Unit Circle

How the unit circle is built

Place a circle of radius 1 in the plane, centered at the origin. Pick any angle θ measured counterclockwise from the positive x-axis. The point on the circle at that angle has coordinates:

(cos θ, sin θ)

That's the entire definition. Sine is the y-coordinate, cosine is the x-coordinate. Tangent is the slope of the line from origin to the point — sin θ / cos θ.

The angle θ is in radians, where one full revolution is 2π. So 0 radians is along the positive x-axis (point (1, 0)). π/2 radians (a quarter turn) is straight up (point (0, 1)). π radians is the negative x-axis (point (-1, 0)). 3π/2 is straight down (point (0, -1)).

Common angles and their values

Angle (rad)	Angle (deg)	cos θ	sin θ	tan θ
0	0°	1	0	0
π/6	30°	√3/2	1/2	√3/3
π/4	45°	√2/2	√2/2	1
π/3	60°	1/2	√3/2	√3
π/2	90°	0	1	undefined (∞)
2π/3	120°	−1/2	√3/2	−√3
3π/4	135°	−√2/2	√2/2	−1
5π/6	150°	−√3/2	1/2	−√3/3
π	180°	−1	0	0
3π/2	270°	0	−1	undefined
2π	360°	1	0	0

Memorize these and you don't need a calculator for most trigonometry problems. The 30-60-90 and 45-45-90 right triangles let you derive these values exactly using only the Pythagorean theorem.

The Pythagorean identity

The unit circle has equation x² + y² = 1. Substitute x = cos θ, y = sin θ:

cos²θ + sin²θ = 1

This is the most fundamental trig identity, and it's just the unit-circle equation in disguise. From it, you can derive all the others by dividing through by cos²θ or sin²θ:

1 + tan²θ = sec²θ           (divide by cos²θ)
cot²θ + 1 = csc²θ           (divide by sin²θ)

Three identities. Same Pythagoras, three different sides of the right triangle inscribed at angle θ.

Symmetries of the unit circle

Transformation	Effect on (cos θ, sin θ)	Identity it gives
Negate angle (reflect across x-axis)	(cos θ, −sin θ)	cos(−θ) = cos θ; sin(−θ) = −sin θ
Add π (point opposite)	(−cos θ, −sin θ)	cos(θ + π) = −cos θ; sin(θ + π) = −sin θ
Add π/2 (quarter turn)	(−sin θ, cos θ)	cos(θ + π/2) = −sin θ; sin(θ + π/2) = cos θ
Reflect across y-axis (π − θ)	(−cos θ, sin θ)	cos(π − θ) = −cos θ; sin(π − θ) = sin θ

Every "trig identity" you've memorized is one of these symmetries applied to the unit circle. Visualizing the picture is faster than memorizing rules.

Parametric form

The unit circle traced as time t increases:

x(t) = cos(t)
y(t) = sin(t)

This is a parametric description — the circle is the set of points (x(t), y(t)) as t varies. Generalizes — circle of radius r centered at (h, k):

x(t) = h + r·cos(t)
y(t) = k + r·sin(t)

Used everywhere — animation paths, planetary orbits, signal processing, GPS waypoint generation.

JavaScript: working with the unit circle

// Convert degrees to radians (radians is what Math.sin / Math.cos expect)
const toRad = deg => deg * Math.PI / 180;
const toDeg = rad => rad * 180 / Math.PI;

// Point on unit circle at angle θ
function unitCirclePoint(theta) {
  return [Math.cos(theta), Math.sin(theta)];
}

// Verify Pythagorean identity (within float precision)
const theta = Math.random() * 2 * Math.PI;
const [x, y] = unitCirclePoint(theta);
console.log(x*x + y*y); // ≈ 1.0

// Animate around the unit circle
let t = 0;
function frame() {
  t += 0.05;
  const [x, y] = unitCirclePoint(t);
  // ... render at (x, y)
  requestAnimationFrame(frame);
}

// Drawing the unit circle on a canvas
function drawUnitCircle(ctx, cx, cy, r) {
  ctx.beginPath();
  for (let theta = 0; theta <= 2 * Math.PI; theta += 0.01) {
    const x = cx + r * Math.cos(theta);
    const y = cy + r * Math.sin(theta);
    if (theta === 0) ctx.moveTo(x, y); else ctx.lineTo(x, y);
  }
  ctx.stroke();
}

Why use radians, not degrees?

Degrees are a human convenience — 360 chosen because it's divisible by many small integers (a Babylonian heritage). Mathematics is cleaner in radians for two reasons:

1. Calculus. The derivative of sin(x) is cos(x) — exactly. In degrees, the derivative is (π/180) cos(x), with the conversion factor sneaking in. Every infinitesimal trigonometric calculation gets cleaner in radians.
2. Arc length. An arc of angle θ on a circle of radius r has length r·θ — when θ is in radians. In degrees, you'd need r·θ·π/180. Length, area, and volume formulas all simplify in radians.

Engineering and everyday practice often use degrees (compass bearings, building angles, weather). Pure math always uses radians. Engineering software typically defaults to degrees with an option to switch — be aware which mode you're in or you'll get nonsense.

Extending to complex numbers — Euler's formula

Euler's formula says e^(iθ) = cos θ + i sin θ. The unit-circle point at angle θ, written as a complex number, is the complex exponential at iθ.

This is why complex multiplication is "rotation in the plane" — multiplying by e^(iθ) rotates a point by angle θ. The unit circle in the complex plane is the set of complex numbers with absolute value 1, and multiplying any two of them adds their angles. This insight is the heart of Fourier analysis, signal processing, and rotation matrices.

Where the unit circle shows up everywhere

• Trigonometry and geometry. Every trig identity. Every angle calculation. The fundamental object.
• Calculus. Trig derivatives, integrals, parametric curves. The smooth periodic functions sin and cos appear in nearly every differential equation.
• Physics. Wave motion (sound, light, water). Simple harmonic motion (pendulums, springs). Circular motion (orbits, rotational mechanics). AC circuits (current and voltage as sinusoids).
• Signal processing. Fourier transforms decompose any signal into sums of sinusoids. The unit circle in the complex plane is where the action happens.
• Computer graphics. Rotation matrices, animation parameter sweeps, 3D camera orbits, smooth interpolation.
• Quantum mechanics. Wave functions are complex amplitudes; their phases live on unit circles.

Common mistakes

• Mixing radians and degrees. Math.sin(30) in JavaScript gives sin(30 radians) ≈ −0.988, not sin(30°) = 0.5. Convert first or use a degree-aware library.
• Confusing sine and cosine with their inverses. sin⁻¹ (or arcsin) gives the angle whose sine is x; sin(x) gives the y-coordinate. Different functions; opposite directions.
• Forgetting that the unit circle is a unit CIRCLE. All Pythagorean identity manipulations require the radius to be 1. For circles of other radii, the identity is x² + y² = r², not 1.
• Memorizing identities instead of visualizing. Trig identities multiply if you memorize. The unit-circle picture generates them all from one symmetry argument.
• Thinking sine ≈ angle for all angles. sin(θ) ≈ θ holds for small θ in radians (the small-angle approximation). For larger angles or in degrees, it's wrong. Use the full sin function.
• Confusing angles measured "from x-axis" vs "from y-axis." Standard convention is counterclockwise from positive x-axis. Surveyors and navigators measure clockwise from north. Always check the convention.