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.
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.