Trigonometric Functions and the Unit Circle

Definition: Trigonometric functions define sine, cosine, and tangent for any angle by using the coordinates of a point on the unit circle (a circle of radius 1 centered at the origin), extending them far beyond the right-triangle ratios they are first introduced with.

How It Works

The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. Every point on it can be written as (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis.
Radians measure an angle by the length of arc it sweeps out on the unit circle itself: one radian is the angle whose arc length equals the radius. A full turn traces the entire circumference, so one full turn is exactly 2π radians.
Degrees and radians convert directly through the relationship π radians = 180°: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.
The unit circle definition sets sin(θ) as the y-coordinate and cos(θ) as the x-coordinate of the point where angle θ meets the circle. Unlike the right-triangle definitions (SOH-CAH-TOA), this works for any angle at all, not just angles between 0° and 90°.
tan(θ) = sin(θ) / cos(θ), which is also the slope of the radius line at angle θ, and equals the length cut off on the vertical line x = 1 when the radius is extended to meet it.
tan(θ) is undefined whenever cos(θ) = 0, which happens at θ = 90° and θ = 270° (and every 180° from either): dividing by zero has no defined result, and tan(θ) grows without bound as θ approaches those angles.
The sign of each function changes by quadrant, summarized by the mnemonic “All Students Take Calculus” (ASTC): quadrant I has all three positive, quadrant II only sine positive, quadrant III only tangent positive, and quadrant IV only cosine positive.
A small set of angles — 0°, 30°, 45°, 60°, 90°, or 0, π/6, π/4, π/3, π/2 in radians — have exact sine and cosine values built from simple fractions and square roots, and are worth memorizing since they appear constantly in later math and physics.
The Pythagorean identity, sin²θ + cos²θ = 1, is not an independent fact to memorize. It falls directly out of the unit circle’s own equation, x² + y² = 1, since x = cos θ and y = sin θ by definition.
Sine and cosine are periodic with period 2π (360°): sin(θ + 360°) = sin(θ) and cos(θ + 360°) = cos(θ) for every θ, because adding a full turn returns the point to the same spot on the circle.
Angles can be negative (measured clockwise instead of counterclockwise) or greater than 360° (more than one full turn); the unit circle handles both without any special-casing, since only the point’s final resting position on the circle determines the function values.

Illustration

θ = 45° (0.785 rad) sin(θ) = 0.707 cos(θ) = 0.707 tan(θ) = 1.000 Quadrant: I

Drag the slider to change θ. The radius, the point (cos θ, sin θ), the dashed sine and cosine drop-lines, and the live readouts below all update together.

θ 45°

var angleTxt = document.getElementById(‘unit-angle-txt’); var sinTxt = document.getElementById(‘unit-sin-txt’); var cosTxt = document.getElementById(‘unit-cos-txt’); var tanTxt = document.getElementById(‘unit-tan-txt’); var quadTxt = document.getElementById(‘unit-quad-txt’);

var thetaIn = document.getElementById(‘unit-theta’); var thetaOut = document.getElementById(‘unit-theta-out’); var resetBtn = document.getElementById(‘unit-reset’);

function fmt(n) { return (Math.round(n * 1000) / 1000).toFixed(3); }

function update() { var deg = parseFloat(thetaIn.value); thetaOut.textContent = deg.toFixed(0) + ’°’;

var rad = deg * Math.PI / 180; // Math.sin/Math.cos need radians, not degrees
var c = Math.cos(rad);
var s = Math.sin(rad);
var p = toPx(c, s);

radiusLine.setAttribute('x2', p[0]);
radiusLine.setAttribute('y2', p[1]);

pointDot.setAttribute('cx', p[0]);
pointDot.setAttribute('cy', p[1]);

sinLine.setAttribute('x1', p[0]);
sinLine.setAttribute('y1', p[1]);
sinLine.setAttribute('x2', p[0]);
sinLine.setAttribute('y2', oy);

cosLine.setAttribute('x1', p[0]);
cosLine.setAttribute('y1', p[1]);
cosLine.setAttribute('x2', ox);
cosLine.setAttribute('y2', p[1]);

// Sample the swept angle as a short polyline near the origin instead of
// an SVG arc command, so it never has to reason about sweep/large-arc flags.
var arcR = 34;
var steps = Math.max(1, Math.round(deg / 3));
var pts = [];
for (var i = 0; i <= steps; i++) {
  var a = (deg * i / steps) * Math.PI / 180;
  var ax = ox + arcR * Math.cos(a);
  var ay = oy - arcR * Math.sin(a);
  pts.push((i === 0 ? 'M' : 'L') + ax.toFixed(1) + ',' + ay.toFixed(1));
}
angleArc.setAttribute('d', pts.join(' '));

var d = Math.round(deg) % 360;
var quadrant;
if (d === 0 || d === 90 || d === 180 || d === 270) quadrant = 'on axis';
else if (d < 90) quadrant = 'I';
else if (d < 180) quadrant = 'II';
else if (d < 270) quadrant = 'III';
else quadrant = 'IV';

angleTxt.textContent = 'θ = ' + deg.toFixed(0) + '° (' + fmt(rad) + ' rad)';
sinTxt.textContent = 'sin(θ) = ' + fmt(s);
cosTxt.textContent = 'cos(θ) = ' + fmt(c);
tanTxt.textContent = 'tan(θ) = ' + (Math.abs(c) < 1e-9 ? 'undefined' : fmt(s / c));
quadTxt.textContent = 'Quadrant: ' + quadrant;

}

thetaIn.addEventListener(‘input’, update); resetBtn.addEventListener(‘click’, function () { thetaIn.value = 45; update(); });

update(); })();

Under the Hood

Deriving the Pythagorean identity directly from the unit circle’s own equation:

x² + y² = 1              the unit circle: every point on it is distance 1 from the origin
x = cos θ                 definition: x-coordinate of the point at angle θ
y = sin θ                 definition: y-coordinate of the point at angle θ
(cos θ)² + (sin θ)² = 1   substitute x and y into the circle's equation
sin²θ + cos²θ = 1         same equation, written in the conventional order

This identity holds for every possible angle θ, not only the special angles below, because it is a direct consequence of the circle’s equation rather than of any particular triangle.

The following angles produce exact sine, cosine, and tangent values (no decimal approximation needed), and are worth memorizing:

Angle (degrees)	Angle (radians)	sin	cos	tan
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined

Worked Example 1: A quadrant II angle. Given: find sin(150°), cos(150°), and tan(150°). Step 1: 150° falls in Quadrant II (between 90° and 180°). Its reference angle is 180° − 150° = 30°. Step 2: In Quadrant II, sine is positive and cosine and tangent are negative (ASTC). The 30° exact values are sin = 1/2, cos = √3/2. Answer: sin(150°) = 1/2, cos(150°) = −√3/2, tan(150°) = −√3/3.

Worked Example 2: A quadrant III angle. Given: find sin(225°), cos(225°), and tan(225°). Step 1: 225° falls in Quadrant III (between 180° and 270°). Its reference angle is 225° − 180° = 45°. Step 2: In Quadrant III, tangent is positive while sine and cosine are both negative. The 45° exact values are sin = cos = √2/2. Answer: sin(225°) = −√2/2, cos(225°) = −√2/2, tan(225°) = 1.

Worked Example 3: A quadrant IV angle. Given: find sin(300°), cos(300°), and tan(300°). Step 1: 300° falls in Quadrant IV (between 270° and 360°). Its reference angle is 360° − 300° = 60°. Step 2: In Quadrant IV, cosine is positive while sine and tangent are negative. The 60° exact values are sin = √3/2, cos = 1/2. Answer: sin(300°) = −√3/2, cos(300°) = 1/2, tan(300°) = −√3.

History

Trigonometry’s roots lie in Greek astronomy: Hipparchus (c. 190–120 BCE) is credited with building the first known trigonometric table, around 150 BCE, to help predict the positions of celestial bodies.
Hipparchus and, later, Ptolemy (2nd century CE, in the Almagest) worked with tables of chords rather than sines, computing the chord length a given angle subtends inside a circle of fixed size.
Indian mathematicians reframed the idea: Aryabhata (c. 476–550 CE) introduced the half-chord — essentially the modern sine function — directly, in his Aryabhatiya of 499 CE, a cleaner formulation than the Greek chord table.
The word “sine” traces an unusually tangled path: Aryabhata’s Sanskrit term “jya” (bowstring) was transliterated into Arabic as “jiba,” later misread by European translators as the unrelated Arabic word “jaib” (bay, or fold), which was then translated into Latin as “sinus” — the direct root of the modern word “sine.”
Islamic Golden Age scholars, including Al-Battani (9th–10th century), extended trigonometric tables and identities and helped transmit the sine-based approach into medieval Europe.
Radian measure, though implicit in the mathematics for far longer, was only formally named in the 19th century: the term “radian” first appeared in print in 1873, in an exam paper by James Thomson (physicist Lord Kelvin’s brother).

Why It Matters

Sound and light are, mathematically, sine waves (or sums of them): a pure musical tone is a sine-shaped pressure oscillation, and the electric and magnetic fields that make up light oscillate sinusoidally as they travel.
Alternating current (AC), the electricity delivered to every home outlet, varies sinusoidally over time; the voltage and current equations used to design AC circuits are built directly from sine and cosine.
GPS and navigation rely on triangulation, using known angles and distances to pinpoint a location, which is trigonometry applied directly; ships and surveyors used the same underlying principle for centuries before satellites existed.
Computer graphics use sine and cosine to rotate objects and animate circular or oscillating motion; every 2D or 3D rotation matrix in a graphics engine is built from sin(θ) and cos(θ).
Music and audio synthesis build complex sounds out of combinations of sine waves at different frequencies and amplitudes, an idea rooted in Fourier’s insight that any periodic signal can be broken into sine components.
Engineering and surveying depend on trigonometry for structural angles, roof pitches, load vectors on beams, and measuring land and distances that can’t be measured directly.

Common Pitfalls

Degree/radian mode confusion is one of the most common real errors with a calculator: computing sin(45) in radians mode gives the sine of 45 radians (about 0.851), not sin(45°) (about 0.707), so it’s essential to check which mode is active.
Forgetting that tan(θ) is undefined at 90° and 270° (and every 180° from those), and mistaking a calculator’s error message or huge output near those angles for a mistake rather than the expected behavior near a vertical asymptote.
Sign errors from not tracking the quadrant: computing a reference angle correctly but then forgetting to apply the ASTC sign for the actual quadrant, for example reporting cos(200°) as positive when Quadrant III requires it to be negative.
Confusing the sine and cosine definitions: mixing up which coordinate belongs to which function, especially since at θ = 0° cos(θ) = 1 while sin(θ) = 0, the opposite of what some people expect.
Assuming the Pythagorean identity sin²θ + cos²θ = 1 means sin θ + cos θ = 1. The squares are essential; dropping them produces a false statement.
Treating a reference angle’s value as the final answer by itself, without adjusting its sign for whichever quadrant the original angle actually lies in.

Comparison

Quadrant	Angle Range	sin	cos	tan
I	0°–90°	+	+	+
II	90°–180°	+	−	−
III	180°–270°	−	−	+
IV	270°–360°	−	+	−

FAQ

Why does the unit circle work for angles greater than 90°, when SOH-CAH-TOA doesn’t? SOH-CAH-TOA defines sine and cosine using the sides of a right triangle, which only has room for angles between 0° and 90°. The unit circle instead defines sin(θ) and cos(θ) as coordinates of a point that can sit anywhere around the full circle, so the definition naturally extends to every angle, including negative angles and angles past 360°.

Why is tangent sometimes described as a length rather than a ratio? If you extend the radius line for angle θ until it crosses the vertical line x = 1 (which is tangent to the circle at the point (1, 0)), the y-coordinate of that crossing point equals tan(θ) — literally the length of a segment tangent to the circle, which is where the function’s name comes from.

Do sine and cosine ever repeat with a shorter period than 2π? No — 2π (360°) is the smallest interval after which both sin(θ) and cos(θ) return to exactly the same value for every θ, so 2π is called the fundamental period. Tangent is a special case: because it’s a ratio of the two, it repeats twice as often, with a fundamental period of just π (180°).

Example

A Ferris wheel is a physical unit circle in action: normalize its radius to 1 and put the center at the origin, and a rider’s height above the center is sin(θ) while their horizontal offset from the center is cos(θ), both changing continuously as the wheel turns at a steady rate. Any smoothly rotating or oscillating system, from a piston to a pendulum’s shadow to an AC voltage, is described by the same two functions for exactly this reason.

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