Arcsine Domain and Range Calculator
Analyze transformed arcsine functions, calculate valid domains and ranges, evaluate values, review steps, graph results, and export your calculations easily.
Calculator Inputs
For real-valued arcsine, the inner expression must remain between −1 and 1.
Example Presets
Understanding Arcsine Domain and Range
The arcsine function, written as arcsin(x) or sin⁻¹(x), reverses the sine function on a carefully restricted interval. Ordinary sine repeats forever, so it cannot have a unique inverse across every real angle. Mathematicians therefore restrict sine to the interval from −π/2 to π/2. On that interval, sine is one-to-one and covers every value from −1 through 1. This creates the standard arcsine domain of [−1, 1] and range of [−π/2, π/2]. In degrees, the range is [−90°, 90°].
Transformed Arcsine Functions
A transformed function can be written as y = a·arcsin(b(x−h))+k. The inner expression controls the allowed x-values. Because arcsine accepts only inputs between −1 and 1, the condition −1 ≤ b(x−h) ≤ 1 determines the domain. When b is nonzero, the domain becomes h−1/|b| ≤ x ≤ h+1/|b|. A larger absolute b compresses the graph horizontally, while a smaller nonzero absolute b stretches it. A negative b reflects the input direction.
Finding the Range
The base arcsine output always lies between −π/2 and π/2. Multiplying by a changes the vertical scale and can reflect the graph. Adding k shifts every output vertically. The transformed range therefore extends from k−|a|π/2 to k+|a|π/2. If a equals zero, the graph becomes the constant y = k. If b equals zero, the arcsine input is always zero, so the whole function is also constant at k for every real x.
Using This Calculator
Enter a, b, h, and k to describe your function. Add an x-value to test whether it belongs to the domain and evaluate the function when valid. Choose radians or degrees, set decimal precision, and calculate. The results show interval notation, inequality notation, transformation details, evaluation output, and step-by-step reasoning. The graph helps connect the formulas with the actual curve. CSV and PDF exports make the calculated values easier to save, share, or include in coursework.
Example Table
| Function | Domain | Range in radians | Main change |
|---|---|---|---|
| y = arcsin(x) | [−1, 1] | [−π/2, π/2] | Standard arcsine |
| y = 2arcsin(x) | [−1, 1] | [−π, π] | Vertical stretch |
| y = arcsin(2x) | [−0.5, 0.5] | [−π/2, π/2] | Horizontal compression |
| y = arcsin(x−3)+2 | [2, 4] | [2−π/2, 2+π/2] | Right and upward shift |
Frequently Asked Questions
What is the domain of arcsin(x)?
The real-valued domain is [−1, 1]. Inputs outside that interval have no real arcsine value.
What is the range of arcsin(x)?
Its principal range is [−π/2, π/2], or [−90°, 90°].
How does b change the domain?
For b ≠ 0, the domain is centered at h and extends 1/|b| in each direction.
How does a change the range?
The range becomes k−|a|π/2 through k+|a|π/2. Negative a also reflects the graph vertically.
What happens when b equals zero?
The inner arcsine input is always zero. The domain becomes all real numbers, and the function equals k.
Can I export my result?
Yes. After calculating, use the CSV or PDF buttons in the results section.