Graph settings
Interactive inverse function graph
Click a curve to inspect its coordinates.
Value table and exports
| x | Transformed input | Function value | Exact or status |
|---|
Formula used
A controls vertical scale and reflection. B controls horizontal scale and reflection. H and K move the graph.
Inverse functions exchange original domains and ranges. Principal restrictions keep each inverse single-valued. The calculator applies those restrictions before plotting any points.
How to use the calculator
- Choose an inverse trigonometric function.
- Enter A, B, H, and K.
- Select radians or degrees.
- Set graph and table limits.
- Choose comparison and display options.
- Press Calculate and graph.
- Inspect points or export results.
Principal domains and ranges
| Function | Domain | Principal range | Restricted original |
|---|---|---|---|
| arcsin(x) | [-1, 1] | [-π/2, π/2] | y = sin(x), restricted to [-π/2, π/2] |
| arccos(x) | [-1, 1] | [0, π] | y = cos(x), restricted to [0, π] |
| arctan(x) | All real numbers | (-π/2, π/2) | y = tan(x), restricted to (-π/2, π/2) |
| arccsc(x) | (-∞, -1] ∪ [1, ∞) | [-π/2, 0) ∪ (0, π/2] | y = csc(x), restricted to [-π/2, 0) ∪ (0, π/2] |
| arcsec(x) | (-∞, -1] ∪ [1, ∞) | [0, π/2) ∪ (π/2, π] | y = sec(x), restricted to [0, π/2) ∪ (π/2, π] |
| arccot(x) | All real numbers | (0, π) | y = cot(x), restricted to (0, π) |
Example data
| Example | A | B | H | K | Purpose |
|---|---|---|---|---|---|
| y = arcsin(x) | 1 | 1 | 0 | 0 | Principal inverse sine |
| y = 2arcsin(x) | 2 | 1 | 0 | 0 | Vertical stretch |
| y = arctan(x − 1) + 2 | 1 | 1 | 1 | 2 | Horizontal and vertical shifts |
| y = −arccos(2x) | −1 | 2 | 0 | 0 | Reflection and horizontal compression |
| y = arcsec(x) | 1 | 1 | 0 | 0 | Two-branch inverse function |
Common mistakes
Inverse notation does not mean reciprocal notation. Arcsin differs from one divided by sine. Always check the selected principal range before interpreting answers.
Horizontal transformations occur inside the function. Their scale behaves inversely compared with vertical scaling. A zero B value cannot produce a valid inverse graph.
Frequently asked questions
What does an inverse trigonometric function return?
It returns an angle for a given ratio. The angle follows a defined principal range. This restriction makes the inverse function single-valued and graphable.
Are arcsin and cosecant the same?
No, arcsin is inverse sine. Cosecant is the reciprocal of sine. Their domains, ranges, and graphs are entirely different functions.
Why must the original function be restricted?
Trigonometric functions repeat their outputs. Unrestricted functions therefore fail the horizontal line test. A principal interval creates a one-to-one original function.
How are original and inverse graphs related?
Their coordinates exchange positions. This creates reflection across the line y equals x. Domain values become range values after applying the inverse.
Can this calculator use degrees?
Yes, choose degrees from angle output. Base inverse values convert before transformations are applied. Vertical shifts then use degrees within the displayed graph.
What happens when an input is outside the domain?
The calculator marks that value undefined. No misleading point appears on the plotted curve. The table also explains the invalid domain condition clearly.
Why does arctangent have horizontal asymptotes?
Its outputs approach limiting principal angles. They never reach those boundary values for finite inputs. Transformations move and scale both horizontal asymptotes accordingly.
How are arcsec and arccsc calculated?
Arcsec uses arccos of one divided by x. Arccsc uses arcsin of one divided by x. Both require absolute input values at least one.
Can I compare several inverse functions?
Yes, select comparison functions before calculating. Each curve receives a separate legend entry. Intersection detection estimates shared points inside the graph window.