Sin, cos, tan and the three reciprocals from any angle in degrees or radians, with the unit circle and the sine/cosine waves drawn alongside.
Enter any angle — positive, negative, or beyond one full turn. The calculator normalizes it to its principal value in [0°, 360°), shows the reference angle and quadrant, and gives the exact form for textbook angles (multiples of 30° or 45°).
Definitions: on the unit circle of radius 1, the point at angle θ measured counter-clockwise from the positive x-axis has coordinates (cos θ, sin θ). All other trigonometric ratios follow: tan θ = sin θ / cos θ, and the reciprocals csc, sec, cot. Reference angle = the acute angle between the terminal side and the x-axis; it tells you which quadrant of the unit circle to mirror the textbook value into.
The unit circle calculator turns a single angle into the entire family of trigonometric quantities that depend on it: the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent), the angle's principal value in [0°, 360°), the matching reference angle, the quadrant, and — when the input is one of the textbook special angles — the closed-form exact value involving fractions and square roots. Two visualizations sit alongside the numbers. The first is the unit circle itself, with the point (cos θ, sin θ) plotted on the rim, the radius drawn from the origin to that point, and dashed perpendiculars dropping to the axes so you can read sin θ and cos θ geometrically. The second is the time-domain pair of curves y = sin x and y = cos x over the interval [0, 2π], with vertical markers at θ that highlight where the angle lands on each curve.
In school we first meet sin, cos and tan as ratios inside a right triangle. That works for acute angles but cannot define the trig functions outside the range (0°, 90°): there is no right triangle with a 120° angle, and no obvious way to talk about the tangent of a negative angle or of 1000°. The unit circle removes this restriction. We anchor an angle θ at the positive x-axis, sweep counter-clockwise (or clockwise for a negative angle), and the terminal side hits the circle of radius 1 at a single point. The x-coordinate of that point is cos θ; the y-coordinate is sin θ. The remaining four functions follow from these two by simple division. Because the circle is symmetric and continuous, this definition extends naturally to every real number, turning trigonometry from a geometric trick for triangles into a fully general theory of periodic functions.
A small handful of angles appear so often in textbooks, physics problems and engineering calculations that their sin, cos and tan values are worth memorizing in exact form. The calculator surfaces these whenever your input matches: 0°, 30°, 45°, 60°, 90° and their reflections in every quadrant up to 360°. The exact values involve integers, halves, and the surds √2/2 and √3/2 — the famous "30-60-90" and "45-45-90" triangle ratios. Recognizing them speeds up mental arithmetic, makes identities cleaner to verify, and helps you spot patterns: for instance, sin 30° and cos 60° share the same value (1/2), an instance of the cofunction identity sin θ = cos(90° − θ) that you can read straight off the symmetry of the circle.
The reference angle of θ is the acute angle between the terminal side and the x-axis. Once you know the reference angle and the quadrant, you know every trig function: the absolute value comes from the reference angle, and the sign comes from the quadrant. The mnemonic ASTC ("All Students Take Calculus") summarizes the sign pattern: in quadrant I all six are positive; in II only sin (and its reciprocal csc) are positive; in III only tan and cot are positive; in IV only cos and sec are positive. Using this, sin 150° drops to sin 30° = 1/2 with a positive sign because 150° lives in quadrant II, while cos 210° is −cos 30° = −√3/2. The calculator computes the reference angle and the quadrant for you, but it is worth practising the lookup by hand a few times: the geometric intuition pays back across calculus, complex numbers and physics.
Plotting sin θ on the vertical axis as θ sweeps once around the unit circle produces the sine wave. The peak at θ = 90° is the moment the point on the circle reaches the top of the y-axis; the zero crossings at θ = 0° and 180° are the moments it crosses the x-axis. Cosine traces the same curve shifted by 90°, which is why sin and cos are sometimes described as the "same wave with a phase difference". This periodic, smooth behaviour is precisely the shape of countless physical signals — alternating current, simple harmonic motion of a pendulum or a spring, the up-and-down position of a piston in an engine, the air pressure in a musical tone, the tide. Behind every Fourier series, every signal-processing filter and every quantum-mechanical wavefunction sits the same unit circle, just re-parametrized.
Physics uses the unit circle to describe rotations and oscillations: angular velocity, phase angles, resonance peaks. Electrical engineering uses cos and sin to model the in-phase and quadrature components of an AC signal, with phasors rotating in the complex plane at the same speed as the point on our circle. Computer graphics rotates 2D and 3D objects with rotation matrices whose entries are sin and cos of the rotation angle. Navigation, surveying and astronomy use trig identities to triangulate positions across a curved earth. Machine learning uses sinusoidal positional encodings in transformer models so that attention layers can talk about relative position without an explicit step counter. None of these domains require the unit circle directly — but every one of them collapses back to it when you push hard enough on the underlying formulae.
Two traps catch beginners. The first is mixing degrees and radians: most mathematical software (and most calculators) defaults to radians, so sin(30) on a fresh calculator returns sin of 30 radians, not 30°. This calculator avoids the trap by asking for the unit explicitly. The second trap is forgetting that tan, sec, csc and cot can be undefined: tan 90° has no value because cos 90° = 0, and csc 0° has no value because sin 0° = 0. The calculator displays "∞" for these poles to make the case visible rather than silent. Beyond these, the rest is practice — try a few negative angles, push beyond 360°, and watch how the normalization wraps the answer back into [0°, 360°) without changing any of the trig values.