Calculate all six trigonometric functions (sin, cos, tan, csc, sec, cot), inverse functions, standard angle exact values, and unit circle reference angles.
Click "Calculate Trigonometry" to evaluate functions.
A civil engineer designing a highway overpass calculates the vertical height of a support pillar given a 30° incline angle and a 50-meter ramp hypotenuse: using right-triangle trigonometry, Height = 50 × sin(30°) = 50 × 0.5 = 25 meters. A game developer programming 2D character rotation uses arctan2(Δy, Δx) to compute exact facing angles in radians. A surveyor evaluating land slopes uses tangent ratios to map terrain elevation changes.
Trigonometry is the branch of mathematics that studies relationships between side lengths and angles of triangles. On the Unit Circle (radius = 1), trigonometric functions extend beyond right triangles to arbitrary positive and negative angles in all four Cartesian quadrants.
The six core trigonometric functions are categorized into primary functions and reciprocal functions:
sin θ / cos θ).1 / sin θ).1 / cos θ).1 / tan θ).This calculator evaluates all six functions in degrees and radians, computes inverse functions, identifies unit circle quadrant locations, and provides exact radical representations for standard benchmark angles (30°, 45°, 60°, 90°). The following guide covers SOH-CAH-TOA definitions, Pythagorean identities, and real-world applications in engineering, physics, and computer graphics.
When an angle input is submitted, the engine converts units if necessary, projects coordinates onto the unit circle, and evaluates functions with high-precision JavaScript IEEE 754 floating-point routines.
1. Degree / Radian Conversion:
Radians = Degrees × (π / 180)
Degrees = Radians × (180 / π)
2. Reciprocal Relationships:
csc θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ = cos θ / sin θ
3. Pythagorean Trigonometric Identity:
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = csc² θ
4. Quadrant Signs (ASTC Rule - All Students Take Calculus):
- Quadrant I (0° to 90°): All functions positive.
- Quadrant II (90° to 180°): Sine & Cosecant positive.
- Quadrant III (180° to 270°): Tangent & Cotangent positive.
- Quadrant IV (270° to 360°): Cosine & Secant positive.
Physics and vector resolution. Physicists resolve force vectors (e.g. gravitational force on an inclined plane) into horizontal (F_x = F cos θ) and vertical (F_y = F sin θ) components.
Architecture and structural engineering. Roof truss designers calculate pitch angles, rafter lengths, and load-bearing stress lines using sine and cosine laws.
Computer graphics and game development. Graphics engines use trigonometric functions for 3D camera rotation, raytracing, and procedural wave animation (sine wave generation).
Acoustics and audio signal processing. Sound engineers use Fourier transform trigonometric series to decompose complex audio signals into fundamental sine wave frequencies.
Toggle between Degrees (°) and Radians (rad) depending on whether you are doing geometric surveying (degrees) or calculus (radians).
Check the Unit Circle Reference card to verify reference angle $\theta_{\text{ref}}$ and quadrant location.
For vector and matrix transformations, pair this tool with our Scientific Calculator.
The calculation engine operates client-side in JavaScript using built-in C-optimized math hardware routines. Calculations evaluate in under 1 millisecond.
| Standard Angle (deg) | Radian Equivalent | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 (0.5) | √3/2 (0.866) | √3/3 (0.577) |
| 45° | π/4 | √2/2 (0.707) | √2/2 (0.707) | 1 |
| 60° | π/3 | √3/2 (0.866) | 1/2 (0.5) | √3 (1.732) |
| 90° | π/2 | 1 | 0 | Undefined (∞) |
Tangent is defined as sin(90°) / cos(90°) = 1 / 0. Division by zero is mathematically undefined, resulting in a vertical asymptote on the tangent graph.
The domain of arcsin(x) and arccos(x) is restricted to [-1.0, +1.0]. Values outside this interval produce a domain error.
Degrees divide a full circle into 360 arbitrary parts. Radians measure angles based on arc length along a circle of radius 1 (a full circle is 2π radians).
Scientific Calculator — Performs advanced logarithms, exponents, and trig computations.
Quadratic Equation Solver — Solves second-order algebraic polynomials.