Trigonometry Calculator

Background.

The Quanta trig calculator evaluates all six standard trigonometric functions — sine, cosine, tangent, cosecant, secant, and cotangent — for any real angle, entered in either degrees or radians. The six functions are the foundational vocabulary of trigonometry, the branch of mathematics that started in ancient astronomy as a way to relate the angles of a triangle to the lengths of its sides and that now powers almost every quantitative description of waves, oscillations, rotations, and periodic phenomena in physics, engineering, and computer science.

The defining setup is the unit circle: draw a circle of radius 1 centred at the origin of the plane, pick an angle θ measured counter-clockwise from the positive x-axis, and mark the point P where the radius at angle θ meets the circle. Then by definition cos(θ) is the x-coordinate of P, sin(θ) is the y-coordinate of P, and the four other functions are simple ratios — tan(θ) = sin(θ)/cos(θ) is the slope of the line from the origin to P, cot(θ) = cos(θ)/sin(θ) is its reciprocal, sec(θ) = 1/cos(θ) is the reciprocal of cosine, and csc(θ) = 1/sin(θ) is the reciprocal of sine. The unit-circle picture instantly explains the most important properties of the six functions. Both sin(θ) and cos(θ) are bounded between −1 and 1 because they are coordinates on a circle of radius 1. Both are periodic with period 2π radians (360°) because rotating one full turn brings P back to itself. They are shifted versions of each other (cos(θ) = sin(θ + π/2)) because rotating the radius by a quarter turn maps the x-coordinate to the y-coordinate. The reciprocals csc, sec, and cot are unbounded — they shoot off to infinity whenever sin or cos passes through zero. And the fundamental Pythagorean identity sin²(θ) + cos²(θ) = 1 falls out immediately because P lies on a circle of radius 1, so its coordinates satisfy x² + y² = 1.

Why do we need all six functions, not just sin, cos, and tan? Historically because Greek and Indian astronomers (Hipparchus, Aryabhata, al-Battani) computed positions of stars and planets using chord, sine, versine, cosine, tangent, and cotangent in different combinations — every reciprocal had a name because dividing by sin or cos was awkward without our modern decimal system, so a pre-computed table of csc, sec, and cot saved a division each time.

Today the reciprocals survive because they show up naturally in identities (1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ are the Pythagorean identities written in their cleanest form), in calculus (the derivative of tan is sec², the integral of sec is ln|sec + tan|, the derivative of −cot is csc²), and in physics (impedance, refraction, and projectile range formulas all involve secant or cosecant terms that would be ugly to write with only sin, cos, and tan).

The two angle units — degrees and radians — both appear in the calculator because both are still in active use. Degrees, with 360 of them in a full turn, descend from Babylonian sexagesimal arithmetic and survive in everyday geometry, surveying, navigation, and architecture because 360 has many divisors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12…) and so divides cleanly into halves, thirds, quarters, fifths, sixths, eighths. Radians, with 2π of them in a full turn, are the natural mathematical unit because arc length on a unit circle equals angle in radians by definition — which makes the derivative of sin(x) come out cleanly as cos(x), with no awkward 180/π conversion factor. Every standard programming library (JavaScript Math.sin, Python math.sin, C math.h) takes radians as input, every calculus textbook uses radians from chapter one onward, and every physics equation involving angular frequency, angular velocity, or wave propagation uses radians. The conversion is π radians = 180 degrees, so 30° = π/6 rad, 45° = π/4 rad, 60° = π/3 rad, 90° = π/2 rad.

Applications span essentially every quantitative discipline. In physics, sine and cosine describe simple harmonic motion (a mass on a spring, a pendulum, a vibrating string), alternating-current voltage and current waveforms, sound waves, light waves, and the projectile-motion equations of classical mechanics — projectile range = v² · sin(2θ) / g is a textbook example. In engineering, the six functions parameterise rotational motion, gear ratios, robot-arm kinematics, signal modulation (AM, FM, QAM), and stress analysis in mechanical structures. In signal processing the Fourier transform decomposes any signal into a sum of sines and cosines, which is why every audio compression scheme (MP3, AAC, Opus), every video codec (JPEG, H.264, AV1), every image filter (blur, sharpen, edge-detect), and every digital communication standard (Wi-Fi, 5G, GPS) ultimately reduces to trigonometric arithmetic. In computer graphics and game development, sin and cos rotate vectors, animate orbits, and parameterise spline curves. In navigation and surveying, the Law of Sines and Law of Cosines extend right-triangle trig to arbitrary triangles, letting a navigator compute a position from two known landmarks. This page walks through the unit-circle definitions, the reciprocal identities, the degrees-to-radians conversion, worked examples for the famous angles 30°, 45°, 60°, and 90°, and the edge cases — like why tan(90°) is undefined and why sin² + cos² = 1 for every angle — that distinguish a calculator that handles trig correctly from one that produces floating-point noise at every multiple of π.

What is trigonometry calculator?

Trigonometry is the branch of mathematics that relates the angles of a triangle to the lengths of its sides, and more generally describes how points move on a circle. The six trigonometric functions — sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) — are the standard vocabulary. The cleanest modern definition uses the unit circle: a circle of radius 1 centred at the origin. For any real angle θ, measured counter-clockwise from the positive x-axis in either degrees or radians, let P = (x, y) be the point where the radius at angle θ meets the unit circle. Then cos(θ) = x, sin(θ) = y, tan(θ) = y/x = sin(θ)/cos(θ), csc(θ) = 1/y = 1/sin(θ), sec(θ) = 1/x = 1/cos(θ), and cot(θ) = x/y = 1/tan(θ). The older right-triangle definition is equivalent for acute angles between 0° and 90°: in a right triangle with one acute angle θ, sin(θ) = opposite / hypotenuse, cos(θ) = adjacent / hypotenuse, tan(θ) = opposite / adjacent, and the three reciprocals follow. The unit-circle definition extends the right-triangle definition to all real angles — negative angles (clockwise rotation), angles greater than 90° (any quadrant), and angles greater than 360° (multiple turns) — which is why every modern textbook starts with the unit circle and treats the right triangle as a special case. The functions are periodic: sin and cos have period 2π radians (360°), so sin(θ + 2π) = sin(θ) for every θ; tan and cot have period π radians (180°). Both sin and cos are bounded between −1 and 1; csc and sec are bounded outside the interval (−1, 1) (so |csc| ≥ 1 and |sec| ≥ 1 wherever they are defined); tan and cot take every real value. The fundamental Pythagorean identity sin²(θ) + cos²(θ) = 1 follows directly from x² + y² = 1 on the unit circle and is the single most-used trig identity in calculus and physics. The two reciprocal Pythagorean identities — 1 + tan²(θ) = sec²(θ) and 1 + cot²(θ) = csc²(θ) — follow by dividing the main identity by cos²(θ) and sin²(θ) respectively. The Quanta trig calculator evaluates all six functions for any real angle, in degrees or radians, and snaps numerical residuals smaller than 1e-12 to exactly 0 (so sin(π) returns 0, not 1.22 × 10⁻¹⁶), which prevents downstream identity-checking calculations from accumulating floating-point noise.

How to use this calculator.

1. 01Enter the angle in the first field. Any real number is valid — positive, negative, fractional, or larger than one full turn. The calculator handles wrap-around automatically (sin(370°) returns the same value as sin(10°), cos(-30°) returns the same value as cos(330°)).
2. 02Pick the unit. Degrees (°) is the everyday unit, with 360 in a full turn. Radians (rad) is the calculus and physics unit, with 2π in a full turn. The conversion appears in the angleRadians and angleDegrees output rows so you can always see both forms of the same angle.
3. 03Read sin(θ) at the top — the primary output, the y-coordinate on the unit circle. Read cos(θ) directly below — the x-coordinate. The other four functions are reciprocals or ratios of these two, and update simultaneously.
4. 04For famous angles, sanity-check against memorised values. sin(0°) = 0, sin(30°) = 0.5, sin(45°) = √2/2 ≈ 0.7071, sin(60°) = √3/2 ≈ 0.8660, sin(90°) = 1. cos(0°) = 1, cos(30°) = √3/2 ≈ 0.8660, cos(45°) = √2/2 ≈ 0.7071, cos(60°) = 0.5, cos(90°) = 0. tan(0°) = 0, tan(45°) = 1, tan(90°) = undefined (Infinity).
5. 05Watch for the Infinity outputs. tan and sec are undefined at 90°, 270°, 450°, … (every odd multiple of π/2 radians) because cos = 0 there; csc and cot are undefined at 0°, 180°, 360°, … (every integer multiple of π radians) because sin = 0 there. The calculator returns Infinity in those cases rather than a misleading large finite number — that is the mathematically correct answer.
6. 06Verify identities with the calculator. Compute sin and cos of any angle, square them, and add — the result is always 1 (the Pythagorean identity). Compute tan(θ) and 1/tan(θ) — the second equals cot(θ). Compute sin(θ + 90°) — the result equals cos(θ). These cross-checks are how you build intuition for which identity applies where.

The formula.

The unit-circle definitions are the source of every other trig formula. For any real angle θ, let P = (cos(θ), sin(θ)) be the point on the unit circle x² + y² = 1 reached by rotating the radius counter-clockwise through θ from the positive x-axis. Then sin(θ) = y-coordinate of P, cos(θ) = x-coordinate of P, tan(θ) = sin(θ) / cos(θ), and the three reciprocal functions are csc(θ) = 1 / sin(θ), sec(θ) = 1 / cos(θ), and cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ). The right-triangle definitions — sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent, csc = hypotenuse/opposite, sec = hypotenuse/adjacent, cot = adjacent/opposite — are equivalent for 0 < θ < 90° and follow by similar-triangles: scale a right triangle with hypotenuse h down to hypotenuse 1, and the legs become exactly the sine and cosine of the angle. The reciprocal identities are direct from the definitions: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ). They are not separate physical relationships — they are renamings, kept because they appear so often in calculus and physics that giving them their own names is a notational convenience. The degrees-to-radians conversion is π radians = 180 degrees, equivalently 1 degree = π/180 radians ≈ 0.01745 rad and 1 radian = 180/π degrees ≈ 57.2958°. The constant π = 3.141592653589793… appears because a full turn around the unit circle has arc length 2π (the circumference of a unit circle), and a radian is by definition the angle subtended by an arc of length 1 on the unit circle — so 2π radians = one full turn = 360 degrees, giving π rad = 180°. The Pythagorean identity sin²(θ) + cos²(θ) = 1 holds for every θ because cos(θ) and sin(θ) are the coordinates (x, y) of a point on the unit circle, and any point on the unit circle satisfies x² + y² = 1 by definition. Dividing both sides by cos²(θ) gives tan²(θ) + 1 = sec²(θ) (the secant Pythagorean identity); dividing both sides by sin²(θ) gives 1 + cot²(θ) = csc²(θ) (the cosecant Pythagorean identity). These three identities, together with the periodicity properties sin(θ + 2π) = sin(θ), cos(θ + 2π) = cos(θ), tan(θ + π) = tan(θ), and the symmetry properties sin(−θ) = −sin(θ) (sine is odd), cos(−θ) = cos(θ) (cosine is even), tan(−θ) = −tan(θ) (tangent is odd), generate every other trigonometric identity (angle-addition formulas, double-angle formulas, half-angle formulas, product-to-sum formulas) through algebra alone. Numerical implementation: the Quanta engine evaluates sin and cos using the platform's IEEE 754 double-precision math library (JavaScript's Math.sin and Math.cos, which compile to hardware FSIN/FCOS instructions on x86 or to the equivalent libm calls on other architectures), then derives tan, csc, sec, cot by simple division. A snap-to-zero threshold of 1e-12 cleans up the floating-point residuals — for example Math.sin(Math.PI) returns 1.22 × 10⁻¹⁶ instead of exactly 0 because π is not exactly representable in binary, so the calculator snaps any |result| < 1e-12 to exactly 0. Division by zero in the reciprocals returns Infinity (the mathematically correct answer for csc(0), sec(π/2), and so on), not NaN, because Infinity correctly signals an asymptote.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST Digital Library of Mathematical Functions, §4 — Elementary Functions, particularly §4.14 (Trigonometric Functions: Definitions and Periodicity) and §4.21 (Identities). The definitive modern reference for the analytic definitions, the standard identities, and the numerical evaluation of sin, cos, tan, csc, sec, cot.
2. [2]Abramowitz, M. and Stegun, I. A. (eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55, 1964 — §4.3 (Circular Functions) tabulates definitions, identities, series expansions, and numerical values for all six trigonometric functions. Still cited by NIST DLMF as a foundational reference.
3. [3]Toomer, G. J. "The Chord Table of Hipparchus and the Early History of Greek Trigonometry." Centaurus 18 (1973) — historical reconstruction of Hipparchus of Nicaea's chord table (circa 160 BC), the first systematic trigonometric table and the direct ancestor of modern sine tables via the Indian half-chord (jya) and the Arabic jiba/jaib mistranslation that produced the Latin sinus.
4. [4]Stewart, James. Calculus: Early Transcendentals, 8th edition. Cengage Learning, 2015 — Chapter 1 (Functions and Models) §1.2 and Appendix D (Trigonometry) develop the unit-circle definitions, the six trig functions, the Pythagorean identities, and the degree-radian conversion as prerequisites for the differential and integral calculus that follows.
5. [5]ISO 80000-2:2019 — Quantities and units, Part 2: Mathematics. International standard fixing the notation sin, cos, tan, csc, sec, cot for the six trigonometric functions, the radian as the SI coherent unit of plane angle (1 rad = 1 m/m), and the degree (1° = π/180 rad) as an accepted non-SI unit.
6. [6]Boyer, Carl B. and Merzbach, Uta C. A History of Mathematics, 3rd edition. Wiley, 2011 — Chapters 7–9 trace the development of trigonometry from Hipparchus (c. 160 BC) through Ptolemy's Almagest chord table (c. 150 AD), the Indian Siddhantas (c. 500 AD), Islamic-world astronomers including al-Battani and al-Khwarizmi (9th–10th c.), and the Renaissance shift to the six functions in their modern form by Regiomontanus and Rheticus.