本讲义涵盖主题 6,三角学(Trigonometry)。标记 (Extended) 的部分只在拓展卷上考查;其他一切对两个层次都适用。角的答案给到一个小数位。
三角学
IGCSE 数学 · 第 6 主题
6.1
勾股定理
大纲
| Subject content | Notes and examples |
|---|---|
| Know and use Pythagoras’ theorem. |
来源:剑桥国际大纲
勾股定理(Pythagoras' theorem)连接一个直角三角形(right-angled triangle)(一个有一个 $90^{\circ}$ 角的三角形(triangle))的三条边。若最长的边(斜边(hypotenuse),在直角对面)是 $c$,那么
$$a^{2} + b^{2} = c^{2}.$$
用它来求一条缺失的边。
Worked example. 一个直角三角形有一条 $13\text{ cm}$ 的斜边和一条 $5\text{ cm}$ 的短边。求另一条短边。
$$b^{2} = 13^{2} - 5^{2} = 169 - 25 = 144, \qquad b = \sqrt{144} = 12\text{ cm}.$$
探索
Pythagoras' theorem
Change the two short sides and see $a^2 + b^2 = c^2$ — the squares on the sides really do add up.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| Pythagoras' theorem | 勾股定理 | gōu gǔ dìng lǐ |
| right-angled triangle | 直角三角形 | zhí jiǎo sān jiǎo xíng |
| triangle | 三角形 | sān jiǎo xíng |
| hypotenuse | 斜边 | xié biān |
6.2
直角三角形
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Know and use the sine, cosine and tangent ratios for acute angles in calculations involving sides and angles of a right-angled triangle. | Angles will be given in degrees and answers should be written in degrees, with decimals correct to one decimal place. |
| 2 Solve problems in two dimensions using Pythagoras’ theorem and trigonometry. | Knowledge of bearings may be required. |
| Subject content | Notes and examples |
|---|---|
| 1 Know and use the sine, cosine and tangent ratios for acute angles in calculations involving sides and angles of a right-angled triangle. | Angles will be given in degrees and answers should be written in degrees, with decimals correct to one decimal place. |
| 2 Solve problems in two dimensions using Pythagoras’ theorem and trigonometry. | Knowledge of bearings may be required. |
| 3 Know that the perpendicular distance from a point to a line is the shortest distance to the line. | |
| 4 Carry out calculations involving angles of elevation and depression. |
来源:剑桥国际大纲
从你正在使用的角标注这些边:对边(opposite)(在角的对面)、邻边(adjacent)(在角旁边),和斜边。这三个比是正弦(sine)、余弦(cosine)和正切(tangent ratio)(sin、cos、tan):
$$\sin\theta = \frac{\text{opp}}{\text{hyp}}, \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}.$$
把它们记作 SOH-CAH-TOA。要求一个角,用反函数($\sin^{-1}$、$\cos^{-1}$、$\tan^{-1}$)。
Worked example (find a side). 在一个直角三角形里斜边是 $10\text{ cm}$ 而角是 $30^{\circ}$。求对边。
$$\text{opp} = 10 \times \sin 30^{\circ} = 10 \times 0.5 = 5\text{ cm}.$$
Worked example (find an angle). 对边是 $4\text{ cm}$ 而邻边是 $3\text{ cm}$。
$$\tan\theta = \frac{4}{3}, \qquad \theta = \tan^{-1}\!\left(\frac{4}{3}\right) = 53.1^{\circ}.$$
探索
Sine, cosine and tangent
Drag the angle on the unit circle to see where sin, cos and tan come from.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| opposite | 对边 | duì biān |
| adjacent | 邻边 | lín biān |
| sine | 正弦 | zhèng xián |
| cosine | 余弦 | yú xián |
| tangent ratio | 正切 | zhèng qiē |
6.2
直角三角形
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Know and use the sine, cosine and tangent ratios for acute angles in calculations involving sides and angles of a right-angled triangle. | Angles will be given in degrees and answers should be written in degrees, with decimals correct to one decimal place. |
| 2 Solve problems in two dimensions using Pythagoras’ theorem and trigonometry. | Knowledge of bearings may be required. |
| 3 Know that the perpendicular distance from a point to a line is the shortest distance to the line. | |
| 4 Carry out calculations involving angles of elevation and depression. |
来源:剑桥国际大纲
仰角(angle of elevation)是从水平向上到你上方一个物体的角。俯角(angle of depression)是从水平向下到你下方一个物体的角。从一个点到一条线的最短距离是垂直(perpendicular)距离。
Worked example. 从距一座塔的脚 $50\text{ m}$ 的一个点,到顶部的仰角是 $40^{\circ}$。求塔的高度。
$$\text{height} = 50 \times \tan 40^{\circ} = 50 \times 0.839 = 42.0\text{ m}.$$
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| angle of elevation | 仰角 | yǎng jiǎo |
| angle of depression | 俯角 | fǔ jiǎo |
| perpendicular | 垂直 | chuí zhí |
6.3
三角函数的精确值
大纲
| Subject content | Notes and examples |
|---|---|
| Know the exact values of: 1 $\sin x$ and $\cos x$ for $x = 0^\circ, 30^\circ, 45^\circ, 60^\circ$ and $90^\circ$. 2 $\tan x$ for $x = 0^\circ, 30^\circ, 45^\circ$ and $60^\circ$. |
来源:剑桥国际大纲
你必须不用计算器就知道这些精确值。
| $x$ | $0^{\circ}$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ |
|---|---|---|---|---|---|
| $\sin x$ | $0$ | $\tfrac{1}{2}$ | $\tfrac{\sqrt{2}}{2}$ | $\tfrac{\sqrt{3}}{2}$ | $1$ |
| $\cos x$ | $1$ | $\tfrac{\sqrt{3}}{2}$ | $\tfrac{\sqrt{2}}{2}$ | $\tfrac{1}{2}$ | $0$ |
| $\tan x$ | $0$ | $\tfrac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ | — |
6.4
三角函数
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Recognise, sketch and interpret the following graphs for $0^\circ \leqslant x \leqslant 360^\circ$: • $y = \sin x$ • $y = \cos x$ • $y = \tan x$. | |
| 2 Solve trigonometric equations involving $\sin x$, $\cos x$ or $\tan x$, for $0^\circ \leqslant x \leqslant 360^\circ$. | e.g. solve: • $\sin x = \frac{\sqrt{3}}{2}$ for $0^\circ \leqslant x \leqslant 360^\circ$ • $2 \cos x + 1 = 0$ for $0^\circ \leqslant x \leqslant 360^\circ$. |
来源:剑桥国际大纲
对于 $0^{\circ} \leqslant x \leqslant 360^{\circ}$:
- $y = \sin x$ 是一个波,在 $0$ 开始、在 $90^{\circ}$ 达到峰值、在 $180^{\circ}$ 回到 $0$、在 $270^{\circ}$ 下到 $-1$。
- $y = \cos x$ 是同样的波但在 $1$ 开始。
- $y = \tan x$ 陡峭地上升并每 $180^{\circ}$ 重复。
一个三角方程(trigonometric equation)在这个范围里常常有不止一个答案。用图(或波的对称)来找到它们全部。
Worked example. 对 $0^{\circ} \leqslant x \leqslant 360^{\circ}$ 解 $\sin x = \tfrac{\sqrt{3}}{2}$。
一个答案是 $x = 60^{\circ}$。正弦波在 $180^{\circ} - 60^{\circ} = 120^{\circ}$ 处也是 $\tfrac{\sqrt{3}}{2}$。所以 $x = 60^{\circ}$ 或 $120^{\circ}$。
Worked example. 对 $0^{\circ} \leqslant x \leqslant 360^{\circ}$ 解 $2\cos x + 1 = 0$。
$$\cos x = -\tfrac{1}{2} \;\Rightarrow\; x = 120^{\circ} \text{ or } 240^{\circ}.$$
探索
Trig graphs & equations
(cos θ, sin θ)
As θ turns, sin and cos trace their waves — and repeat every 360°.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| trigonometric equation | 三角方程 | sān jiǎo fāng chéng |
6.5
非直角三角形
大纲
| Subject content | Notes and examples |
|---|---|
| 1 Use the sine and cosine rules in calculations involving lengths and angles for any triangle. | Includes problems involving obtuse angles and the ambiguous case. |
| 2 Use the formula $\text{area of triangle} = \frac{1}{2} ab \sin C$. | The sine and cosine rules and the formula for area of a triangle are given in the List of formulas. |
来源:剑桥国际大纲
对于任何三角形(不只是直角的),边 $a, b, c$ 在角 $A, B, C$ 的对面:
$$\text{sine rule:}\quad \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C},$$
$$\text{cosine rule:}\quad a^{2} = b^{2} + c^{2} - 2bc\cos A.$$
当你有一条边和它的对角时用正弦定理(sine rule)。当你有两条边和它们之间的角、或全部三条边时用余弦定理(cosine rule)。用正弦定理时,注意两解情况(ambiguous case),那里一个角可能是锐角或钝角。
任何三角形的面积是
$$\text{area} = \tfrac{1}{2}ab\sin C.$$
Worked example. 一个三角形有 $b = 7\text{ cm}$、$c = 8\text{ cm}$ 而它们之间的角 $A = 40^{\circ}$。求边 $a$。
$$a^{2} = 7^{2} + 8^{2} - 2(7)(8)\cos 40^{\circ} = 113 - 112 \times 0.766 = 27.2,$$
$$a = \sqrt{27.2} = 5.2\text{ cm}.$$
探索
Sine & cosine rule
Two sides and the angle between them fix the triangle: the cosine rule finds the third side, the sine rule the other angles.
探索
What sin and cos mean
Spin the angle on the unit circle: the horizontal leg is cos θ and the vertical leg is sin θ — the same ratios the sine and cosine rules use.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| sine rule | 正弦定理 | zhèng xián dìng lǐ |
| cosine rule | 余弦定理 | yú xián dìng lǐ |
| ambiguous case | 两解情况 | liǎng jiě qíng kuàng |
6.6
三维中的勾股定理与三角学
大纲
| Subject content | Notes and examples |
|---|---|
| Carry out calculations and solve problems in three dimensions using Pythagoras' theorem and trigonometry, including calculating the angle between a line and a plane. |
来源:剑桥国际大纲
在三维里,在立体里面找到一个直角三角形,然后对它用勾股定理或三角学。一个常见的任务是一条线和一个平坦表面(一个平面(plane))之间的角。
Worked example. 一个盒子有一个 $6\text{ cm}$ 乘 $8\text{ cm}$ 的底和 $5\text{ cm}$ 的高。求一条空间对角线和底之间的角。
先是底对角线:$\sqrt{6^{2} + 8^{2}} = \sqrt{100} = 10\text{ cm}$。这条对角线和高构成一个直角三角形,所以与底的角 $\theta$ 是
$$\tan\theta = \frac{5}{10} = 0.5, \qquad \theta = \tan^{-1}(0.5) = 26.6^{\circ}.$$
探索
Pythagoras' theorem
In a right-angled triangle a² + b² = c². The same idea, applied twice, gives lengths inside 3-D solids.
词汇表
训练
| 英文 | 中文 | 拼音 |
|---|---|---|
| plane | 平面 | píng miàn |
6.6
考试技巧
- 当没有角时用勾股定理($a^2 + b^2 = c^2$);当涉及一个角时用 SOH-CAH-TOA。
- 斜边总是在直角对面。相对于你正在使用的角标注这些边(对边、邻边、斜边)。
- 要求一个角,用反函数($\sin^{-1}$、$\cos^{-1}$、$\tan^{-1}$),并检查你的计算器设置为度。
- 对于一个没有直角的三角形,用正弦定理或余弦定理——当你知道两条边和它们之间的角时用余弦定理。
本主题的互动课程
逐步学习,并即时检测练习。