The ArcSine (Inverse Sine) Function
ArcSine returns the angle whose sine equals the given value. arcsin(x) = θ where sin(θ) = x. Domain: [-1, 1], Range: [-π/2, π/2] (principal value).
Why This Mathematical Concept Matters
Why: ArcSine is used to find angles from ratios — surveying, navigation, game development, and solving equations like sin(θ) = x.
How: arcsin(x) returns the unique angle in [-π/2, π/2] such that sin(θ) = x. The range is restricted so the inverse is a true function (one output per input).
- ●arcsin(sin(θ)) = θ only when θ ∈ [-π/2, π/2]. For θ = 150°, sin(150°)=0.5 but arcsin(0.5)=30°.
- ●arcsin(x) + arccos(x) = π/2 for all x ∈ [-1, 1] — they are complementary angles.
- ●d/dx[arcsin(x)] = 1/√(1-x²); the derivative is essential for integration.
Examples — Click to Load
Inverse Trig Value Breakdown
All Related Inverse Values
arcsin vs arccos (Complementary)
Calculation Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
arcsin is used to find angles in right triangles when you know opposite/hypotenuse ratio.
— Khan Academy
arcsin(sin(x)) ≠ x in general! The principal value is always in [-90°, 90°].
— Paul's Notes
Key Takeaways
- • arcsin(x) returns the angle whose sine is x. Domain: [-1, 1], Range: [-π/2, π/2] (principal value)
- • arcsin(0)=0°, arcsin(0.5)=30°, arcsin(√2/2)=45°, arcsin(1)=90°. arcsin(-x) = -arcsin(x) (odd function)
- • arcsin(x) + arccos(x) = π/2 — they are complementary angles for x ∈ [-1, 1]
- • sin(arcsin(x)) = x for x ∈ [-1, 1]. Note: arcsin(sin(θ)) = θ only when θ ∈ [-π/2, π/2]
- • The range [-π/2, π/2] is chosen so arcsin is a true function (one output per input)
Did You Know?
How ArcSine Works
arcsin(x) finds the unique angle θ in [-π/2, π/2] such that sin(θ) = x. Since sine is not one-to-one on its full domain, we restrict the range to obtain a well-defined inverse.
Principal Value
The principal value of arcsin is the angle in [-90°, 90°]. For sin(θ) = 0.5, both 30° and 150° work, but arcsin(0.5) = 30° by convention.
asin(sin(x)) Considerations
arcsin(sin(x)) = x only when x ∈ [-π/2, π/2]. For x = 2π/3, sin(2π/3)=√3/2 but arcsin(√3/2)=π/3. The inverse "undoes" only within the principal range.
Complementary Identity
arcsin(x) + arccos(x) = π/2 for all x ∈ [-1, 1]. They represent complementary angles on the unit circle — the two angles that sum to 90°.
Expert Tips
Memorize Key Values
arcsin(0)=0°, arcsin(0.5)=30°, arcsin(√2/2)=45°, arcsin(√3/2)=60°, arcsin(1)=90°. Use the Unit Circle Calculator to visualize.
Watch Domain Restrictions
arcsin is undefined for |x| > 1. For complex numbers, arcsin extends via analytic continuation. See this calculator for real inputs.
Use arccos for Complementary
If you need the angle in [0, π], use arccos(x) instead. arcsin(x) + arccos(x) = 90° always. Try the ArcCosine Calculator.
Verify with sin(arcsin(x))
Always check: sin(arcsin(x)) = x. This verification catches domain errors. Use the Sine Calculator to confirm.
Inverse Trig Calculator Comparison
| Feature | ArcSine | ArcCosine | ArcTangent |
|---|---|---|---|
| Domain | [-1, 1] | [-1, 1] | All reals |
| Range | [-π/2, π/2] | [0, π] | (-π/2, π/2) |
| Principal value | Yes | Yes | Yes |
| Odd/Even | Odd | Neither | Odd |
| Complement | arccos(x) | arcsin(x) | arccot(x) |
| Undefined for | |x| > 1 | |x| > 1 | Never |
| Common use | Opp/hyp ratio | Adj/hyp ratio | Slope angle |
| atan2 alternative | No | No | Yes (atan2) |
Frequently Asked Questions
Why is arcsin only defined for [-1, 1]?
Because sine only outputs values between -1 and 1. There is no real angle whose sine equals 2 or -1.5. The domain of an inverse function equals the range of the original function.
What is arcsin(-0.5)?
-30° (or -π/6 rad). Arcsine is odd: arcsin(-x) = -arcsin(x). The negative input gives a negative angle in the fourth quadrant.
Why does arcsin(sin(150°)) = 30°?
sin(150°) = 0.5, and arcsin(0.5) returns the principal value 30°, not 150°. The inverse only "undoes" within the range [-90°, 90°].
What is arcsin(x) + arccos(x)?
Always π/2 (90°) for x ∈ [-1, 1]. They are complementary angles — the two acute angles in a right triangle sum to 90°.
Can arcsin output angles in Q2 or Q3?
No. The principal range is [-π/2, π/2], which covers only Q1 and Q4. For angles in Q2 or Q3, use arccos or adjust with π.
How is arcsin used in calculus?
d/dx[arcsin(x)] = 1/√(1-x²). The integral ∫ dx/√(1-x²) = arcsin(x) + C. Essential for trigonometric substitution.
What about arcsin for complex numbers?
arcsin extends to complex inputs via arcsin(z) = -i ln(iz + √(1-z²)). For real inputs, we use the principal real branch.
Why is the range [-π/2, π/2] chosen?
Sine is strictly increasing on [-π/2, π/2], so the inverse is unique. This interval is the "principal branch" — the standard convention in math and programming.
ArcSine by the Numbers
Official & Educational Sources
Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. For mission-critical applications (aerospace, medical devices), always verify with certified computational tools. Inputs outside [-1, 1] are undefined for real arcsin.