Visualize arcsin, arccos, arctan, arccsc, arcsec, and arccot on their restricted domains. Each graph shows labeled π-fraction axes, open/closed endpoint circles, asymptote dashes, and a movable gold evaluation point. Toggle the parent trig function overlay to see the reflection relationship across y = x.
The sine function is not one-to-one on all of ℝ — infinitely many angles share the same sine value. For example, sin(π/6) = sin(5π/6) = 0.5, so the horizontal line y = 0.5 crosses the sine graph in infinitely many places.
To define an inverse, we restrict the domain to an interval where sin(x) is strictly increasing (or strictly decreasing) — so each output value comes from exactly one input. We choose the interval that contains 0 and still covers the full output range [−1, 1].
This is the horizontal line test: a function has an inverse if and only if every horizontal line crosses its graph at most once.
| Function | Domain | Range (Principal) | Endpoints |
|---|---|---|---|
| arcsin(x) | [−1, 1] | [−π/2, π/2] | Closed both |
| arccos(x) | [−1, 1] | [0, π] | Closed both |
| arctan(x) | (−∞, ∞) | (−π/2, π/2) | Open — asymptotes |
| arccsc(x) | (−∞,−1]∪[1,∞) | [−π/2,0)∪(0,π/2] | Closed at |x|=1, open at 0 |
| arcsec(x) | (−∞,−1]∪[1,∞) | [0,π/2)∪(π/2,π] | Closed at |x|=1, open at π/2 |
| arccot(x) | (−∞, ∞) | (0, π) | Open — asymptotes |
Our tutors make restricted domains, principal values, and inverse function graphs click — from the unit circle all the way to calculus applications.