Arcsine Calculator (sin−1)

Calculate the inverse sine ( arcsin, sin−1 ) in degrees, radians, and exact π multiples. Explore interactive unit circle coordinate projections, right triangle ratios, and calculus identities.

Reviewed by Sanjay Samanta & Academic Review Board
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Last Updated: September 2026
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Trigonometric & Analytical Proof Verified

Arcsine ( sin−1 ) Calculator

Calculate inverse sine in degrees, radians, and exact π multiples with interactive unit circle visualization.

Special Exact Value Presets: Click to auto-evaluate
x ∈ [−1, 1]
Principal Arcsine Output (−90° ≤ θ ≤ 90°)
Degrees (°) 30° 30° 00′ 00″
Exact π Radians π / 6 Exact Fractional Multiple
Decimal Radians 0.5236 rad Standard SI Radians
Grradians (grad) 33.3333 grad Grade / Centesimal
Trigonometric Function Values at Angle θ:
sin(θ) 0.5000
cos(θ) 0.8660
tan(θ) 0.5774
csc(θ) 2.0000
sec(θ) 1.1547
cot(θ) 1.7321
Unit Circle Visualizer (Radius = 1) Range: [−π/2, π/2]
x y (1,0) (-1,0)
Orange dashed line: vertical sine projection y = 0.50
Full Circle Solutions (0° ≤ θ < 360°)
Quadrant I / IV (Principal Solution): 30.00° (π/6 rad)
Quadrant II / III (Second Solution = 180° − θ): 150.00° (5π/6 rad)
Evaluation Steps:
1. Input sine value: x = 0.50
2. θ = arcsin(0.50) = 30° = π/6 radians
3. Verified: sin(30°) = 0.5000 ✓
Direct Answer & Overview
Verified Educational Guide

How to Calculate Arcsine (Inverse Sine)

To calculate the arcsine of a number x, find the angle θ whose sine equals x: θ = arcsin(x) or θ = sin⁻¹(x). The input value x must satisfy -1 ≤ x ≤ 1. The principal output angle θ is restricted to [-π/2, π/2] radians (or [-90°, 90°] in degrees).

Primary Mathematical Formula Standard Mathematical Model
Standard Equation
ƒ(x)
Q.E.D.
θ = arcsin(x) = sin⁻¹(x) | sin(θ) = x, where -1 ≤ x ≤ 1 and -π/2 ≤ θ ≤ π/2
Evaluated with exact mathematical formulation • Rigorously verified
Exact Formula
Input Parameters
Required
1
Sine value x within real interval [-1, 1], or triangle Opposite/Hypotenuse ratio
Expected Outputs
Calculated
Angle θ in Degrees (°), exact π Radians, decimal Radians, Grradians, and 6 companion trig ratios
Worked Numerical Example
Instant Verification
Evaluate θ = arcsin(0.5)
→ sin(30°) = 1/2 = 0.5 → arcsin(0.5) = 30° = π/6 radians
θ = 30° (π / 6 rad | 0.5236 rad)

Mathematical Definition, Domain & Principal Range

The standard sine function f(θ) = sin(θ) maps an angle θ to a real number in the interval [−1, 1]. Because sine is periodic with period 2π, it fails the horizontal line test across the entire real number line. To define a true single-valued inverse function arcsine, mathematicians restrict the domain of sine to the interval [−π/2, π/2]:

Domain Constraints

Input Domain: [−1, 1]

Because |sin(θ)| ≤ 1, arcsine is only defined for real numbers satisfying:

−1 ≤ x ≤ 1
Principal Range

Output Range: [−π/2, π/2]

The principal value always returns an angle in Quadrant I or Quadrant IV:

−π/2 ≤ θ ≤ π/2  (−90° ≤ θ ≤ 90°)

Exact Special Values Reference Table

Standard trigonometry curricula and standardized tests (AP Calculus, SAT, ACT, GRE) require mastery of exact radical values of arcsine without a calculator:

Input x (Exact) Decimal x Principal Angle θ (°) Principal Angle θ (Radians) Quadrant
−1 −1.0000 −90° −π / 2 Negative Y-Axis
−√3 / 2 −0.8660 −60° −π / 3 Quadrant IV
−√2 / 2 −0.7071 −45° −π / 4 Quadrant IV
−1 / 2 −0.5000 −30° −π / 6 Quadrant IV
0 0.0000 0° 0 Origin / Positive X
1 / 2 0.5000 30° π / 6 Quadrant I
√2 / 2 0.7071 45° π / 4 Quadrant I
√3 / 2 0.8660 60° π / 3 Quadrant I
1 1.0000 90° π / 2 Positive Y-Axis

Inverse Trigonometric Identities & Calculus Rules

Composition Cancellation
sin(arcsin(x)) = x

Holds for all x ∈ [−1, 1].

arcsin(sin(θ)) = θ

Holds ONLY when θ ∈ [−π/2, π/2].

Complementary Cofunction Angle
arcsin(x) + arccos(x) = π / 2

The sum of arcsine and arccosine of any value in [−1, 1] always equals 90° (π/2 rad).

Calculus Derivative Rule
d/dx [arcsin(x)] = 1 / √(1 − x²)

For all −1 < x < 1 (non-differentiable at endpoints ±1).

Indefinite Integration Formula
∫ arcsin(x) dx = x arcsin(x) + √(1 − x²) + C

Derived via integration by parts with u = arcsin(x) and dv = dx.

Step-by-Step Worked Numerical Solutions

Example 1: Right Triangle Angle Finding

Find Angle θ with Opposite = 5 and Hypotenuse = 10

1. Compute sine ratio: sin(θ) = Opp / Hyp = 5 / 10 = 0.50
2. Apply inverse sine: θ = arcsin(0.50)
3. Recall unit circle: sin(30°) = 0.50
4. Result: θ = 30° (π/6 radians)
Example 2: Negative Input Evaluation

Evaluate θ = arcsin(−0.7071)

1. Recognize radical form: −0.7071 ≈ −√2 / 2
2. Because arcsine is an odd function: arcsin(−x) = −arcsin(x)
3. θ = −arcsin(√2 / 2) = −45° (−π/4 rad)
4. Result: θ = −45° (−π/4 rad)

Physics, Optics & Engineering Applications

1. Optics & Snell's Law of Refraction

When light passes from medium 1 (index n1) to medium 2 (index n2), Snell's Law states n1 sin(θ1) = n2 sin(θ2). To solve for the angle of refraction: θ2 = arcsin((n1/n2) sin(θ1)). The critical angle for total internal reflection is θcrit = arcsin(n2 / n1).

2. Ballistics & Projectile Launch Angles

To hit a target at distance R with initial velocity v0 under gravity g, the launch angle θ satisfies R = (v0² / g) sin(2θ). Solving for θ yields θ = 0.5 × arcsin(g R / v0²).

3. Robotics Kinematics & Game Development

Robotic inverse kinematics and 3D game engines calculate joint angles and pitch orientations from directional vector heights using pitch = arcsin(y / length).

Frequently Asked Questions

What is arcsine (sin⁻¹) in trigonometry?
Arcsine, written as arcsin(x) or sin⁻¹(x), is the inverse function of the sine trigonometric function. Given a real number x between -1 and 1 representing the sine of an angle, arcsine returns the principal angle θ whose sine equals x: sin(θ) = x.
What are the domain and range of the arcsine function?
The domain of arcsin(x) is [-1, 1] because the sine of any real angle always oscillates between -1 and 1. The principal range of arcsin(x) is restricted to [-π/2, π/2] in radians, or [-90°, 90°] in degrees (covering Quadrants I and IV), ensuring that arcsine is a well-defined single-valued mathematical function.
Why does arcsin(x) produce an error for inputs like 1.5 or -2?
In real Euclidean geometry, the sine of an angle in a right triangle is the ratio of the Opposite side to the Hypotenuse (Opposite / Hypotenuse). Because the hypotenuse is always the longest side, this ratio can never exceed 1 or drop below -1. Any value |x| > 1 lies outside the real domain of arcsine.
What is the difference between sin⁻¹(x) and 1/sin(x)?
sin⁻¹(x) denotes the inverse function arcsin(x), which calculates an angle from a ratio. In contrast, 1/sin(x) is the reciprocal trigonometric function known as cosecant, csc(x) or (sin(x))⁻¹. They are fundamentally different mathematical concepts.
How do you find all solutions to sin(θ) = x on a full 360° circle?
For any sine value x, there are generally two solutions in [0°, 360°): the principal solution θ₁ = arcsin(x) (adjusted to [0°, 360°)), and the symmetric solution θ₂ = 180° - θ₁ (or π - θ₁ in radians). The general solutions across all integers k are θ = θ₁ + 360°k and θ = θ₂ + 360°k.
What is the derivative and integral of arcsin(x)?
The first derivative with respect to x is d/dx[arcsin(x)] = 1 / √(1 - x²) for -1 < x < 1. The indefinite integral is ∫ arcsin(x) dx = x · arcsin(x) + √(1 - x²) + C.

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