AnyTool
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How do I use an interactive unit circle?

Drag the point around AnyTool’s Interactive Unit Circle — or click the ring, tap a common-angle button, drag the slider, or type an angle in degrees or radians — and it shows everything at once: the angle in degrees and as a multiple of π, the point (cos θ, sin θ), all six trig functions, the reference angle and the quadrant. At the 16 standard angles it gives the exact value (such as √3/2, −1/2 or 1) instead of just a decimal, and it honestly marks tan, sec, csc and cot as “undefined” where they have no value. With snapping on, the point locks onto clean angles. A reference chart lists all 16 standard angles with exact sin, cos and tan, and everything runs in your browser, so nothing is uploaded.

  • Drag the point, click the ring, slide, or type degrees/radians
  • Live sin, cos, tan, csc, sec and cot for any angle
  • Exact values at the 16 standard angles (e.g. √3/2, −1/2, undefined)
  • Reference angle, quadrant and (cos θ, sin θ) point shown live
  • Reference chart of all 16 angles; 100% in your browser — nothing uploaded

What is

Unit circle

The unit circle is the circle of radius 1 centred at the origin, on which a point at angle θ (measured counter-clockwise from the positive x-axis) has coordinates (cos θ, sin θ); the remaining trig functions follow as tan θ = sin θ / cos θ, csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = cos θ / sin θ. AnyTool draws it as an interactive SVG: dragging the point sets θ and the tool computes every function locally, showing exact values at the standard angles and honestly flagging the angles where a function is undefined.

Calculators

Related terms

Reference angleQuadrantRadianSpecial anglesASTC sign rule

Frequently Asked Questions

A point at angle θ on the unit circle has coordinates (cos θ, sin θ), because the radius is 1.

On the unit circle the radius is exactly 1, so the point reached by sweeping an angle θ counter-clockwise from the positive x-axis sits at (cos θ, sin θ). That is why the unit circle defines sine and cosine geometrically: the horizontal coordinate is the cosine and the vertical coordinate is the sine. AnyTool draws both as projections from the point down to the x-axis and across to the y-axis, so you can watch cos θ and sin θ change as you drag the point, and at standard angles it shows the exact values such as (√2/2, √2/2) at 45°.

Because tan θ = sin θ / cos θ and cos 90° = 0, dividing by zero leaves tan 90° with no value — so it is undefined.

Tangent is defined as sin θ divided by cos θ. At 90° (and at 270°) the cosine is 0, so the division has no result and tan is undefined; the same happens to sec θ = 1/cos θ at those angles. Symmetrically, csc θ = 1/sin θ and cot θ = cos θ / sin θ are undefined where sin θ = 0, namely at 0°, 180° and 360°. AnyTool prints “undefined” at exactly these angles instead of showing a fake very large number, and it cleans tiny floating-point dust like cos 90° ≈ 6 × 10⁻¹⁷ to a true zero so the result is honest.

The reference angle is the acute angle to the x-axis: θ in Q1, 180°−θ in Q2, θ−180° in Q3, 360°−θ in Q4 — and AnyTool shows it and the quadrant automatically.

The reference angle is the smallest positive angle between the terminal side and the x-axis, always between 0° and 90°. In Quadrant I it equals θ, in Quadrant II it is 180°−θ, in Quadrant III it is θ−180°, and in Quadrant IV it is 360°−θ. The reference angle gives the magnitude of every trig value, while the quadrant gives the sign (the ASTC rule: All positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4). AnyTool displays the quadrant and the reference angle in both degrees and radians for whatever angle you set.

Yes — every calculation runs in your browser with no server, no signup and no tracking, and the page keeps working offline after first load.

AnyTool’s Interactive Unit Circle is completely client-side: the trigonometry comes from a small pure engine running in JavaScript, the circle is drawn as an in-page SVG, and nothing you do is sent anywhere. There is no account, no tracking and no upload, and once the page has loaded it keeps working without a connection. It is a visual reference and calculator — it shows exact values at the standard angles and rounded decimals elsewhere, and it does not solve equations or do symbolic algebra beyond the unit-circle functions.

Detailed Explanation

Methodology

How the Interactive Unit Circle Computes Trig Values

AnyTool’s Interactive Unit Circle is a draggable in-page SVG backed by a small pure trigonometry engine. When you drag the point, click the ring, slide, tap a common-angle button or type an angle, the pointer position is converted to an angle θ with atan2, optionally snapped to the nearest of the 16 standard angles, and normalised to the [0°, 360°) range. The engine then computes the point (cos θ, sin θ) and all six functions — tan θ = sin θ / cos θ, csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = cos θ / sin θ — using the reciprocal form so true singularities surface as infinities rather than huge finite numbers. Tiny floating-point dust such as cos 90° ≈ 6 × 10⁻¹⁷ is cleaned to exact 0, 1 or −1. The radian measure is formatted as a tidy multiple of π (π/6, 3π/4, and so on) by testing small denominators. Every calculation is deterministic and runs locally; nothing is sent to a server.

  • Pointer → angle via atan2, optional snap to 16 standard angles
  • Six functions from sin and cos; reciprocals expose true singularities
  • Float dust (e.g. cos 90°) cleaned to exact 0, 1 or −1
  • Radians formatted as multiples of π (π/6, 3π/4, …)
  • Pure, unit-testable engine — no eval(), no server
How It Works

Exact Values, Reference Angle, Quadrant and Chart

At each of the 16 standard angles the tool shows the exact value of every function — for example sin 60° = √3/2, cos 120° = −1/2, tan 45° = 1 — alongside a rounded decimal, rather than only a decimal approximation. For any angle it reports the (cos θ, sin θ) point, the reference angle in both degrees and radians, and the quadrant, and it draws the angle ray, the sweep arc and the cosine (horizontal) and sine (vertical) projections so the geometry behind the numbers is visible. A reference chart lists all 16 standard angles with their exact sin, cos and tan, and tapping a row jumps the circle to that angle. The whole result can be copied to the clipboard, snapping can be toggled, and the figure adapts to dark mode.

  • Exact values at 16 standard angles (√3/2, −1/2, 1, undefined, …)
  • Reference angle (deg + rad), quadrant and (cos θ, sin θ) point
  • Angle ray, sweep arc and cos/sin projections drawn live
  • Reference chart of all 16 angles; tap a row to jump there
  • Copy result, snap toggle, dark-mode-aware SVG
Limitations

Undefined Values and What It Is Not

Some trig functions genuinely have no value at certain angles, and the tool states this plainly. Because tan θ = sin θ / cos θ and sec θ = 1/cos θ, both are undefined where cos θ = 0 (at 90° and 270°); csc θ = 1/sin θ and cot θ = cos θ / sin θ are undefined where sin θ = 0 (at 0°, 180° and 360°). AnyTool prints “undefined” at exactly those angles instead of a fake large number. Outside the 16 standard angles the values are decimals rounded to four places, so they are approximations rather than exact surds. The sign of each function follows the quadrant by the ASTC rule. The tool is a visual unit-circle reference and calculator — it does not solve trigonometric equations, do symbolic simplification beyond the standard exact values, or handle inverse-function ranges.

  • tan/sec undefined at 90° and 270° (cos θ = 0)
  • csc/cot undefined at 0°, 180°, 360° (sin θ = 0) — shown as “undefined”
  • Non-standard angles give decimals rounded to 4 places
  • Signs follow the quadrant (ASTC rule)
  • A reference/visualiser — not an equation solver or CAS
Privacy & Security

Privacy and Offline Use

All trigonometry and rendering happen in the browser, so the angle you choose and every value computed stay on your device — there is no server call, no account and no tracking, and the page keeps working offline after first load. The current angle and toggles live only in the page and reset on reload. The trigonometry engine is a pure, unit-testable module, so its values are consistent and verifiable.

Unit circle tools: in-browser (AnyTool) vs typical online unit circles
CapabilityAnyToolTypical online unit circles
InteractionDrag point, click ring, slider, type deg/radOften dropdown or static chart
Functions shownAll six (sin, cos, tan, csc, sec, cot)Frequently only sin/cos/tan
Exact valuesYes, at 16 standard anglesOften decimals only
Undefined handling“undefined” at true singularitiesOften shows ∞ or a huge number
Reference angle & quadrantShown liveSometimes missing
Reference chartAll 16 angles, tap to jumpStatic image or separate page
Works offlineYes (PWA)No
Cost / signupFree, no signupOften ad-heavy or gated

AnyTool computes every value locally with a pure trigonometry engine and uploads nothing.