Inverse Trigonometric Functions
Introduced| English |
|---|
| sine/saɪn/ |
| inverse/ɪnˈvɜːs/ |
| one-to-one/wʌn tuː wʌn/ |
| domain/dəˈmeɪn/ |
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…
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?
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)?
$\sin\tfrac{\pi}{2} = 1$, so $\arcsin(1) = \tfrac{\pi}{2}$ — the angle in the restricted range.
Select all true statements about inverse trig functions.
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 $y = x$.
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.