What is inverse sine (arcsin)?

Inverse sine, denoted as arcsin(x) or sin⁻¹(x), is the inverse function of the sine function. It returns the angle whose sine is x. The domain of arcsin is [-1, 1] and the range is [-π/2, π/2] radians (or [-90°, 90°] degrees).

What is the domain and range of arcsin?

The domain of arcsin is [-1, 1] because sine values are always between -1 and 1. The range is [-π/2, π/2] in radians or [-90°, 90°] in degrees. This ensures arcsin is a function (one-to-one).

How do you calculate arcsin?

To calculate arcsin(x), find the angle θ such that sin(θ) = x. For example, arcsin(0.5) = 30° (or π/6 radians) because sin(30°) = 0.5. Use a calculator's arcsin function or look up values in trigonometric tables.

What are special values of inverse sine?

Common special values: arcsin(0) = 0°, arcsin(0.5) = 30° (π/6), arcsin(√2/2) = 45° (π/4), arcsin(√3/2) = 60° (π/3), arcsin(1) = 90° (π/2), arcsin(-1) = -90° (-π/2). These correspond to common trigonometric angles.

How is arcsin different from sin?

sin(θ) takes an angle and returns a ratio, while arcsin(x) takes a ratio and returns an angle. They are inverse functions: sin(arcsin(x)) = x for x in [-1, 1], and arcsin(sin(θ)) = θ only when θ is in [-π/2, π/2].

Can arcsin be greater than 90 degrees?

No, the principal value of arcsin is always between -90° and 90° (or -π/2 and π/2 radians). However, there are infinite angles with the same sine value, but arcsin returns only the principal value in this range.

What is the general solution for arcsin?

For arcsin(x) = θ, the general solution is: θ = arcsin(x) + 2πn OR θ = π - arcsin(x) + 2πn, where n is any integer. In degrees: θ = arcsin(x)° + 360°n OR θ = 180° - arcsin(x)° + 360°n.

How do you verify an arcsin calculation?

To verify arcsin(x) = θ, check that sin(θ) = x. For example, if arcsin(0.5) = 30°, then sin(30°) should equal 0.5. This verification confirms the calculation is correct.

What is the relationship between arcsin and arccos?

arcsin(x) and arccos(x) are complementary: arcsin(x) + arccos(x) = π/2 (90°). This relationship follows from the identity sin²(θ) + cos²(θ) = 1 and the fact that complementary angles have related sine and cosine values.

Where is inverse sine used?

Inverse sine is used in trigonometry, calculus, physics (wave analysis, projectile motion), engineering (signal processing, control systems), computer graphics (angle calculations), and navigation. It's essential for solving equations involving sine functions.

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Trigonometry Tool

Inverse Sine Calculator - Arcsin Calculator & Sin Inverse Calculator

Free inverse sine calculator & arcsin calculator. Calculate arcsin(x) for values in the range [-1, 1]. Get results in radians and degrees with step-by-step solutions, special value recognition, and trigonometric analysis.

Last updated: February 2, 2026

Calculate arcsin in radians and degrees

Special value recognition (0, 0.5, √2/2, etc.)

Step-by-step solutions with verification

Need a custom math calculator for your educational platform? Get a Quote

Inverse Sine Calculator

Calculate arcsin(x) with results in radians and degrees. Input must be in range [-1, 1].

Result

arcsin(0.5) = 0.523599 rad
= 30.00°

Special Value:

π/6 = 30°

Radians

0.523599

Degrees

30.00°

Quadrant & Range:

Quadrant: I

Range: [-π/2, π/2]

General Solution:

θ = 30.00° + 360°n OR θ = 150.00° + 360°n (where n ∈ ℤ)

Step-by-Step Solution

Step 1: Check domain - Input value 0.5 is in valid range [-1, 1]
Step 2: Calculate arcsin(0.5)
Step 3: arcsin(0.5) = 0.523599 radians
Step 4: Convert to degrees: 0.523599 × (180/π) = 30.000000°
Step 5: Quadrant analysis - Result is in Quadrant I (0° to 90°)
Principal value range: [-π/2, π/2]
Step 6: Special value detected!
arcsin(0.5) = π/6 = 30°
Step 7: General solution (all angles with this sine value):
In radians: θ = 0.5236 + 2πn OR θ = 2.6180 + 2πn (where n ∈ ℤ)
In degrees: θ = 30.00° + 360°n OR θ = 150.00° + 360°n (where n ∈ ℤ)
Step 8: Verification: sin(arcsin(0.5)) = sin(0.523599) = 0.500000
✓ Verification successful: Result matches input value

Inverse Sine Calculator Features

Arcsin Calculator

Calculate arcsin(x) for values in domain [-1, 1]

Domain & Range

Domain: [-1, 1]
Range: [-π/2, π/2]

Returns angle whose sine equals input value

Special Value Recognition

Recognizes common trigonometric values

Special Values

0, 0.5, √2/2, √3/2, ±1

Shows exact values like π/6, π/4, π/3, π/2

General Solution

Find all angles with the same sine value

Formula

θ = arcsin(x) + 2πn
OR
θ = π - arcsin(x) + 2πn

All solutions including periodic repetitions

Domain Validation

Automatic check for valid input range

Valid Range

-1 ≤ x ≤ 1

Error messages for invalid inputs

Automatic Verification

Verify that sin(arcsin(x)) = x

Verification

sin(arcsin(x)) = x

Confirms calculation accuracy

Step-by-Step Solutions

Detailed explanations for each calculation step

Includes

All calculation steps

Learn the arcsin calculation process

Quick Example Result

arcsin(0.5)

Radians

π/6 ≈ 0.5236

Degrees

30°

How the Inverse Sine Calculator Works

Our inverse sine calculator computes arcsin(x) using the standard mathematical definition. The calculator validates the input domain, calculates the principal value, identifies special values, and provides complete step-by-step solutions. The result is verified by checking that sin(arcsin(x)) = x.

Inverse Sine Definition

Definition: arcsin(x) = θ such that sin(θ) = x
Domain: x ∈ [-1, 1]
Range: θ ∈ [-π/2, π/2] radians or [-90°, 90°] degrees
Property: sin(arcsin(x)) = x (for x in domain)

The range restriction ensures arcsin is a function (one-to-one), returning only the principal value.

Mathematical Foundation

Inverse sine is the inverse function of sine, meaning it "undoes" the sine operation. Since sine is not one-to-one (multiple angles have the same sine value), we restrict the range to [-π/2, π/2] to make arcsin a proper function. The inverse sine function is odd: arcsin(-x) = -arcsin(x).

arcsin is the inverse of sin on the restricted domain
Principal value is always in [-π/2, π/2] or [-90°, 90°]
arcsin(-x) = -arcsin(x) (odd function property)
arcsin(x) + arccos(x) = π/2 (complementary relationship)
General solution includes all angles with the same sine value
Derivative: d/dx[arcsin(x)] = 1/√(1-x²)

Special Values of Inverse Sine

arcsin(0) = 0° = 0 radians
arcsin(0.5) = 30° = π/6 radians
arcsin(√2/2) = 45° = π/4 radians
arcsin(√3/2) = 60° = π/3 radians
arcsin(1) = 90° = π/2 radians
arcsin(-1) = -90° = -π/2 radians

Applications of Inverse Sine

Inverse sine is essential in many fields:

Trigonometry: Solving equations involving sine functions
Calculus: Integration techniques and derivative calculations
Physics: Wave analysis, harmonic motion, projectile motion angles
Engineering: Signal processing, control systems, phase calculations
Computer Graphics: Angle calculations for rotations and transformations
Navigation: Calculating angles and directions from coordinate systems

Sources & References

Trigonometry - Ron Larson (10th Edition)Comprehensive coverage of inverse trigonometric functions
Precalculus: Mathematics for Calculus - Stewart, Redlin, Watson (7th Edition)Detailed explanation of inverse functions and their properties
Khan Academy - Inverse Trigonometric FunctionsInteractive lessons on arcsin and other inverse trig functions

Need help with other trigonometric functions? Try our inverse tangent calculator or tangent calculator.

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Inverse Sine Calculator Examples

Example: arcsin(0.5)

Calculate the inverse sine of 0.5

Given:

Input: x = 0.5
Domain check: 0.5 ∈ [-1, 1] ✓
Task: Find θ such that sin(θ) = 0.5

Step-by-Step Solution:

Check domain: 0.5 is in [-1, 1]
Calculate: arcsin(0.5) = π/6 radians
Convert: π/6 = 30°
Verify: sin(30°) = 0.5 ✓
Special value: π/6 = 30°

Result: arcsin(0.5) = 30° = π/6 radians

This is a special value corresponding to a common trigonometric angle.

Example: arcsin(1)

arcsin(1) = 90° = π/2

Maximum value of arcsin

Example: arcsin(-0.5)

arcsin(-0.5) = -30° = -π/6

Negative input gives negative angle

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Trigonometry Tool

Inverse Sine Calculator - Arcsin Calculator & Sin Inverse Calculator

Last updated: February 2, 2026

Calculate arcsin in radians and degrees

Special value recognition (0, 0.5, √2/2, etc.)

Step-by-step solutions with verification

Need a custom math calculator for your educational platform? Get a Quote

Inverse Sine Calculator

Calculate arcsin(x) with results in radians and degrees. Input must be in range [-1, 1].

Result

arcsin(0.5) = 0.523599 rad
= 30.00°

Special Value:

π/6 = 30°

Radians

0.523599

Degrees

30.00°

Quadrant & Range:

Quadrant: I

Range: [-π/2, π/2]

General Solution:

θ = 30.00° + 360°n OR θ = 150.00° + 360°n (where n ∈ ℤ)

Step-by-Step Solution

Step 1: Check domain - Input value 0.5 is in valid range [-1, 1]
Step 2: Calculate arcsin(0.5)
Step 3: arcsin(0.5) = 0.523599 radians
Step 4: Convert to degrees: 0.523599 × (180/π) = 30.000000°
Step 5: Quadrant analysis - Result is in Quadrant I (0° to 90°)
Principal value range: [-π/2, π/2]
Step 6: Special value detected!
arcsin(0.5) = π/6 = 30°
Step 7: General solution (all angles with this sine value):
In radians: θ = 0.5236 + 2πn OR θ = 2.6180 + 2πn (where n ∈ ℤ)
In degrees: θ = 30.00° + 360°n OR θ = 150.00° + 360°n (where n ∈ ℤ)
Step 8: Verification: sin(arcsin(0.5)) = sin(0.523599) = 0.500000
✓ Verification successful: Result matches input value

Inverse Sine Calculator Features

Arcsin Calculator

Calculate arcsin(x) for values in domain [-1, 1]

Domain & Range

Domain: [-1, 1]
Range: [-π/2, π/2]

Returns angle whose sine equals input value

Special Value Recognition

Recognizes common trigonometric values

Special Values

0, 0.5, √2/2, √3/2, ±1

Shows exact values like π/6, π/4, π/3, π/2

General Solution

Find all angles with the same sine value

Formula

θ = arcsin(x) + 2πn
OR
θ = π - arcsin(x) + 2πn

All solutions including periodic repetitions

Domain Validation

Automatic check for valid input range

Valid Range

-1 ≤ x ≤ 1

Error messages for invalid inputs

Automatic Verification

Verify that sin(arcsin(x)) = x

Verification

sin(arcsin(x)) = x

Confirms calculation accuracy

Step-by-Step Solutions

Detailed explanations for each calculation step

Includes

All calculation steps

Learn the arcsin calculation process

Quick Example Result

arcsin(0.5)

Radians

π/6 ≈ 0.5236

Degrees

30°

How the Inverse Sine Calculator Works

Inverse Sine Definition

Definition: arcsin(x) = θ such that sin(θ) = x
Domain: x ∈ [-1, 1]
Range: θ ∈ [-π/2, π/2] radians or [-90°, 90°] degrees
Property: sin(arcsin(x)) = x (for x in domain)

The range restriction ensures arcsin is a function (one-to-one), returning only the principal value.

Mathematical Foundation

arcsin is the inverse of sin on the restricted domain
Principal value is always in [-π/2, π/2] or [-90°, 90°]
arcsin(-x) = -arcsin(x) (odd function property)
arcsin(x) + arccos(x) = π/2 (complementary relationship)
General solution includes all angles with the same sine value
Derivative: d/dx[arcsin(x)] = 1/√(1-x²)

Special Values of Inverse Sine

arcsin(0) = 0° = 0 radians
arcsin(0.5) = 30° = π/6 radians
arcsin(√2/2) = 45° = π/4 radians
arcsin(√3/2) = 60° = π/3 radians
arcsin(1) = 90° = π/2 radians
arcsin(-1) = -90° = -π/2 radians

Applications of Inverse Sine

Inverse sine is essential in many fields:

Trigonometry: Solving equations involving sine functions
Calculus: Integration techniques and derivative calculations
Physics: Wave analysis, harmonic motion, projectile motion angles
Engineering: Signal processing, control systems, phase calculations
Computer Graphics: Angle calculations for rotations and transformations
Navigation: Calculating angles and directions from coordinate systems

Sources & References

Trigonometry - Ron Larson (10th Edition)Comprehensive coverage of inverse trigonometric functions
Precalculus: Mathematics for Calculus - Stewart, Redlin, Watson (7th Edition)Detailed explanation of inverse functions and their properties
Khan Academy - Inverse Trigonometric FunctionsInteractive lessons on arcsin and other inverse trig functions