Arcsin Calculator
Free Arcsin Calculator for trigonometry. Enter values to get step-by-step solutions with formulas and graphs. See charts, tables, and visual results.
Calculator
Adjust values & calculateEnter a value between -1 and 1
Formula
The inverse sine function returns the angle (in radians from -pi/2 to pi/2, or degrees from -90 to 90) whose sine equals the input value x. The input must be between -1 and 1 inclusive.
Last reviewed: December 2025
Worked Examples
Example 1: Finding an Angle from a Sine Value
Example 2: Snell Law Refraction Angle
Background & Theory
The Arcsin Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Arcsin Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
arcsin(x) = angle where sin(angle) = x, for x in [-1, 1]
The inverse sine function returns the angle (in radians from -pi/2 to pi/2, or degrees from -90 to 90) whose sine equals the input value x. The input must be between -1 and 1 inclusive.
Worked Examples
Example 1: Finding an Angle from a Sine Value
Problem: The sine of an angle is 0.866. Find the angle in degrees and radians.
Solution: arcsin(0.866) = 59.9971 degrees (approximately 60 degrees)\nIn radians: 60 x pi/180 = 1.0472 radians = pi/3\n\n0.866 is approximately sqrt(3)/2\nThis is the sine of 60 degrees in a 30-60-90 triangle\n\nVerification: sin(60 degrees) = sin(pi/3) = sqrt(3)/2 = 0.8660\ncos(60 degrees) = 0.5\ntan(60 degrees) = sqrt(3) = 1.7321
Result: arcsin(0.866) = 60 degrees = pi/3 radians
Example 2: Snell Law Refraction Angle
Problem: Light enters water (n=1.33) from air (n=1.0) at 45 degrees. Find the refraction angle using arcsin.
Solution: Snell Law: n1 x sin(theta1) = n2 x sin(theta2)\n1.0 x sin(45) = 1.33 x sin(theta2)\nsin(theta2) = sin(45) / 1.33 = 0.7071 / 1.33 = 0.5317\ntheta2 = arcsin(0.5317) = 32.12 degrees\n\nThe light bends toward the normal (45 to 32.12 degrees)\nbecause it enters a denser medium.
Result: Refraction angle = 32.12 degrees (light bends toward normal)
Frequently Asked Questions
What is arcsin (inverse sine)?
Arcsin, also written as sin^(-1) or asin, is the inverse function of the sine function. Given a value x between -1 and 1, arcsin(x) returns the angle whose sine equals x. The output is restricted to the range [-90, 90] degrees or [-pi/2, pi/2] radians, which is called the principal value range. This restriction ensures the function returns exactly one unique answer for each input. For example, arcsin(0.5) = 30 degrees because sin(30 degrees) = 0.5. The arcsin function is fundamental in trigonometry and is used extensively in physics, engineering, signal processing, and computer graphics whenever you need to find an angle from a known sine ratio.
What is the domain and range of arcsin?
The domain of arcsin is the closed interval [-1, 1], meaning you can only compute arcsin of values between -1 and 1 inclusive. Any input outside this range is mathematically undefined because sine values never exceed 1 or go below -1. The range (output) of arcsin is [-pi/2, pi/2] radians or equivalently [-90, 90] degrees. At the boundary values: arcsin(-1) = -90 degrees, arcsin(0) = 0 degrees, and arcsin(1) = 90 degrees. The function is strictly increasing throughout its domain, meaning larger input values always produce larger angles. This monotonic property makes arcsin a well-defined inverse function within its principal value range.
What is the relationship between arcsin and arccos?
Arcsin and arccos are complementary functions, satisfying the identity arcsin(x) + arccos(x) = pi/2 (or 90 degrees) for all x in [-1, 1]. This means if you know one, you can easily find the other by subtracting from 90 degrees. For example, arcsin(0.5) = 30 degrees and arccos(0.5) = 60 degrees, and 30 + 60 = 90. This relationship comes from the complementary angle identity in trigonometry: sin(theta) = cos(90 - theta). So if sin(theta) = x, then cos(90 - theta) = x, meaning arcsin(x) = theta and arccos(x) = 90 - theta. The domains of both functions are identical [-1, 1], but their ranges differ: arcsin outputs [-90, 90] while arccos outputs [0, 180].
What are the common arcsin values to know?
The key arcsin values correspond to special angles on the unit circle. arcsin(0) = 0 degrees, arcsin(1/2) = 30 degrees, arcsin(sqrt(2)/2) = 45 degrees, arcsin(sqrt(3)/2) = 60 degrees, and arcsin(1) = 90 degrees. For negative values: arcsin(-1/2) = -30 degrees, arcsin(-sqrt(2)/2) = -45 degrees, arcsin(-sqrt(3)/2) = -60 degrees, and arcsin(-1) = -90 degrees. The decimal approximations are: arcsin(0.5) = 30 degrees, arcsin(0.7071) = 45 degrees, arcsin(0.8660) = 60 degrees. These values appear constantly in physics problems involving projectile motion, wave mechanics, and optics, as well as in geometry and calculus courses.
How is arcsin used in physics?
Arcsin appears in many physics applications. In optics, Snell law states n1 times sin(theta1) = n2 times sin(theta2), and finding the refraction angle requires arcsin: theta2 = arcsin(n1 times sin(theta1) / n2). In projectile motion, the launch angle for a given range R at velocity v is theta = arcsin(Rg / v^2) / 2, where g is gravitational acceleration. In wave physics, arcsin determines phase angles and interference patterns. In mechanics, arcsin is used to find angles of inclined planes from force components. In electrical engineering, arcsin appears in AC circuit analysis for finding phase differences between voltage and current. In astronomy, arcsin helps calculate declination angles and the altitude of celestial objects from their coordinates.
What is the derivative and integral of arcsin?
The derivative of arcsin(x) with respect to x is 1 / sqrt(1 - x^2). This derivative is positive for all x in (-1, 1), confirming that arcsin is strictly increasing. The derivative becomes infinite at x = -1 and x = 1, corresponding to vertical tangent lines at the endpoints of the domain. The integral of arcsin(x) dx is x times arcsin(x) + sqrt(1 - x^2) + C, where C is the constant of integration. This integral is derived using integration by parts. Additionally, the integral of 1/sqrt(1-x^2) dx = arcsin(x) + C, which is one of the standard integral formulas memorized in calculus. These results are essential for solving differential equations and computing areas under curves in advanced mathematics.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy