Arcsin Calculator (Inverse Sine)
Calculate arcsin(x) - the inverse sine function
Arcsin is only defined for values between -1 and 1
Common Arcsin Values
arcsin(0)
0°
0
arcsin(0.5 (1/2))
30°
π/6
arcsin(√2/2 ≈ 0.707)
45°
π/4
arcsin(√3/2 ≈ 0.866)
60°
π/3
arcsin(1)
90°
π/2
arcsin(-0.5)
-30°
-π/6
arcsin(-√2/2)
-45°
-π/4
arcsin(-√3/2)
-60°
-π/3
arcsin(-1)
-90°
-π/2
How to Use This Calculator
Enter Sine Value
Input the sine value (x) you want to find the angle for. This value must be between -1 and 1.
Calculate
Click the "Calculate Arcsin" button to get the angle in both degrees and radians.
Review Results
View the angle result and verify it by checking that sin(angle) equals your input value.
Formula
arcsin(x) = θ, where sin(θ) = x
Principal value range: -90° to 90° (-π/2 to π/2 radians)
Definition:
The arcsin function (also written as sin⁻¹) is the inverse of the sine function. Given a sine value x, arcsin(x) returns the angle θ such that sin(θ) = x.
Domain and Range:
- Domain: -1 ≤ x ≤ 1 (only these values are valid inputs)
- Range: -90° ≤ arcsin(x) ≤ 90° (-π/2 ≤ arcsin(x) ≤ π/2 radians)
- The principal value is always in the first or fourth quadrant
Relationship to Sine:
sin(arcsin(x)) = x, for -1 ≤ x ≤ 1
Key Properties:
- arcsin(-x) = -arcsin(x) (odd function)
- arcsin(0) = 0°
- arcsin(1) = 90°
- arcsin(-1) = -90°
About Arcsin Calculator
The Arcsin Calculator (Inverse Sine Calculator) is a powerful tool that finds the angle whose sine equals a given value. Arcsin is the inverse function of sine, meaning it "undoes" the sine operation. This calculator helps you quickly determine angles from sine values, which is essential in trigonometry, geometry, and various engineering applications.
What is Arcsin?
Arcsin, written as arcsin(x) or sin⁻¹(x), is the inverse trigonometric function of sine. Given a sine value x between -1 and 1, arcsin(x) returns the angle (in the principal range of -90° to 90°) whose sine is x. It answers the question: "What angle has a sine of x?"
When to Use This Calculator
- Triangle Problems: Find angles when you know side ratios (opposite/hypotenuse)
- Physics: Calculate angles in projectile motion, waves, and oscillations
- Engineering: Solve problems involving angles in mechanical and electrical systems
- Navigation: Determine angles from elevation and distance measurements
- Signal Processing: Analyze phase relationships and frequency components
- Verification: Check solutions to trigonometric equations
Why Use Our Calculator?
- ✅ Instant Results: Get accurate angle values immediately in both degrees and radians
- ✅ Input Validation: Automatically checks that inputs are in valid range (-1 to 1)
- ✅ Verification: Shows that sin(result) equals your input to confirm accuracy
- ✅ Easy to Use: Simple interface requiring only one input value
- ✅ 100% Free: No registration or payment required
- ✅ Mobile Friendly: Works perfectly on all devices
- ✅ Educational: Helps understand the relationship between sine and arcsin
Important Notes
- Restricted Domain: Arcsin only accepts values between -1 and 1 (the range of sine)
- Principal Value: The calculator returns the principal value between -90° and 90°
- Multiple Solutions: For any sine value, there are infinitely many angles with that sine (due to periodicity), but arcsin returns only the principal value
- Quadrants: Arcsin returns angles in quadrants I (0°-90°) and IV (-90° to 0°)
Frequently Asked Questions
What is arcsin(1)?
arcsin(1) = 90° (or π/2 radians). This is because sin(90°) = 1. The arcsin of 1 gives the angle with maximum sine value.
What is arcsin(0)?
arcsin(0) = 0°. This is because sin(0°) = 0. The arcsin of 0 gives the angle at the rightmost point of the unit circle.
Why is arcsin only defined for values between -1 and 1?
Arcsin is the inverse of sine. Since sine only produces values between -1 and 1 (as sine represents the y-coordinate on a unit circle), arcsin can only "reverse" values in that range. If you input a value outside -1 to 1, there is no real angle with that sine value.
What's the difference between arcsin and sin⁻¹?
They are the same function! Both arcsin(x) and sin⁻¹(x) represent the inverse sine function. The notation sin⁻¹ is more common in some contexts, while arcsin is clearer in distinguishing it from 1/sin(x), which is the reciprocal (cosecant), not the inverse.
Can arcsin return angles greater than 90°?
No, the principal value of arcsin is always between -90° and 90° (-π/2 and π/2 radians). However, in general, there are infinitely many angles with the same sine value (like 150° has the same sine as 30°), but arcsin specifically returns the principal value in quadrants I or IV.
How is arcsin different from arccos?
Arcsin finds angles from sine values and returns angles in the range [-90°, 90°], while arccos finds angles from cosine values and returns angles in the range [0°, 180°]. They are complementary: arcsin(x) + arccos(x) = 90° for x in [0, 1].