↩️ Inverse Trigonometry Calculator
Calculate inverse trigonometric functions (arcsin, arccos, arctan) to find angles from known ratios. Get results in both degrees and radians with domain validation.
Understanding Inverse Trigonometric Functions
Inverse trigonometric functions, also called arc functions, are the reverse operations of the standard trigonometric functions. While sin(30°) = 0.5, the inverse function arcsin(0.5) = 30°. These functions are essential for finding angles when you know the ratio of sides in a right triangle or the coordinates of a point on the unit circle.
Each inverse trigonometric function has a specific domain (valid input range) and range (output values). For example, arcsin and arccos only accept inputs between -1 and 1, while arctan accepts any real number. Understanding these limitations is crucial for correct application of inverse trig functions in mathematics and engineering.
Our calculator automatically validates inputs and provides results in both degrees and radians. It also indicates the quadrant of the resulting angle and calculates the reference angle, helping you understand the geometric meaning of the results.
Calculate Inverse Trigonometric Functions
Valid Input Ranges:
Inverse Trigonometric Function Reference
arcsin (sin⁻¹)
Domain: [-1, 1]
Range: [-90°, 90°]
Use: Find angle from sine ratio
Example: arcsin(0.5) = 30°
arccos (cos⁻¹)
Domain: [-1, 1]
Range: [0°, 180°]
Use: Find angle from cosine ratio
Example: arccos(0.5) = 60°
arctan (tan⁻¹)
Domain: All real numbers
Range: (-90°, 90°)
Use: Find angle from tangent ratio
Example: arctan(1) = 45°
arccot (cot⁻¹)
Domain: All real numbers
Range: (0°, 180°)
Use: Find angle from cotangent ratio
Example: arccot(1) = 45°
arcsec (sec⁻¹)
Domain: |x| ≥ 1
Range: [0°, 180°] - {90°}
Use: Find angle from secant ratio
Example: arcsec(2) = 60°
arccsc (csc⁻¹)
Domain: |x| ≥ 1
Range: [-90°, 90°] - {0°}
Use: Find angle from cosecant ratio
Example: arccsc(2) = 30°
Common Inverse Trigonometric Values
| Value | arcsin | arccos | arctan |
|---|---|---|---|
| 0 | 0° | 90° | 0° |
| 0.5 | 30° | 60° | 26.57° |
| √2/2 ≈ 0.707 | 45° | 45° | 35.26° |
| √3/2 ≈ 0.866 | 60° | 30° | 40.89° |
| 1 | 90° | 0° | 45° |
Frequently Asked Questions
Why do inverse trig functions have restricted domains?
Trigonometric functions are periodic and many-to-one, meaning multiple angles can produce the same ratio. To make inverse functions well-defined (one-to-one), mathematicians restrict the domain to principal values. For example, arcsin only returns angles between -90° and 90°, even though sine equals the same value at multiple angles.
What's the difference between sin⁻¹ and 1/sin?
sin⁻¹(x) is the inverse function (arcsin), which finds the angle whose sine is x. In contrast, (sin x)⁻¹ = 1/sin x = csc x is the reciprocal function. These are completely different operations: sin⁻¹(0.5) = 30°, but 1/sin(30°) = 1/0.5 = 2.
How do I find all angles with a given trigonometric value?
Inverse trig functions only give principal values. To find all solutions, use the principal value plus the function's period. For example, if arcsin(0.5) = 30°, then sin⁻¹(0.5) also equals 150° (180° - 30°) in the range [0°, 360°]. For complete solutions, add multiples of 360° to each.