Inverse Trigonometric Functions · 反三角函数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| sine/saɪn/ | 正弦 | zhèng xián |
| inverse/ɪnˈvɜːs/ | 反函数 | fǎn hán shù |
| one-to-one/wʌn tuː wʌn/ | 一一对应 | yī yī duì yìng |
| domain/dəˈmeɪn/ | 定义域 | dìng yì yù |
From a ratio back to an angle
- Sine turns an angle into a ratio; sometimes we need to run that backward.
- "Which angle has a sine of $0.5$?" is a question only an inverse can answer.
- Ramps, satellite dishes, and navigation all need the angle from a measured ratio.
- The inverse trig functions do exactly this reversal.
从比值回到角
- 正弦把角变成比值;有时我们需要反着来。
- "哪个角的正弦是 $0.5$?"这个问题只有反函数能回答。
- 斜坡、卫星天线和导航,都需要从测得的比值反求角度。
- 反三角函数做的正是这种反向操作。
What the inverse does
- The inverse 反函数 of sine, written $\arcsin$ or $\sin^{-1}$, takes a value and returns an angle.
- $\arcsin(0.5) = 30°$, because $\sin 30° = 0.5$.
- Likewise $\arccos$ and $\arctan$ reverse cosine and tangent.
- Each undoes its trig function, recovering the angle that produced a value.
反函数做什么
- 正弦的反函数(inverse),记作 $\arcsin$ 或 $\sin^{-1}$,接收一个值,返回一个角。
- $\arcsin(0.5) = 30°$,因为 $\sin 30° = 0.5$。
- 同样,$\arccos$ 和 $\arctan$ 把余弦和正切倒过来。
- 每一个都撤销它的三角函数,恢复产生某个值的那个角。
The inverse function $\arcsin$ (or $\sin^{-1}$) takes a value and returns… · 反函数 $\arcsin$(或 $\sin^{-1}$)接收一个值,返回……
The inverse · 反函数 of sine reverses it: $\arcsin(0.5)$ is the angle whose sine is $0.5$, namely $30°$. · 正弦的反函数把它倒过来:$\arcsin(0.5)$ 是正弦等于 $0.5$ 的那个角,即 $30°$。
Why we must restrict
- Sine is not one-to-one 一一对应: countless angles share the same sine value.
- If we reversed all of them, the "inverse" would give many answers at once — not a function.
- So we restrict the domain to one rising piece, from $-\tfrac{\pi}{2}$ to $\tfrac{\pi}{2}$.
- On that restricted domain 定义域, each output comes from exactly one angle.
为什么必须限制
- 正弦不是一一对应(one-to-one)的:无数个角有相同的正弦值。
- 如果把它们全都倒过来,"反函数"会一次给出许多答案——那就不是函数了。
- 所以我们把定义域限制到一段上升的部分,从 $-\tfrac{\pi}{2}$ 到 $\tfrac{\pi}{2}$。
- 在那段受限的定义域(domain)上,每个输出都恰好来自一个角。
From a ratio back to an angle · 从一个比值回到一个角
An inverse trig function answers the reverse question — given the sine or cosine value, which angle produced it? · 反三角函数回答反过来的问题——给定正弦或余弦的值,是哪个角产生了它?
Why must sine be restricted before it can have an inverse function? · 为什么正弦在能有反函数之前必须被限制?
Many angles share the same sine, so sine is not one-to-one. Restricting its domain fixes a single output per input. · 许多角有相同的正弦,所以正弦不是一一对应的。限制它的定义域使每个输入对应唯一的输出。
What is $\arcsin(1)$ (in radians)? · $\arcsin(1)$ 是多少(用弧度)?
$\sin\tfrac{\pi}{2} = 1$, so $\arcsin(1) = \tfrac{\pi}{2}$ — the angle in the restricted range. · $\sin\tfrac{\pi}{2} = 1$,所以 $\arcsin(1) = \tfrac{\pi}{2}$——限制范围内的那个角。
Select all · 所有 true statements about inverse trig functions. · 选出关于反三角函数的所有正确说法。
An inverse returns just one · 一个 angle (in the restricted range), not all of them. The other three are correct. · 反函数只返回一个角(在限制范围内),而不是全部。其余三条正确。
The inverse graphs
- Because they are inverses, the graphs reflect the restricted trig graphs across $y = x$.
- $\arcsin$ has domain $[-1, 1]$ (all valid sine 正弦 values) and range $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$.
- $\arccos$ has the same domain but range $[0, \pi]$.
- Their inputs are ratios; their outputs are angles.
反函数的图像
- 因为它们互为反函数,图像是受限三角函数图像关于 $y = x$ 的反射。
- $\arcsin$ 的定义域是 $[-1, 1]$(所有合法的正弦(sine)值),值域是 $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$。
- $\arccos$ 定义域相同,但值域是 $[0, \pi]$。
- 它们的输入是比值;输出是角。
The graph of an inverse trig function is the restricted trig graph reflected across $y = x$. · 反三角函数的图像是被限制的三角函数图像关于 $y = x$ 的反射。
As with any inverse, swapping input and output mirrors the graph across the line $y = x$. · 和任何反函数一样,交换输入和输出使图像关于直线 $y = x$ 镜像。
Reading a value
- To find $\arccos(0)$, ask "which angle in $[0,\pi]$ has cosine $0$?" — the answer is $\tfrac{\pi}{2}$.
- The inverse returns the one angle inside the restricted range, called the principal value.
- Other angles may also work, but the inverse function reports just one.
- Add full turns or use symmetry if you need the rest.
读一个值
- 要求 $\arccos(0)$,就问"$[0,\pi]$ 里哪个角的余弦是 $0$?"——答案是 $\tfrac{\pi}{2}$。
- 反函数返回受限范围内那一个角,叫做主值。
- 其他角可能也成立,但反函数只报告一个。
- 如果你需要其余的,就加上整圈或利用对称性。
The inverse gives only the principal angle, not every solution. $\arcsin(0.5) = 30°$, but $150°$ also has sine $0.5$. When solving an equation you must find all the angles yourself — the inverse hands you just one.
反函数只给出主角,而不是每一个解。$\arcsin(0.5) = 30°$,但 $150°$ 的正弦也是 $0.5$。解方程时你必须自己找出所有的角——反函数只交给你一个。
A ramp rises $3$ m over a slope length of $6$ m. What angle does it make?
- The sine of the angle is $\tfrac{\text{rise}}{\text{slope}} = \tfrac{3}{6} = 0.5$.
- So the angle is $\arcsin(0.5) = 30°$.
- The inverse turned the measured ratio back into the angle we wanted.
一条斜坡在 $6$ 米的坡长上升高 $3$ 米。它构成多大的角?
- 角的正弦是 升高 ÷ 坡长 $= \tfrac{3}{6} = 0.5$。
- 所以角是 $\arcsin(0.5) = 30°$。
- 反函数把测得的比值变回了我们想要的角。
An inverse trig function returns an angle from a ratio: $\arcsin$, $\arccos$, $\arctan$. Since trig functions are not one-to-one, we restrict their domain first. The inverse graphs reflect the restricted sine/cosine/tangent across $y = x$, and report just the principal angle.
反三角函数从一个比值返回一个角:$\arcsin$、$\arccos$、$\arctan$。由于三角函数不是一一对应的,我们先限制它们的定义域。反函数的图像是受限正弦/余弦/正切关于 $y = x$ 的反射,且只报告主角。