Conversions
degrees = radians × 180 / π
radians = degrees × π / 180
turns = radians / 2π
Radian angle analyzer
Enter decimal values, fractions, π expressions, degrees, gradians, turns, milliradians, or DMS.
Scientific expression evaluator
Inverse trigonometric functions
atan2(y, x)
Arc, sector, chord, and segment
Angular motion
Triangle calculator in radian mode
Interactive trigonometric graph
Move the pointer over the graph. Drag to pan and use the mouse wheel to zoom.
Built-in example library
Choose an example
pi/4
Local calculation history
Stored only in this browser.
Formula used
Core relationships used throughout the calculator.
Unit circle
x = cos(θ), y = sin(θ)
The terminal point provides sine and cosine directly.
Coterminal angles
θₖ = θ + 2πk
Any integer k produces the same terminal side.
Arc geometry
s = rθ
A = ½r²θ
c = 2r sin(θ/2)
Angular motion
ω = θ/t
v = rω
α = Δω/Δt
Triangles
c² = a² + b² − 2ab cos(C)
a/sin(A) = b/sin(B) = c/sin(C)
How to use this calculator
Choose an input unit
Use radians for decimal values or π expressions. Select degrees, gradians, turns, milliradians, or DMS when the source measurement uses another system.
Enter an expression
Type a number, fraction, power, root, constant, or supported function. The dedicated parser accepts only known mathematical tokens and functions.
Set normalization
Map an angle into zero to 2π, negative π to positive π, or zero to π according to the application.
Read the result
Review exact π notation, decimal radians, degrees, DMS, quadrant, reference angle, unit-circle coordinates, and all six trigonometric functions.
Use specialized tools
Open separate modules for inverse functions, arc geometry, rotational motion, triangles, graphing, worked examples, and local history.
Export information
Copy or print the result, download the graph, and export locally stored calculation history as a CSV file.
Understanding radians
Natural circular measurement
A radian compares arc length with radius. When an arc has the same length as its circle’s radius, the central angle is one radian. A full turn is 2π radians, a half turn is π radians, and a quarter turn is π/2 radians. Because the length units cancel, radians are dimensionless and fit naturally into formulas for circular geometry.
Why mathematics and physics use radians
Calculus identities take their simplest form in radian mode. The derivative of sine is cosine without an extra conversion factor. Small-angle approximations such as sin(θ) ≈ θ also require radians. Rotational motion, wave equations, oscillation, signal processing, robotics, and navigation therefore commonly use radians.
Reference angles and signs
The reference angle is the smallest angle between the terminal side and the x-axis. It determines trigonometric magnitudes. The quadrant determines signs: sine is positive in Quadrants I and II, cosine is positive in I and IV, and tangent is positive in I and III.
Floating-point tolerance
Computers store π approximately. Directly calculating cosine at π/2 may produce a tiny nonzero value instead of exact zero. The calculator applies a small tolerance before deciding whether reciprocal or quotient functions are undefined.
Frequently asked questions
Can I type the π symbol?
Yes. Both pi and π are accepted, along with expressions such as 3*pi/4 and sqrt(2)*pi.
Why is tangent sometimes undefined?
Tangent equals sine divided by cosine. It is undefined where cosine is zero, including odd multiples of π/2.
Does this calculator use eval()?
No. The PHP expression engine tokenizes input and uses a recursive-descent parser with a strict function whitelist.
How is exact π notation found?
The calculator approximates radians divided by π as a bounded rational number and checks the approximation error.
Where is history saved?
History is saved in browser local storage. It is not written to a database or transmitted to another service.
Which PHP versions are supported?
PHP 8.0 or newer is recommended. The code is designed for current PHP 8 and PHP 8.3 deployments.