Unit Circle Calculator

What is Unit Circle Calculator?

A Unit Circle Calculator automates computations on the circle of radius 1 centered at the origin, where every angle \(\theta\) maps to a point \((x,y)\) with coordinates equal to the cosine and sine of that angle. This makes it a compact engine for trigonometry: once \((x,y)\) is known, all six trig functions follow immediately, along with quadrant signs, reference angles, and periodic identities. Core relationships include the circle equation, the coordinate–trig correspondence, and degree–radian conversion.

About the Unit Circle Calculator

Enter an angle in degrees or radians and the calculator returns the exact point \((\cos\theta,\sin\theta)\) when available, plus decimal approximations. It shows quadrant, reference angle, and signs; highlights undefined cases (e.g., \(\tan\tfrac{\pi}{2}\)); and lists identities that hold for the same input. You can toggle between equivalent angles using periodicity and see the effect instantly. For study and verification, the tool displays friendly exact values—such as \(\tfrac{\sqrt{2}}{2}\) and \(\tfrac{\sqrt{3}}{2}\)—alongside numeric forms.

How to Use this Unit Circle Calculator

1. Choose angle units (degrees or radians) and enter \(\theta\).
2. Optionally enable exact values to show radicals and fractions when possible.
3. Click calculate to obtain \((\cos\theta,\sin\theta)\), the six trig values, and quadrant info.
4. Use periodic controls to view equivalent angles: \(\theta\pm2\pi\) or \(\theta\pm360^\circ\).
5. Copy the formatted results and identities for homework, tests, or lesson notes.

Examples

Example 1: \(\theta=\tfrac{\pi}{3}\)

Special-angle values follow directly from the unit circle.

Example 2: \(210^\circ\)

Quadrant III has both sine and cosine negative; tangent is positive.

Example 3: \(\theta=-\tfrac{\pi}{4}\)

Negative angles rotate clockwise; values match the reference angle with signs by quadrant.

Example 4: \(\theta=\tfrac{13\pi}{6}\)

Reduce modulo \(2\pi\) to \(\tfrac{\pi}{6}\).

FAQs

What does the unit circle represent?

All points at distance 1 from the origin, mapping angles to cosine and sine coordinates for trigonometric evaluation.

How do I convert degrees to radians?

Multiply degrees by \(\pi/180\). To convert back, multiply radians by \(180/\pi\).

Why is tangent undefined at \(90^\circ\) and \(270^\circ\)?

Because \(\cos\theta=0\) there and \(\tan\theta=\sin\theta/\cos\theta\) requires division by zero.

How do I find the quadrant quickly?

Reduce the angle to \([0,360^\circ)\) or \([0,2\pi)\) and check signs: QI (+,+), QII (−,+), QIII (−,−), QIV (+,−).

Can the calculator show exact radical values?

Yes, for special angles it outputs forms like \(\tfrac{\sqrt{2}}{2}\) and \(\tfrac{\sqrt{3}}{2}\) alongside decimals.

Will it handle angles beyond one full turn?

Yes. Periodicity reduces any input using \(\theta\mapsto\theta\pm2\pi\) or \(\pm360^\circ\).

Does sign change with negative angles?

Signs follow the quadrant after reduction; sine is odd and cosine is even: \(\sin(-\theta)=-\sin\theta\), \(\cos(-\theta)=\cos\theta\).

What are reciprocal trig functions?

\(\sec=1/\cos\), \(\csc=1/\sin\), \(\cot=\cos/\sin\); they inherit domains from their denominators.

Can it show the tangent line approximation?

For small \(\theta\), \(\sin\theta\approx\theta\) and \(\cos\theta\approx1-\theta^2/2\); the tool can display these expansions.

How precise are decimal results?

Precision depends on the chosen rounding; exact forms remain exact regardless of display precision.

Will it compute arc length on the circle?

Yes. On radius 1, arc length equals the angle in radians: \(s=\theta\).

What’s the connection to complex numbers?

Euler’s formula gives \(e^{i\theta}=\cos\theta+i\sin\theta\), placing the unit circle at the heart of complex rotation.

Does it support inverse trig?

Yes, with principal values and domain restrictions, returning angles consistent with standard conventions.