Inverse Trigonometric Functions
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| sine/saɪn/ | hàm sin |
| inverse/ɪnˈvɜːs/ | hàm ngược |
| one-to-one/wʌn tuː wʌn/ | đơn ánh |
| domain/dəˈmeɪn/ | miền xác định |
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.
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.
The inverse function $\arcsin$ (or $\sin^{-1}$) takes a value and returns…
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The inverse of sine reverses it: $\arcsin(0.5)$ is the angle whose sine is $0.5$, namely $30°$.
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.
From a ratio back to an angle
An inverse trig function answers the reverse question — given the sine or cosine value, which angle produced it?
Why must sine be restricted before it can have an inverse function?
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Many angles share the same sine, so sine is not one-to-one. Restricting its domain fixes a single output per input.
What is $\arcsin(1)$ (in radians)?
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$\sin\tfrac{\pi}{2} = 1$, so $\arcsin(1) = \tfrac{\pi}{2}$ — the angle in the restricted range.
Select all true statements about inverse trig functions.
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An inverse returns just one angle (in the restricted range), not all of them. The other three are correct.
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.
The graph of an inverse trig function is the restricted trig graph reflected across $y = x$.
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As with any inverse, swapping input and output mirrors the graph across the line $y = x$.
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.
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.
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.
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.
Write from an angle of pi over 4
- Find arctangent of 1.
- The result is an angle, not a ratio.
What is arctangent of 1, in the standard range?
The principal angle whose tangent is 1 is pi over 4.
What angle does arctangent of 1 return?
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Arctangent undoes tangent on its restricted domain. Other options — 1 (the ratio left as an angle); pi over 2 (the right angle).