Arcsin
Free arcsin (inverse sine) calculator -- compute sin⁻¹(x) in degrees and radians for values between -1 and 1. Interactive arcsin function graph and detailed trigonometry guide.
Enter a number between -1 and 1
About This Calculator
About the Arcsin (Inverse Sine) Calculator
This arcsin calculator (also known as inverse sine calculator or sin⁻¹ calculator) computes the inverse of the sine function for any real number in the domain [-1, 1]. Given a value x, the calculator returns the angle theta such that sin(theta) = x, where theta lies in the principal range of [-90 deg, 90 deg] (or [-pi/2, pi/2] radians). Results are displayed in both degrees and radians, and an interactive arcsin function curve graph helps visualize the relationship between input and output.
Arcsin Formula & Methodology
The arcsin function is defined as the inverse of the sine function with a restricted domain. Since sine is periodic and not one-to-one over all real numbers, its domain is restricted to [-pi/2, pi/2] to make the inverse a proper function. The calculation is:
y = arcsin(x) iff x = sin(y), where -pi/2 <= y <= pi/2
Degrees: arcsin(x) = sin⁻¹(x) x (180/pi)
Radians: arcsin(x) = sin⁻¹(x)
Domain Restrictions
The arcsin function is only defined for inputs in the closed interval [-1, 1]. This is because the sine function only outputs values between -1 and 1. Any value outside this range will produce an error -- the result would be a complex number, which is beyond the scope of a real-valued calculator.
Key Trigonometric Relationships
- Sine: sin(arcsin(x)) = x for all x in [-1, 1]
- Cosine: cos(arcsin(x)) = sqrt(1 - x^2)
- Tangent: tan(arcsin(x)) = x / sqrt(1 - x^2)
- Negative argument: arcsin(-x) = -arcsin(x) (odd function)
- Complementary: arcsin(x) + arccos(x) = pi/2
Application Areas
The arcsin function is widely used in physics (angle of incidence, refraction, projectile motion), engineering (signal processing, structural analysis), forensic science (angle of impact analysis), computer graphics (camera angles, 3D rotations), and navigation (bearing calculations). In education, it is fundamental to high school and college trigonometry curricula worldwide.
Frequently Asked Questions
What is arcsin (inverse sine)?
Arcsin (also written as sin⁻¹ or asin) is the inverse function of the sine function. For any value x in the range [-1, 1], arcsin(x) returns the angle theta (in degrees or radians) whose sine is x. In formula terms, arcsin(x) = y iff x = sin(y), where y lies in the principal range [-90 deg, 90 deg] or [-pi/2, pi/2] radians.
What is the domain and range of arcsin?
For real-valued results, the domain of arcsin is [-1, 1] -- meaning you can only compute arcsin for inputs between -1 and 1 inclusive. The principal range (output) of arcsin is [-90 deg, 90 deg] in degrees or [-pi/2, pi/2] in radians. Attempting arcsin outside the domain returns undefined (complex) results.
How is arcsin used in right triangle calculations?
In a right triangle, arcsin is used to find an unknown angle when the opposite side and hypotenuse lengths are known. If side a is opposite angle alpha and the hypotenuse is c, then sin(alpha) = a/c and therefore alpha = arcsin(a/c). This is essential in trigonometry, engineering, physics, and construction for determining angles from known side ratios.
What are the common arcsin values?
Common arcsin values include: arcsin(0) = 0 deg (0 rad), arcsin(0.5) = 30 deg (pi/6 rad), arcsin(sqrt2/2) = 45 deg (pi/4 rad), arcsin(sqrt3/2) = 60 deg (pi/3 rad), arcsin(1) = 90 deg (pi/2 rad), arcsin(-0.5) = -30 deg (-pi/6 rad), arcsin(-1) = -90 deg (-pi/2 rad). These correspond to standard angles on the unit circle.
What is the relationship between arcsin and arccos?
Arcsin and arccos are complementary inverse trigonometric functions. For any x in [-1, 1], arcsin(x) + arccos(x) = 90 deg (pi/2 radians). This identity is useful for converting between sine and cosine inverse values. For example, arcsin(0.6) = 36.87 deg, then arccos(0.6) = 53.13 deg, and 36.87 deg + 53.13 deg = 90 deg.
What is the derivative and integral of arcsin?
The derivative of arcsin(x) is d/dx arcsin(x) = 1 / sqrt(1 - x^2), which exists for all x in (-1, 1). The indefinite integral of arcsin(x) is integral arcsin(x) dx = x · arcsin(x) + sqrt(1 - x^2) + C. These formulas are fundamental in calculus for solving problems involving inverse trigonometric functions.
What is the difference between arcsin and sin⁻¹?
Arcsin and sin⁻¹ are two notations for the same inverse sine function. Arcsin is the preferred notation to avoid confusion, because sin⁻¹(x) could be misinterpreted as 1/sin(x) (which is actually csc(x), the cosecant). In computer programming, the function is often called asin(). Regardless of notation, arcsin(x) always returns the angle whose sine is x.