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.
Reviewed by Manoj Kumar, Mathematics Educator
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy