Inverse Sine Calculator

Inverse Sine Calculator — Calculate arcsin(x) in Degrees or Radians

Compute the inverse sine (arcsin) of any value between −1 and 1 with step-by-step explanation. Learn the domain, range, graph, and real-world applications of arcsin with comprehensive FAQs. For related trigonometry tools, see our Pythagorean theorem calculator or the distance calculator.

Table of Contents

1. 1What Is Inverse Sine?
2. 2Domain and Range
3. 3Common arcsin Values
4. 4Graph of arcsin(x)
5. 5arcsin in Right Triangles
6. 6Properties of arcsin
7. 7Degrees vs. Radians
8. 8Step-by-Step Examples
9. 9Relationship with arccos
10. 10Common Mistakes
11. 11Real-World Applications
12. 12Practice Problems
13. 13Tips for Mastery
14. 14Final Thoughts
15. 15FAQ: Common Questions

Calculate Inverse Sine (arcsin)

Enter a value between −1 and 1. The calculator will compute arcsin(x) in your chosen unit and show the steps.

Value (x) — must be between −1 and 1

Output UnitDegrees (°)Radians (rad)

arcsin(x) will appear here.

Note: arcsin(x) = sin⁻¹(x) returns the angle whose sine is x. The result is always between −90° and 90° (−π/2 and π/2).

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Inverse Sine Function – Complete Explanation

The inverse sine function, written as arcsin(x) or sin⁻¹(x), answers the question: "What angle has a sine of x?" It's the inverse operation of the sine function, restricted to a specific domain and range to make it a proper function.

Inverse Sine Definition

arcsin(x) = θ  where  sin(θ) = x

Domain: −1 ≤ x ≤ 1
Range: −π/2 ≤ θ ≤ π/2 (−90° ≤ θ ≤ 90°)

1. What Is Inverse Sine?

Inverse sine (arcsin) finds the angle when you know the sine value. Since sine is periodic and many angles have the same sine value, the inverse sine function is restricted to return only one value — the principal value between −90° and 90°.

Key characteristics:

• Domain: −1 ≤ x ≤ 1
• Range: −π/2 ≤ arcsin(x) ≤ π/2 (−90° to 90°)
• Notation: arcsin(x) or sin⁻¹(x)
• Principal value: Always returns the angle closest to 0°.

2. Domain and Range

Domain: −1 ≤ x ≤ 1

Range: −π/2 ≤ arcsin(x) ≤ π/2

In degrees: −90° ≤ arcsin(x) ≤ 90°

Values outside the domain (like 2 or −1.5) are undefined for arcsin.

3. Common arcsin Values

arcsin(0) = 0°

arcsin(1/2) = 30° (π/6)

arcsin(√2/2) = 45° (π/4)

arcsin(√3/2) = 60° (π/3)

arcsin(1) = 90° (π/2)

arcsin(−1/2) = −30° (−π/6)

arcsin(−1) = −90° (−π/2)

4. Graph of arcsin(x)

The arcsin graph is a smooth increasing curve that starts at (−1, −π/2), passes through (0, 0), and ends at (1, π/2).

Key features: Domain = [−1, 1], Range = [−π/2, π/2], Continuous, Increasing, Symmetric about origin (odd function).

5. arcsin in Right Triangles

In a right triangle, arcsin finds the angle when you know the opposite side and hypotenuse.

If sin(θ) = opposite/hypotenuse

Then θ = arcsin(opposite/hypotenuse)

Example: opposite = 3, hypotenuse = 5 → θ = arcsin(3/5) = arcsin(0.6) ≈ 36.87°

6. Properties of arcsin

• Odd function: arcsin(−x) = −arcsin(x)
• Inverse relationship: sin(arcsin(x)) = x for all x in domain
• Complementary identity: arcsin(x) + arccos(x) = π/2 (90°)
• Derivative: d/dx arcsin(x) = 1/√(1−x²)
• Integral: ∫ arcsin(x) dx = x·arcsin(x) + √(1−x²) + C

7. Degrees vs. Radians

The result of arcsin can be expressed in either degrees or radians.

Radians: natural unit for mathematics (arcsin(1) = π/2)

Degrees: more intuitive for real-world measurements (arcsin(1) = 90°)

Convert: degrees = radians × 180/π

8. Step-by-Step Examples

Example 1: arcsin(√3/2)

Step 1: Recognize that sin(60°) = √3/2.
Step 2: Since √3/2 is positive and within [−1, 1],
        arcsin(√3/2) = 60°.

Answer: 60° or π/3 radians

Example 2: arcsin(−0.5)

Step 1: Recognize that sin(−30°) = −0.5.
Step 2: Since −0.5 is negative, the angle is negative.
        arcsin(−0.5) = −30°.

Answer: −30° or −π/6 radians

9. Relationship with arccos

arcsin and arccos are complementary functions:

arcsin(x) + arccos(x) = π/2  (90°)

For example: arcsin(0.5) = 30°, arccos(0.5) = 60°, and 30° + 60° = 90°.

10. Common Mistakes

• Input outside domain: arcsin(2) is undefined. Only values between −1 and 1 are valid.
• Confusing arcsin with 1/sin: arcsin(x) ≠ 1/sin(x). The notation sin⁻¹(x) means inverse function, not reciprocal.
• Forgetting the range restriction: arcsin only returns values between −90° and 90°.
• Using wrong unit: Always check whether your calculator is in degree or radian mode.
• Ignoring negative values: arcsin(−x) = −arcsin(x).

11. Real-World Applications

• Physics: Finding angles from known sine ratios (projectile motion, wave analysis).
• Engineering: Calculating angles in trusses, ramps, and mechanical linkages.
• Navigation: Determining elevation angles from distance measurements.
• Computer graphics: Converting between coordinate systems.
• Surveying: Finding angles from measured distances.
• Sound processing: Analyzing phase relationships.

12. Practice Problems

Problem 1: Find arcsin(1). → 90° (π/2)

Problem 2: Find arcsin(0). → 0° (0)

Problem 3: In a right triangle, the opposite side is 4 and the hypotenuse is 5. Find the angle. → arcsin(4/5) ≈ 53.13°

Problem 4: Find arcsin(−√2/2). → −45° (−π/4)

13. Tips for Mastery

• Memorize the common arcsin values (0, ±1/2, ±√2/2, ±√3/2, ±1).
• Remember the domain: −1 to 1. Anything else is invalid.
• Remember the range: −90° to 90°.
• arcsin is an odd function: arcsin(−x) = −arcsin(x).
• Use the relationship arcsin(x) + arccos(x) = 90° to check your work.

14. Final Thoughts

The inverse sine function is essential in trigonometry and geometry. Understanding its domain, range, and relationship to the sine function is crucial for solving triangles and real-world problems. Use the calculator above to check your work and practice until arcsin becomes second nature.

15. Frequently Asked Questions

The domain is −1 ≤ x ≤ 1. Values outside this range are undefined.

The range is −π/2 to π/2 (−90° to 90°).

No! arcsin(x) is the inverse function. 1/sin(x) is the reciprocal (cosecant).

arcsin(0.5) = 30° or π/6 radians.

Because arcsin is an odd function, symmetric about the origin.

arcsin(x) + arccos(x) = π/2 (90°). They are complementary.

No. The principal value is unique and always between −90° and 90°.

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Conclusion

The inverse sine function is a cornerstone of trigonometry. Master its domain, range, and applications, and you'll be well-equipped for advanced mathematical studies.

Readers who wish to continue practicing can explore the companion learning resource for guided exercises.