Math Calculator

Unit Circle Calculator

One angle. The complete circle.

Explore an angle

Locate its terminal point and all six trig values.

Angle

Use signed finite decimals, scientific e notation or fractions of integers. Complete turns are reduced before numerical evaluation.

Input angle convention

Changes how the retained number is interpreted. All normalized output conventions are shown together.

Explore and compare

Load a special angle

Loads the selected angle in degrees, with exact values at fifteen-degree intervals.

Explore one turn in degrees

Keyboard arrows change one degree. Moving this slider switches the source convention to degrees.

Rounded decimal places

Exact radicals stay unchanged. Nonzero values below the chosen decimal scale retain scientific notation.

Worked examples

Show unit circle diagramShow special-angle referenceShow step-by-step solution

Finite inputs support forty significant digits per number. Nonzero source magnitudes range from $10^{-40}$ through $10^{12}$ . Ordinary decimal radians are capped at $10^6$ in magnitude to control numerical range reduction. Exact angular reductions precede trigonometry; general trig values have double-precision numerical limits.

Quadrant I. Exact special-angle values with separately rounded decimal views.

Interpreted source angle

\theta=\left(30\right)^\circ

Terminal point

P=\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)

Quadrant I. Signs follow the actual terminal point.

Equal axis scale;

x=\cos\theta

y=\sin\theta

and radius

r=1

. Positive rotation is counterclockwise. Near an axis, the small separation may be below display resolution; use the values and location label rather than the pixels.

Angle relationships

Exact angle relationships are separate from rounded radian views. Near a boundary, a rounded label can coincide with the axis value without changing the actual location.

Normalized angle in degrees: $\left(30\right)^\circ$ Exact relationship
Normalized angle in pi radians: $\frac{1}{6}\pi\,\mathrm{rad}$ Exact relationship
Normalized angle in ordinary radians: $\approx0.523599\,\mathrm{rad}$ Numerical view
Normalized angle in turns: $\frac{1}{12}\,\mathrm{turn}$ Exact relationship
Reference angle in degrees: $\left(30\right)^\circ$ Exact relationship
One additional positive coterminal angle: $\left(390\right)^\circ$ Exact relationship
One negative coterminal angle: $\left(-330\right)^\circ$ Exact relationship

All six trigonometric functions

Function	Exact value or domain	Rounded decimal
Sine $\sin\theta$	$\frac{1}{2}$	$\approx0.5$
Cosine $\cos\theta$	$\frac{\sqrt{3}}{2}$	$\approx0.866025$
Tangent $\tan\theta$	$\frac{\sqrt{3}}{3}$	$\approx0.57735$
Cosecant $\csc\theta$	$2$	$\approx2$
Secant $\sec\theta$	$\frac{2\sqrt{3}}{3}$	$\approx1.154701$
Cotangent $\cot\theta$	$\sqrt{3}$	$\approx1.732051$

Decimal precision changes the rounded view, never the domain or the exact expression. Nonzero tiny values use scientific notation. General-angle values are numerical approximations, even when a rounded decimal looks like an integer.

Step-by-step solution

1. Interpret the supplied angle
The source convention controls the meaning of the retained input. A symbolic pi radian expression retains its exact coefficient.
$\theta=\left(30\right)^\circ$
2. Remove complete rotations
Use a floor quotient, including for negative sources, to leave a terminal position in one counterclockwise revolution. General ordinary-radian conversions are numerical.
$k=0,\quad t=\frac{1}{12},\quad 0\leq t<1$
3. Identify the quadrant and reference angle
Measure the acute reference angle to the nearest horizontal semiaxis. The terminal point determines the signs of the coordinate ratios.
$\alpha=\left(30\right)^\circ$
4. Read the coordinate functions
This reference angle is a multiple of fifteen degrees, so its signed radical values remain exact.
$P=(\cos\theta,\sin\theta)=\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$
5. Apply ratio definitions only on their domains
Tangent and secant require a nonzero cosine. Cosecant and cotangent require a nonzero sine. No large finite pole estimate is substituted for an exactly undefined ratio.
$\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\sec\theta=\frac{1}{\cos\theta},\quad\csc\theta=\frac{1}{\sin\theta},\quad\cot\theta=\frac{\cos\theta}{\sin\theta}$
6. Check the unit radius
The displayed point comes from the same reduced angle as the ratios. Its squared radius checks the circular geometry within floating-point precision; this is not an exact algebraic certificate for general angles.
$\cos^2\theta+\sin^2\theta=1,\quad x^2+y^2\approx1$

The reference-angle rule assumes $0^\circ\leq\theta_0<360^\circ$ . It is $\alpha=\theta_0$ in the first quadrant, $\alpha=180^\circ-\theta_0$ in the second, $\alpha=\theta_0-180^\circ$ in the third and $\alpha=360^\circ-\theta_0$ in the fourth.

Complete special-angle reference

Every fifteen-degree interval in one closed revolution. The final row repeats the starting point after a complete turn. Undefined functions retain their original denominator restrictions.

Degrees	Radians	Sine	Cosine	Tangent	Cosecant	Secant	Cotangent
At $0^\circ$	$0\,\mathrm{rad}$	$0$	$1$	$0$	$\text{undefined}$	$1$	$\text{undefined}$
At $15^\circ$	$\frac{1}{12}\pi\,\mathrm{rad}$	$\frac{\sqrt{6}-\sqrt{2}}{4}$	$\frac{\sqrt{6}+\sqrt{2}}{4}$	$2-\sqrt{3}$	$\sqrt{6}+\sqrt{2}$	$\sqrt{6}-\sqrt{2}$	$2+\sqrt{3}$
At $30^\circ$	$\frac{1}{6}\pi\,\mathrm{rad}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$	$2$	$\frac{2\sqrt{3}}{3}$	$\sqrt{3}$
At $45^\circ$	$\frac{1}{4}\pi\,\mathrm{rad}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
At $60^\circ$	$\frac{1}{3}\pi\,\mathrm{rad}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2\sqrt{3}}{3}$	$2$	$\frac{\sqrt{3}}{3}$
At $75^\circ$	$\frac{5}{12}\pi\,\mathrm{rad}$	$\frac{\sqrt{6}+\sqrt{2}}{4}$	$\frac{\sqrt{6}-\sqrt{2}}{4}$	$2+\sqrt{3}$	$\sqrt{6}-\sqrt{2}$	$\sqrt{6}+\sqrt{2}$	$2-\sqrt{3}$
At $90^\circ$	$\frac{1}{2}\pi\,\mathrm{rad}$	$1$	$0$	$\text{undefined}$	$1$	$\text{undefined}$	$0$
At $105^\circ$	$\frac{7}{12}\pi\,\mathrm{rad}$	$\frac{\sqrt{6}+\sqrt{2}}{4}$	$-\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)$	$-\left(2+\sqrt{3}\right)$	$\sqrt{6}-\sqrt{2}$	$-\left(\sqrt{6}+\sqrt{2}\right)$	$-\left(2-\sqrt{3}\right)$
At $120^\circ$	$\frac{2}{3}\pi\,\mathrm{rad}$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2\sqrt{3}}{3}$	$-2$	$-\frac{\sqrt{3}}{3}$
At $135^\circ$	$\frac{3}{4}\pi\,\mathrm{rad}$	$\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$-1$	$\sqrt{2}$	$-\sqrt{2}$	$-1$
At $150^\circ$	$\frac{5}{6}\pi\,\mathrm{rad}$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$	$2$	$-\frac{2\sqrt{3}}{3}$	$-\sqrt{3}$
At $165^\circ$	$\frac{11}{12}\pi\,\mathrm{rad}$	$\frac{\sqrt{6}-\sqrt{2}}{4}$	$-\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)$	$-\left(2-\sqrt{3}\right)$	$\sqrt{6}+\sqrt{2}$	$-\left(\sqrt{6}-\sqrt{2}\right)$	$-\left(2+\sqrt{3}\right)$
At $180^\circ$	$1\pi\,\mathrm{rad}$	$0$	$-1$	$0$	$\text{undefined}$	$-1$	$\text{undefined}$
At $195^\circ$	$\frac{13}{12}\pi\,\mathrm{rad}$	$-\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)$	$-\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)$	$2-\sqrt{3}$	$-\left(\sqrt{6}+\sqrt{2}\right)$	$-\left(\sqrt{6}-\sqrt{2}\right)$	$2+\sqrt{3}$
At $210^\circ$	$\frac{7}{6}\pi\,\mathrm{rad}$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$	$-2$	$-\frac{2\sqrt{3}}{3}$	$\sqrt{3}$
At $225^\circ$	$\frac{5}{4}\pi\,\mathrm{rad}$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$1$	$-\sqrt{2}$	$-\sqrt{2}$	$1$
At $240^\circ$	$\frac{4}{3}\pi\,\mathrm{rad}$	$-\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$\sqrt{3}$	$-\frac{2\sqrt{3}}{3}$	$-2$	$\frac{\sqrt{3}}{3}$
At $255^\circ$	$\frac{17}{12}\pi\,\mathrm{rad}$	$-\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)$	$-\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)$	$2+\sqrt{3}$	$-\left(\sqrt{6}-\sqrt{2}\right)$	$-\left(\sqrt{6}+\sqrt{2}\right)$	$2-\sqrt{3}$
At $270^\circ$	$\frac{3}{2}\pi\,\mathrm{rad}$	$-1$	$0$	$\text{undefined}$	$-1$	$\text{undefined}$	$0$
At $285^\circ$	$\frac{19}{12}\pi\,\mathrm{rad}$	$-\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)$	$\frac{\sqrt{6}-\sqrt{2}}{4}$	$-\left(2+\sqrt{3}\right)$	$-\left(\sqrt{6}-\sqrt{2}\right)$	$\sqrt{6}+\sqrt{2}$	$-\left(2-\sqrt{3}\right)$
At $300^\circ$	$\frac{5}{3}\pi\,\mathrm{rad}$	$-\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$-\sqrt{3}$	$-\frac{2\sqrt{3}}{3}$	$2$	$-\frac{\sqrt{3}}{3}$
At $315^\circ$	$\frac{7}{4}\pi\,\mathrm{rad}$	$-\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$-1$	$-\sqrt{2}$	$\sqrt{2}$	$-1$
At $330^\circ$	$\frac{11}{6}\pi\,\mathrm{rad}$	$-\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$	$-2$	$\frac{2\sqrt{3}}{3}$	$-\sqrt{3}$
At $345^\circ$	$\frac{23}{12}\pi\,\mathrm{rad}$	$-\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)$	$\frac{\sqrt{6}+\sqrt{2}}{4}$	$-\left(2-\sqrt{3}\right)$	$-\left(\sqrt{6}+\sqrt{2}\right)$	$\sqrt{6}-\sqrt{2}$	$-\left(2+\sqrt{3}\right)$
At $360^\circ$	$2\pi\,\mathrm{rad}$	$0$	$1$	$0$	$\text{undefined}$	$1$	$\text{undefined}$

What Can You Create?

Connect rotations, coordinates and exact values

Follow the same point through different rotations

Explore clockwise sources, counterclockwise sources and multiple complete turns without confusing angular travel with terminal position. Every source retains its interpretation in the working. Normalized degrees, ordinary radians, pi radians and turns describe the same endpoint together. A separate pair of coterminal examples makes periodicity visible without replacing the angle that was originally entered or silently discarding the signed rotation record.

Build a complete trigonometric comparison

Compare sine and cosine with tangent and all three reciprocal functions at the same position. Each function retains its own denominator restriction. The exact table extends through all four quadrants and includes every fifteen-degree interval. Numerical values sit beside exact certificates where available, allowing a learner to compare signs, magnitudes and approximations without mistaking the rounded display for a proof of an exact relationship.

Create a reusable study record

Combine the literal source, convention, reference angle, terminal point and ratio definitions in one saved calculation. A hidden reference table remains available in CSV. A visible reference table becomes part of the printable record. Sharing preserves the supported input and display choices. These options help a teacher or student revisit the same problem rather than trying to reconstruct an angle from an isolated screenshot of its final decimal values.

Unit-circle formulas and worked examples

P=(x,y)=(\cos\theta,\sin\theta),\qquad x^2+y^2=1

\theta_0=\theta-360^\circ\left\lfloor\frac{\theta}{360^\circ}\right\rfloor,\qquad0^\circ\leq\theta_0<360^\circ

\theta_{\mathrm{rad}}=\frac{\pi}{180}\theta_{\mathrm{deg}},\qquad1\,\mathrm{turn}=2\pi\,\mathrm{rad}=360^\circ

At $150^\circ=5\pi/6\,\mathrm{rad}$ , the reference angle is $30^\circ$ . The second quadrant gives $P=(-\sqrt{3}/2,1/2)$ and $\tan\theta=-\sqrt{3}/3$ . At $90^\circ$ , cosine is zero, so tangent and secant are undefined while cotangent remains zero.

At $-45^\circ$ , adding one turn gives the normalized angle $315^\circ$ . Its terminal point is $P=(\sqrt{2}/2,-\sqrt{2}/2)$ . The negative sine and positive cosine make tangent negative. An ordinary radian source of $1\,\mathrm{rad}$ has no claimed short radical form in this tool and is evaluated numerically.

Unit circle editorial diagram labeling common degree and radian angles with coordinate values. — Common axis and first-quadrant angles, where cosine is the x-coordinate and sine is the y-coordinate.

Quadrants, signs, and coordinates

Read every standard angle from one coordinate rule

A unit-circle point is not a list to memorize in isolation. At an angle $\theta$ , the point is $(\cos\theta,\sin\theta)$ . The first-quadrant magnitudes repeat through the other quadrants, while the x- and y-signs change with the point's location.

Sign pattern by quadrant

Angles start on the positive x-axis and increase counterclockwise. Tangent follows $\tan\theta=\frac{\sin\theta}{\cos\theta}$ , so its sign comes from the sine and cosine signs.

Quadrant I: cosine positive, sine positive, tangent positive.
Quadrant II: cosine negative, sine positive, tangent negative.
Quadrant III: cosine negative, sine negative, tangent positive.
Quadrant IV: cosine positive, sine negative, tangent negative.
Horizontal axis: sine is zero, so tangent is zero.
Vertical axis: cosine is zero, so tangent is undefined.

Use a reference angle for magnitude

A reference angle is the acute angle between the terminal side and the x-axis. Standard angles such as 30, 45, and 60 degrees provide the same absolute coordinate values in every quadrant.

For 150 degrees, the reference angle is 30 degrees. The magnitude pair is therefore $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$ , and quadrant II changes only the x-sign: $\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)$ .

Connect degrees, radians, and turns

A half turn is 180 degrees or $\pi\,\mathrm{rad}$ , and a quarter turn is 90 degrees or $\frac{\pi}{2}\,\mathrm{rad}$ . Adding a full $2\pi$ -radian turn returns to the same coordinate.

Use radian labels when working with arc length, calculus, or identities. Use degrees when a diagram or instruction expresses familiar portions of a turn. The coordinate does not depend on which label system names the angle.

Interpret tangent as a ratio

Tangent equals sine divided by cosine, or $\frac{y}{x}$ on the unit circle. At 45 degrees both coordinates have the same nonzero magnitude, so the ratio is 1.

At 90 and 270 degrees, $x=0$ . Division by zero is undefined, which is why the reference table does not show a finite tangent value for the vertical-axis angles.

Worked check: locate 150 degrees

1Find the reference angle: $180^\circ-150^\circ=30^\circ$ .
2Start with the 30-degree magnitudes $\cos30^\circ=\frac{\sqrt{3}}{2}$ and $\sin30^\circ=\frac{1}{2}$ .
3Apply quadrant II signs to get $\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)$ ; tangent is $\frac{1/2}{-\sqrt{3}/2}=-\frac{\sqrt{3}}{3}$ .

Reference boundaries

The visible table highlights common axis and first-quadrant angles. Apply reference-angle and sign rules for standard angles such as 120, 135, 150, 210, 225, 240, 300, 315, and 330 degrees.
Exact radicals are preferable in symbolic work. Decimal values can be useful for measurement or software inputs, but their precision depends on the rounding you choose.
The unit circle defines trigonometric values for angles; it does not by itself solve a triangle. A triangle problem also needs side, angle, and geometry constraints.

Sources and definitions

NIST SP 330, Section 5: angle units
Reference for degree and radian conventions used in the table and diagram.

Continue the task

Related next steps

Degrees-radians converterTranslate any angle between the two notations.Trigonometry calculatorsUse the values in triangle and theorem tools.Polar grid generatorPlot angles on labeled radial paper.

Why Users Love This Tool

Why units, signs and domains remain visible

The mathematical model stays explicit

The circle has unit radius and equal horizontal and vertical scales.
Axes are named as boundaries rather than assigned to a neighboring quadrant.
Exact reduction handles negative and additional rotations before numerical evaluation.
Undefined functions follow their original sine or cosine denominators.

Approximation is an explained choice

Exact fifteen-degree reference values retain their signed radicals.
Ordinary radian decimals are never guessed to be symbolic pi inputs.
Tiny nonzero numerical values retain a scientific view at low display precision.
Angle relationships, decimal views and physical drawing resolution are distinguished.

Perfect For

A practical reference for connected trigonometry

Students learning beyond the first quadrant

Start with the familiar positive-coordinate triangle, then move into the other quadrants while keeping the same circle. The reference angle explains the magnitude, and the coordinate signs explain which functions become negative. Compare vertical and horizontal boundaries separately. This approach connects a memorized table to a geometric definition and makes the different domains of tangent and cotangent easier to understand.

Teachers explaining exact and measured angles

Contrast a symbolic pi source with an ordinary decimal radian measurement that is merely close to it. Follow the ordered equations to show why a numerical answer can approach an axis without becoming an exact axis case. Use the fifteen-degree examples to introduce angle-difference identities, then compare a general input whose terminal coordinates require numerical evaluation rather than a short radical expression.

Learners revising circular motion foundations

Use a normalized terminal position to describe direction while retaining complete rotations in the source record. The circle is dimensionless and does not introduce a physical radius, speed or elapsed time. For an applied motion problem, those quantities need an additional model. This page supplies the angular and trigonometric relationships that such a model can build on, together with the assumptions needed to interpret them correctly.

How It Works

How it works in three quick steps.

Identify the source angle convention

Enter a signed angle in degrees, ordinary radians, multiples of pi radians or turns. Use finite decimals, scientific e notation or integer fractions. In radians mode, pi syntax also accepts expressions such as pi/6 and -2pi/3. Selecting another source convention reinterprets the retained number; the normalized outputs show conversions of the solved angle.

Locate the terminal point and inspect the domains

Read the normalized angle, location and reference angle before inspecting the six function values. The drawing keeps equal horizontal and vertical units. Exact fifteen-degree reference values remain radical expressions, while general numerical values are labelled approximate. At exact axes, the original sine or cosine denominator determines which functions are undefined.

Compare a full revolution and retain the working

Use the special-angle selector or the keyboard-accessible degree slider to explore another terminal position. Review the complete reference table and ordered working. Download CSV with hidden reference rows, copy a readable summary, share all display choices or print the currently visible calculation and table to a PDF through the browser.

Download & Print

Keep the angle, domains and working together

Download a complete CSV

The spreadsheet-friendly record includes both the active calculation and all special-angle reference values. Exact expressions are converted to readable text because a CSV cell cannot render visual LaTeX. Formula-like source cells are protected against spreadsheet execution. The export retains whether a value is exact, numerical or undefined, so the meaning survives after the presentation format changes.

Print a focused mathematical PDF

Choose Print / Save PDF and use the browser's PDF destination. The print layout removes controls and supporting article sections, expands the result to the printable width and keeps visible mathematics available. Enable the diagram, working and reference table before printing if you want them included. Portrait A4 or Letter works for the main calculation; the wide reference uses a compact print presentation.

Share or copy the supported problem

A shareable fragment keeps the source convention, literal angle, precision and visibility switches. Loading a malformed fragment leaves the current problem unchanged and reports the issue. A copied summary includes the same active problem and complete working as readable plain text. If the browser does not offer clipboard access, CSV provides a separate way to keep the calculation without losing its assumptions.

FAQ

Frequently Asked Questions

What is a unit circle calculator?

A unit circle calculator connects a signed rotation to the point where its terminal side meets a circle of radius

1

centered at the origin. The horizontal coordinate is

\cos\theta

and the vertical coordinate is

\sin\theta

. This tool also reports their ratios and reciprocals, the normalized angle, the reference angle and the quadrant or axis. Its reference table covers every fifteen-degree interval rather than only the first quadrant.

Can I enter radians using pi instead of a decimal?

Yes. Select radians and enter supported literal syntax such as pi/6, -pi/2, 2pi/3 or 1.5*pi. Alternatively, select multiples of pi radians and enter only a rational coefficient, such as 1/6. A decimal radian measurement is interpreted as that finite number, not guessed to be a symbolic multiple of

\pi

. Arbitrary expressions and function calls are rejected, and the calculator never executes input as code.

How are negative angles and extra rotations handled?

Positive rotation is counterclockwise and negative rotation is clockwise. The calculator removes complete signed turns using a floor quotient, leaving a normalized position in

0^\circ\leq\theta_0<360^\circ

. Consequently, a negative forty-five-degree source reaches the same point as positive three hundred fifteen degrees. Extra rotations remain part of the source record and working even though sine, cosine and all other defined terminal-point functions repeat at the same position.

Which trigonometric values are exact?

Symbolic degree, turn and pi-radian sources retain exact angular reduction. If the reference angle is a multiple of fifteen degrees, all defined trigonometric functions retain signed rational or radical expressions. This includes the familiar thirty-, forty-five- and sixty-degree values and the additional fifteen- and seventy-five-degree values. Other reference angles use numerical trigonometry. A displayed decimal matching a familiar value does not turn a numerical evaluation into an exact certificate.

Why is tangent undefined at a vertical axis?

Tangent is

\tan\theta=\sin\theta/\cos\theta

, so its original denominator excludes every exact vertical-axis position where cosine is zero. Secant has the same restriction. At a horizontal-axis position, sine is zero, so cosecant and cotangent are undefined instead. The word undefined describes a missing real function value, not a very large real number. An angle merely close to an axis can still have a large finite value.

Is cotangent always the reciprocal of tangent?

The reliable unit-circle definition is

\cot\theta=\cos\theta/\sin\theta

on the domain where sine is nonzero. Writing cotangent as the reciprocal of tangent requires extra care because tangent itself is undefined at the vertical axes. Cotangent is nevertheless zero at those positions. This calculator checks the original sine denominator directly, which preserves that zero instead of incorrectly propagating tangent's undefined status into a function with a different domain.

What does the reference angle mean on an axis?

Inside a quadrant, the reference angle is the acute angle between the terminal ray and the nearest horizontal semiaxis. An axis is outside all four open quadrants, so the output uses the separate label horizontal-axis distance there. That distance is zero on a horizontal axis and a right angle on a vertical axis. This explicit boundary convention avoids calling a right angle acute or assigning an axis to an arbitrary neighboring quadrant.

Does increasing decimal precision change the answer?

The precision selector changes only the rounded numerical view. It does not alter the retained input, exact radical, denominator restriction or geometric location. Nonzero values below the chosen decimal scale use scientific notation instead of silently appearing as zero. Numerical functions still have ordinary double-precision limits, so selecting more places is a display preference rather than a promise of arbitrary-precision trigonometry or extra accuracy in a measured source.

Why does the diagram sometimes look as if the point is on an axis?

The diagram uses a single equal scale for both coordinates, so a sufficiently small angular separation occupies less than one screen pixel. A point close to an axis can therefore overlap that axis visually even though its coordinates and function values remain different. The numerical location and exact source relationship are the authoritative records. The arc shows the normalized counterclockwise position and does not attempt to stack the original number of complete rotations.

What is included when I save or share a calculation?

CSV retains the literal source, convention, precision, location, angle relationships, six function values, rounded views, ordered working and the complete special-angle reference, even when that table is hidden on screen. A shared fragment preserves every validated display choice. Copy uses readable plain text where visual LaTeX is unavailable. Browser printing retains the visible result, diagram, working and reference table, while hiding editing controls and supporting article content for a focused calculation record.

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About This Tool

A full-circle explanation rather than a first-quadrant lookup

The unit circle is a compact model that joins a geometric direction to functions used throughout trigonometry. Its center is the origin and its radius is $1$ , so its coordinates satisfy $x^2+y^2=1$ . Starting on the positive horizontal axis, a signed rotation determines a terminal ray. The intersection of that ray with the circle gives cosine horizontally and sine vertically. This definition continues through every quadrant and complete revolution, where a right-triangle-only picture needs additional sign rules or a carefully chosen reference angle.

This calculator keeps the rotation and terminal position related but distinct. A source of several complete turns has a different angular travel from its normalized counterpart, even though both reach the same point. Exact rational reduction preserves that distinction for degree, turn and symbolic pi-radian sources. Ordinary decimal radians remain numerical measurements; an eighty-four-decimal-place pi constant controls range reduction before double-precision trigonometry is evaluated on a small argument. The interface states the input bounds, approximation status and plotting resolution rather than claiming that all decimal digits are exact.

The six functions do not share one universal domain. Their original coordinate denominators must be checked separately, especially at axes. The reference table and interactive output therefore retain undefined values as domain statements. Every defined fifteen-degree special value has a rational or radical expression; other angles receive a numerical view. The diagram uses the same reduced model as the computed point and keeps both physical axis units equal. Together, the source record, angle relationships, ordered working and export options make the page useful for explanation and revision as well as a quick numerical check.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

The editorial team checks definitions, examples, linked tools, and cited references. Dates shown here come from the page's maintained publication record.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

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Calculators hub

Unit Circle Calculator

Explore an angle

Explore and compare

Interpreted source angle

Terminal point

Angle relationships

All six trigonometric functions

Step-by-step solution

1. Interpret the supplied angle

2. Remove complete rotations

3. Identify the quadrant and reference angle

4. Read the coordinate functions

5. Apply ratio definitions only on their domains

6. Check the unit radius

Complete special-angle reference

Connect rotations, coordinates and exact values

Follow the same point through different rotations

Build a complete trigonometric comparison

Create a reusable study record

Unit-circle formulas and worked examples

Sign pattern by quadrant

Use a reference angle for magnitude

Connect degrees, radians, and turns

Interpret tangent as a ratio

Worked check: locate 150 degrees

Reference boundaries

Sources and definitions

Related next steps

Why units, signs and domains remain visible

The mathematical model stays explicit

Approximation is an explained choice

A practical reference for connected trigonometry

Students learning beyond the first quadrant

Teachers explaining exact and measured angles

Learners revising circular motion foundations

How it works in three quick steps.

Identify the source angle convention

Locate the terminal point and inspect the domains

Compare a full revolution and retain the working

Keep the angle, domains and working together

Download a complete CSV

Print a focused mathematical PDF

Share or copy the supported problem

Frequently Asked Questions

Related calculators

Right Triangle Calculator

Circle Calculator

Degrees to Radians Converter

A full-circle explanation rather than a first-quadrant lookup

Who maintains this page

Ownership and review

Methodology

Dates and sources

Explore more calculators