Exact values, graphs and trig equations · 精确值、图象与三角方程
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| exact values/eɡˈzækt ˈvæljuːz/ | 精确值 | jīng què zhí |
| trig graphs/trɪɡ ɡræfz/ | 三角函数图 | sān jiǎo hán shù tú |
| asymptotes/ˈæsɪmptəʊts/ | 渐近线 | jiàn jìn xiàn |
| symmetry/ˈsɪmətri/ | 对称 | duì chèn |
The values you must know by heart
- Some trig values are so important that you need them without a calculator. They come from two special right triangles: the $45$-$45$-$90$ and the $30$-$60$-$90$.
你必须熟记于心的值
- 一些三角值如此重要,以至于你需要它们而不用计算器。它们来自两个特殊的直角三角形:$45$-$45$-$90$ 和 $30$-$60$-$90$。
The exact values 精确值 table
| $x$ | $0^{\circ}$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ |
|---|---|---|---|---|---|
| $\sin x$ | $0$ | $\dfrac{1}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{\sqrt{3}}{2}$ | $1$ |
| $\cos x$ | $1$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{1}{2}$ | $0$ |
| $\tan x$ | $0$ | $\dfrac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ | undefined |
Memory trick for sine: $0, 1, 2, 3, 4$ under the root and over $2$: $\;\dfrac{\sqrt{0}}{2}, \dfrac{\sqrt{1}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{4}}{2}$. Cosine is the reverse.
A Ferris wheel: a point on the rim traces a sine curve
精确值表
| $x$ | $0^{\circ}$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ |
|---|---|---|---|---|---|
| $\sin x$ | $0$ | $\dfrac{1}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{\sqrt{3}}{2}$ | $1$ |
| $\cos x$ | $1$ | $\dfrac{\sqrt{3}}{2}$ | $\dfrac{\sqrt{2}}{2}$ | $\dfrac{1}{2}$ | $0$ |
| $\tan x$ | $0$ | $\dfrac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ | undefined |
正弦的记忆技巧:$0, 1, 2, 3, 4$ 在根号下并除以 $2$:$\;\dfrac{\sqrt{0}}{2}, \dfrac{\sqrt{1}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{4}}{2}$。余弦是反过来的。
一个摩天轮:轮缘上的一个点描出一条正弦曲线
Trig graphs & equations · 三角函数图象与方程
(cos θ, sin θ)
As θ turns, sin and cos trace their waves — and repeat every 360°.
What is the exact value of sin 30°? (as a decimal)
sin 30° = 1/2 = 0.5.
What is the exact value of tan 45°?
tan 45° = 1 (opposite = adjacent in a 45-45-90 triangle).
cos 60° = 1/2.
cos 60° = 1/2 is one of the key exact values to memorise.
Trig graphs 三角函数图
- $y = \sin x$ and $y = \cos x$ are waves oscillating between $-1$ and $1$.
- $y = \tan x$ repeats every $180^{\circ}$ and has vertical asymptotes 渐近线 at $90^{\circ}$ and $270^{\circ}$.
Trig graphs are waves — sine and cosine oscillate between $-1$ and $1$, tangent repeats every $180^{\circ}$.
Two solutions. A trig equation usually has two answers in $0^{\circ}$–$360^{\circ}$. For $\sin x = \dfrac{\sqrt{3}}{2}$: $x = 60^{\circ}$ or $x = 180 - 60 = 120^{\circ}$. Don't forget the second answer.
三角函数图象
- $y = \sin x$ 和 $y = \cos x$ 是在 $-1$ 和 $1$ 之间振荡的波。
- $y = \tan x$ 每 $180^{\circ}$ 重复,并在 $90^{\circ}$ 和 $270^{\circ}$ 有竖直渐近线。
三角函数图象是波——正弦和余弦在 $-1$ 和 $1$ 之间振荡,正切每 $180^{\circ}$ 重复。
两个解。 一个三角方程在 $0^{\circ}$–$360^{\circ}$ 中通常有两个答案。对 $\sin x = \dfrac{\sqrt{3}}{2}$:$x = 60^{\circ}$ 或 $x = 180 - 60 = 120^{\circ}$。不要忘记第二个答案。
Solving trig equations
- Find the first solution using the inverse function.
- Use the graph's symmetry 对称 to find the second:
- For a positive sine value in $0^{\circ}\le x<360^{\circ}$, the second solution is $180^{\circ}-x$; for negative values use the lower-half graph.
- $\cos x = k$: second solution is $360^{\circ} - x$.
The two special triangles give the exact values: the $45^\circ$ triangle and the $30^\circ$-$60^\circ$ triangle
求解三角方程
- 用反函数找第一个解。
- 用图象的对称性找第二个:
- 对于$0^{\circ}\le x<360^{\circ}$中正弦值为正的情况，第二个解为$180^{\circ}-x$；若为负值，请使用下半部分的图像。
- $\cos x = k$:第二个解是 $360^{\circ} - x$。
两个特殊三角形给出精确值:$45^\circ$ 三角形和 $30^\circ$-$60^\circ$ 三角形
Solve sin x = √3/2 for 0–360°. One answer is 60°. What is the other (degrees)?
The sine wave is symmetric about 90°: 180 − 60 = 120°.
Solve 2cos x + 1 = 0 for 0–360°. One answer is 120°. What is the other (degrees)?
cos x = −1/2 gives x = 120° or 360 − 120 = 240°.
If sin x = k has one answer x = 40°, the second answer is 180 − ______ = ______°.
The second sine solution is 180° − 40° = 140°.
Worked examples
- $\sin x = \dfrac{\sqrt{3}}{2} \Rightarrow x = 60^{\circ}$ or $120^{\circ}$.
- $2\cos x + 1 = 0 \Rightarrow \cos x = -\dfrac{1}{2} \Rightarrow x = 120^{\circ}$ or $240^{\circ}$.
例题
- $\sin x = \dfrac{\sqrt{3}}{2} \Rightarrow x = 60^{\circ}$ 或 $120^{\circ}$。
- $2\cos x + 1 = 0 \Rightarrow \cos x = -\dfrac{1}{2} \Rightarrow x = 120^{\circ}$ 或 $240^{\circ}$。
Negative trig values and tangent equations
- In $0^{\circ}\le x<360^{\circ}$, $\sin x=-1/2$ is negative below the horizontal axis: $x=210^{\circ},330^{\circ}$. A calculator value $-30^{\circ}$ must be replaced by its equivalent $330^{\circ}$ inside the required interval.
- Tangent repeats after $180^{\circ}$: $\tan x=1$ gives $x=45^{\circ},225^{\circ}$ in this interval. At $0$, $90$, $180$, $270$, $360^{\circ}$ check endpoints and undefined values rather than assuming every equation has exactly two solutions.
负三角函数值与正切方程
- 在$0^{\circ}\le x<360^{\circ}$中，$\sin x=-1/2$位于水平轴下方时为负：$x=210^{\circ},330^{\circ}$。计算器得出的值$-30^{\circ}$必须替换为其在指定区间内的等效值$330^{\circ}$。
- 正切函数每$180^{\circ}$重复一次：在此区间内，$\tan x=1$对应$x=45^{\circ},225^{\circ}$。在$0$、$90$、$180$、$270$、$360^{\circ}$处，需检查端点和无定义值，而非假设每个方程恰好有两个解。
Find the smaller solution of sin x = −1/2 for 0° ≤ x < 360°.
Sine is negative in quadrants III and IV; the solutions are 210° and 330°.
You've got it
- know the exact values: $\sin 30^{\circ} = \dfrac{1}{2}$, $\cos 60^{\circ} = \dfrac{1}{2}$, $\tan 45^{\circ} = 1$
- sine and cosine waves run between $-1$ and $1$
- a trig equation usually has two answers in $0^{\circ}$–$360^{\circ}$
你掌握了
- 知道精确值:$\sin 30^{\circ} = \dfrac{1}{2}$,$\cos 60^{\circ} = \dfrac{1}{2}$,$\tan 45^{\circ} = 1$
- 正弦和余弦波在 $-1$ 和 $1$ 之间运行
- 一个三角方程在 $0^{\circ}$–$360^{\circ}$ 中通常有两个答案