1/2, sin45=\sqrt{2}/2, sin60=\sqrt{3}/2; cos30=\sqrt{3}/2, cos45=\sqrt{2}/2, cos60=1/2; tan30=1/\sqrt{3}, tan45=1, tan60=\sqrt{3} |
| CAST Diagram | Q1: All +ve; Q2: Sin +ve; Q3: Tan +ve; Q4: Cos +ve |
| sin x / cos x properties | Period = 360° (2π), Amplitude = 1, Range = [-1, 1] |
| tan x properties | Period = 180° (π), No amplitude, Range = all real numbers, Asymptotes at 90°+180°n |

## 下一步

Now that you have mastered trigonometric ratios and their graphs, you are ready to move on to core trigonometric identities, which build on your knowledge of sin, cos and tan relationships to simplify expressions and solve exam-style problems. This topic is a foundational prerequisite for all subsequent trigonometry content in the Edexcel IGCSE Further Pure Math syllabus, including solving trigonometric equations and applying addition formulae. Make sure you practise sketching graphs and using the CAST diagram regularly to retain these core skills for your exam.

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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-trigonometric-ratios-and-their-graphs/