Interactive Unit Circle

Why the unit circle makes sine and cosine visual

A point at angle θ on a radius-1 circle has coordinates (cos θ, sin θ). The horizontal coordinate is cosine and the vertical coordinate is sine, which is why the same point can be traced onto cosine and sine waves without introducing a new formula.

How exact values are built instead of guessed from decimals

The 16 standard angles use exact fractions and radicals such as 1/2, √2/2, and √3/2. WebToolArc keeps those symbolic values in a verified lookup table. Angles outside that standard set are shown as numerical approximations and are not relabeled as exact radical answers.

Reference angles and quadrant signs

Reduce the angle to its acute reference angle, recall the first-quadrant magnitudes, then apply the quadrant signs. In Quadrant II only sine is positive, in Quadrant III tangent is positive, and in Quadrant IV cosine is positive. Axis angles are handled directly because one coordinate is zero.

Why tangent, secant, cosecant, or cotangent can be undefined

Tangent and secant divide by cos θ, so they are undefined where cos θ = 0. Cosecant and cotangent divide by sin θ, so they are undefined where sin θ = 0. The studio displays “undefined” rather than a huge floating-point artifact near those axes.

How to practise the circle efficiently

Start with degrees and radians, then coordinates, then sine/cosine, and finally tangent plus quadrant/reference-angle questions. Switch the circle labels to Blank after you can recognize the 16 landmarks. Saved practice sessions are optional and local; scores are never placed in a share URL.

Scope and accuracy boundary

This is a trigonometry learning and calculation tool, not a computer algebra system. Exact symbolic values are guaranteed only for the listed standard angles. Arbitrary-angle values come from browser floating-point trigonometry and are shown as approximations. The interactive diagram is explanatory rather than a proof by scale.