Inverse Trig Calculator | Arcsin, Arccos
Calculate arcsin, arccos, arctan and get angles from trig ratios.
What Is the Inverse Trig Calculator | Arcsin, Arccos?
This calculator evaluates all six inverse trigonometric functions plus three inverse hyperbolic functions. Enter a value, select the function, and get the angle in degrees, radians, and gradians simultaneously, plus a quick-reference table of all six trig ratios for common angles.
- ›10 functions: arcsin, arccos, arctan, arccsc, arcsec, arccot, arcsinh, arccosh, arctanh, and atan2(y, x) for full quadrant-aware angle computation.
- ›Triple output: Every result shown simultaneously in degrees, radians, and gradians, with fraction-of-π notation where clean (e.g., π/6, π/4, π/3, π/2).
- ›All-6 reference table: Shows sin, cos, tan, csc, sec, cot values for 0, 30°, 45°, 60°, and 90° for quick lookup during problem solving.
- ›Domain validation: Clear error messages when input is outside the domain (e.g., arcsin(1.5) is undefined, calculator explains why).
- ›Presets: Six instant examples, arcsin(0.5), arccos(0), arctan(1), atan2(3,4), arcsinh(1), arctanh(0.5).
Formula
Standard Inverse Trig Functions
arcsin(x) = θ iff sin(θ) = x and θ ∈ [−π/2, π/2]
arccos(x) = θ iff cos(θ) = x and θ ∈ [0, π]
arctan(x) = θ iff tan(θ) = x and θ ∈ (−π/2, π/2)
Key Identities
arcsin(x) + arccos(x) = π/2 for x ∈ [−1, 1]
arctan(x) + arctan(1/x) = π/2 for x > 0
arcsin(−x) = −arcsin(x) (arcsin is odd)
arccos(−x) = π − arccos(x)
| Function | Domain | Range (Principal) |
|---|---|---|
| arcsin(x) | [−1, 1] | [−90°, 90°] = [−π/2, π/2] |
| arccos(x) | [−1, 1] | [0°, 180°] = [0, π] |
| arctan(x) | all reals | (−90°, 90°) = (−π/2, π/2) |
| arccsc(x) | |x| ≥ 1 | [−90°,0)∪(0°,90°] |
| arcsec(x) | |x| ≥ 1 | [0°,90°)∪(90°,180°] |
| arccot(x) | all reals | (0°, 180°) = (0, π) |
| arcsinh(x) | all reals | all reals |
| arccosh(x) | x ≥ 1 | [0, ∞) |
| arctanh(x) | |x| < 1 | all reals |
How to Use
- 1Select the function: Choose from the dropdown, arcsin, arccos, arctan, arccsc, arcsec, arccot, or the hyperbolic/atan2 variants.
- 2Enter the value: For arcsin/arccos: enter a number between −1 and 1. For arctan: any real number. For arccsc/arcsec: any value with |x| ≥ 1.
- 3For atan2: Enter both x and y coordinates. atan2(y, x) gives the full four-quadrant angle without the x/y ambiguity of arctan(y/x).
- 4Choose your output unit: Select degrees, radians, or gradians to see results in your preferred angular measure. All three are shown.
- 5Try a preset: Click any preset, arcsin(0.5), arctan(1), etc., to see a worked example with the result highlighted.
- 6Press Enter or Calculate: Results appear with the exact decimal value, fraction-of-π where applicable, and the co-function companion value.
- 7Check the reference table: Expand the all-6 trig values table to see sin/cos/tan/csc/sec/cot for all standard angles for quick comparison.
Example Calculation
Find arcsin(0.5), what angle has a sine of 0.5?
Question: sin(θ) = 0.5, find θ (principal value)
Recall the unit circle: sin(30°) = 0.5
arcsin(0.5) = 30° = π/6 rad ≈ 0.5236 rad = 33.33 grad
Co-function companion:
arccos(0.5) = 60° = π/3 (since arcsin + arccos = π/2)
Answer: arcsin(0.5) = 30.0000° = π/6 rad
arcsin(x) vs sin⁻¹(x), same thing, different notation
arcsin(x) and sin⁻¹(x) are equivalent notations for the inverse sine. The superscript −1 means "inverse function", not "reciprocal". The reciprocal of sin(x) is csc(x) = 1/sin(x). This is a common source of confusion on calculators where the sin⁻¹ key may be labelled differently.
Understanding Inverse Trig | Arcsin, Arccos
Why Do Inverse Trig Functions Have Restricted Ranges?
The sine function is not one-to-one over all real numbers, sin(30°) = sin(150°) = 0.5. A function must be one-to-one to have an inverse. By convention, arcsin is restricted to the principal value range [−90°, 90°], where sine is one-to-one. This gives a unique angle for each ratio.
Different textbooks and calculators always return the principal value, but the general solution to sin(θ) = x includes infinitely many angles: θ = arcsin(x) + 360°k and θ = 180° − arcsin(x) + 360°k for any integer k.
The atan2 Function, Why It Matters
arctan(y/x) only returns angles in (−90°, 90°) and loses information about the quadrant when x is negative. The atan2(y, x) function takes both coordinates separately and returns the correct angle in (−180°, 180°]. This is essential in computer graphics, robotics, navigation, and physics when converting from Cartesian to polar coordinates.
Inverse Hyperbolic Functions
arcsinh, arccosh, and arctanh are the inverses of the hyperbolic functions sinh, cosh, and tanh. Unlike circular trig functions, hyperbolic functions are defined in terms of exponentials: sinh(x) = (eˣ − e⁻ˣ)/2. Their inverses have logarithmic forms:
- ›arcsinh(x) = ln(x + √(x² + 1)) for all x
- ›arccosh(x) = ln(x + √(x² − 1)) for x ≥ 1
- ›arctanh(x) = ½ ln((1+x)/(1−x)) for |x| < 1
Real-World Applications
- ›Structural engineering: finding the angle of inclination of a ramp or beam from its rise and run, arctan(rise/run).
- ›Optics and Snell's Law: the refracted angle θ₂ = arcsin(n₁ sin θ₁ / n₂) when light passes between media.
- ›Robotics: inverse kinematics uses arctan/atan2 to find joint angles from desired end-effector position.
- ›Navigation: bearing from north computed as atan2(Δeast, Δnorth) gives correct quadrant without extra conditionals.
- ›Computer graphics: atan2(y, x) computes the orientation angle of a vector, used in every 2D game and UI framework.
Frequently Asked Questions
Why is the domain of arcsin limited to [−1, 1]?
The sine function produces values only in [−1, 1]. Since arcsin must "undo" sine, it can only accept inputs that sine can produce.
If you try arcsin(1.5) you get an error, there is no real angle whose sine is 1.5, because sine never exceeds 1 in magnitude.
Why does arccos return values in [0°, 180°] only?
Cosine is one-to-one on [0°, 180°], for each cosine value, there is exactly one angle in that range. This is the principal value convention.
This means arccos(−0.5) = 120°, not 240°. If your problem requires the angle in a different quadrant, you need to determine that from context and adjust the principal value accordingly.
What is the difference between arctan and atan2?
arctan(y/x) loses information because division can make different (y,x) pairs look identical. For example, arctan(−1/1) = arctan(1/−1) = −45°, but these points are in opposite quadrants (270° apart).
atan2(y, x) uses both values separately: atan2(−1, 1) = −45° (fourth quadrant) while atan2(1, −1) = 135° (second quadrant). It returns angles in (−180°, 180°].
How do I find all angles, not just the principal value?
The general solution for sin(θ) = x is: θ = arcsin(x) + 360°k or θ = 180° − arcsin(x) + 360°k, for any integer k.
For cos(θ) = x: θ = ±arccos(x) + 360°k.
For tan(θ) = x: θ = arctan(x) + 180°k.
What is arcsin(0), arcsin(1), and arcsin(−1)?
These are the three most common arcsin values:
- arcsin(0) = 0° = 0 rad (sin(0°) = 0)
- arcsin(1) = 90° = π/2 rad (sin(90°) = 1)
- arcsin(−1) = −90° = −π/2 rad (sin(−90°) = −1)
What are inverse hyperbolic trig functions used for?
Inverse hyperbolic functions appear in:
- Calculus integration: ∫ 1/√(1+x²) dx = arcsinh(x) + C
- Catenary curves (hanging cables): the shape involves cosh(x)
- Special relativity: rapidity (a measure of velocity in relativity) involves arctanh(v/c)
- Conformal mapping in complex analysis
Is sin⁻¹(x) the same as 1/sin(x)?
No. This is one of the most common mistakes in trigonometry.
sin⁻¹(x) means arcsin(x), the inverse function that returns an angle.
1/sin(x) is the cosecant function csc(x). The −1 superscript on a trig function denotes the inverse function, not the reciprocal.