Beautiful Math
Path

Trigonometry · Lesson 3

The unit circle

On a circle of radius 1, cosine and sine become the x and y coordinates of a point. That single picture extends trigonometry to every angle.

Core ~17 minDeep study ~48 minWhy radius 1?How can sine be negative?

Before this: Right-triangle ratios

How this idea was born

Wheels turn past a right angle. Seasons cycle. Ships and stars keep moving after an acute corner is no longer enough. Right-triangle sine and cosine know only acute angles; the world of turns is larger: obtuse angles, full rotations, negative orientations, endless spinning.
Astronomers needed a handbook that could follow circular motion all the way around. In the second century, Claudius Ptolemy's Almagest carried chord tables and geometric methods that made spherical astronomy computable for centuries. MacTutor sketches the system-building ethic: gather geometric tools into a handbook other minds can run with. Different notation than modern sine, same ambition: turn angles into lengths you can tabulate and reuse.
The unit circle is the modern picture that extends those ratios to every oriented angle. On a circle of radius 1, cosine and sine become coordinates of a point. Triangle ratios and circular motion become the same objects.
For a modern rereader, once you can read a point on the circle as a cosine-sine address, obtuse angles stop feeling like exceptions and negative sine stops feeling like a bug.

First principles

The unit circle is x2+y2=1x^2 + y^2 = 1. From the positive xx-axis, turn an oriented angle θ\theta. The landing point PP has coordinates
P=(cos⁡θ, sin⁡θ)P = (\cos\theta,\ \sin\theta)
θcos θsin θ
On the unit circle, the point at angle θ is (cos θ, sin θ).
Why radius 1? Because then "adjacent/hypotenuse" collapses to the xx-coordinate itself, and "opposite/hypotenuse" collapses to yy. Triangle ratios with hypotenuse normalized away become coordinates, then extend to every oriented angle.
In the first quadrant both sine and cosine are positive. Elsewhere signs follow the signs of xx and yy. That is how obtuse and reflex angles stay meaningful, and how sine can be negative without drama.
Quadrant sign cheat that is not superstition: ask "Is xx positive or negative? That is the sign of cosine." Then "Is yy positive or negative? That is the sign of sine." The picture decides; you do not need a chant.

Worked intuition

Coordinates are the ratios

On radius 1, the old acute-angle definitions are not replaced. They are completed. Draw the right triangle from the origin to PP to the xx-axis when PP is in the first quadrant: you recover adjacent =x= x, opposite =y= y, hypotenuse =1= 1.
Example - θ=π/3\theta = \pi/3 (60°). From the special triangle, P=(cos⁡(π/3),sin⁡(π/3))=(1/2,3/2)P = (\cos(\pi/3), \sin(\pi/3)) = (1/2, \sqrt{3}/2). Both coordinates positive. First quadrant.
Example - θ=2π/3\theta = 2\pi/3 (120°). Same yy-coordinate as 60∘60^\circ but xx flips negative: P=(−1/2,3/2)P = (-1/2, \sqrt{3}/2). Cosine negative, sine positive. Second quadrant.
Example - θ=−π/2\theta = -\pi/2. A clockwise quarter turn from the positive xx-axis lands at (0,−1)(0, -1). Sine is −1-1; cosine is 00. The point sits on the negative yy-axis.

Anchor points worth knowing

| Angle | Radians | (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) | |-------|---------|----------------------------| | 0∘0^\circ | 00 | (1,0)(1, 0) | | 90∘90^\circ | π/2\pi/2 | (0,1)(0, 1) | | 180∘180^\circ | π\pi | (−1,0)(-1, 0) | | 270∘270^\circ | 3π/23\pi/2 | (0,−1)(0, -1) | | 360∘360^\circ | 2π2\pi | (1,0)(1, 0) again |
Full turns return to the start. That is periodicity in coordinate form.

Pythagorean identity

From x2+y2=1x^2 + y^2 = 1:
cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1
Geometry lesson Pythagoras, rewritten in trig clothing. Algebrica's Pythagorean identity page is a clean companion to this fact.
Using the identity: If sin⁡θ=3/5\sin\theta = 3/5 and θ\theta is in the second quadrant, then cos⁡2θ=1−9/25=16/25\cos^2\theta = 1 - 9/25 = 16/25, so cos⁡θ=−4/5\cos\theta = -4/5 (negative because x<0x < 0 in QII). The identity gives magnitude; the quadrant gives sign.

Reference angles

Every point on the unit circle shares yy-magnitude (sine magnitude) or xx-magnitude (cosine magnitude) with some acute angle called the reference angle. Example: sin⁡(150∘)=sin⁡(30∘)=1/2\sin(150^\circ) = \sin(30^\circ) = 1/2, but cos⁡(150∘)=−cos⁡(30∘)\cos(150^\circ) = -\cos(30^\circ) because xx is negative in QII.

Circular motion read sideways

A wheel's position is an angle. The height of a point on the rim as the wheel turns is a sine wave. Circular motion and oscillation are one picture. Strogatz's "Sine Qua Non" leans on that Ferris-wheel intuition: watch a point go around; watch its height oscillate.

Common confusions

Thinking the unit circle is a new subject. It is right-triangle trig with hypotenuse 1, then extended by orientation and continuation around the circle.
Fear of negative sine or cosine. Signs are quadrant coordinates, not moral failures of the definition.
Memorizing quadrant signs without the picture. If you can see xx and yy, you do not need a separate superstition chart.
Forgetting radians on the unit circle. Arc length equals θ\theta only when θ\theta is in radians and radius is 1. That is a feature, not a trivia item.
Confusing chord traditions with modern sine. Historical tables often tabulated chords; modern sine is a half-chord in that lineage. Notation changed; the ambition (angle to length) endured.
Treating tan⁡θ\tan\theta as always defined. On the unit circle, tan⁡θ=sin⁡θ/cos⁡θ\tan\theta = \sin\theta/\cos\theta. When cos⁡θ=0\cos\theta = 0 (at π/2\pi/2 and 3π/23\pi/2), tangent blows up. The geometry is vertical; the ratio has no finite value.
Assuming sine and cosine can exceed 1. On the unit circle, both coordinates live in [−1,1][-1, 1]. If your calculator gives sin⁡θ=1.2\sin\theta = 1.2, check mode, labeling, or whether you confused sine with something else.

History and stories

Ptolemy's Almagest (Claudius Ptolemy, c. 100-170) carried chord tables and geometric methods that made spherical astronomy computable for centuries. MacTutor emphasizes the handbook ethic: gather geometric tools so other minds can run with them.
Different notation than modern sine, same ambition: turn angles into lengths you can tabulate. Hipparchus earlier, Ptolemy's systematization later, Indian half-chord (sine) tables in Aryabhata's tradition: a long craft, not a single invention moment.
Popular stories that flatten Ptolemy into either pure hero or pure villain of astronomy miss the mathematical point for this lesson: chord methods as computational geometry.
Ptolemy's chord of an angle is related to modern sine by a factor of 2 and a half-angle shift. You do not need the exact conversion formula to appreciate the continuity: tables for the sky, updated notation, same computational hunger.

Real world

Any repeating cycle can be drawn as motion around a circle:
  • Wheel and gear positions
  • AC electricity phase
  • Seasonal models and circadian abstractions
  • Animation and robotics joint angles
The double view (point on a circle, coordinate oscillating in time) is the whole subject in one mental gif.
AC power: voltage and current trace sine waves 90∘90^\circ out of phase in ideal inductor-capacitor circuits. The phase angle is a position on the unit circle, measured in radians or degrees depending on the engineer's habit.
Color wheels: hue is often modeled as an angle around a circle. Saturation and lightness are separate knobs, but hue is pure turning.
Planetarium software: star positions use spherical coordinates (declination, right ascension). The unit circle is the equatorial slice of that larger sphere.

Open when you want the machine-learning connection. Skip freely.

Field notes, extra examples, and glossary live here when you want more than the core path.

Carry this forward

  • Unit circle: trig functions as coordinates on radius 1.
  • Identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 is Pythagoras.
  • Signs follow quadrants; oriented angles unlock the full line of inputs.
  • Reference angles connect every quadrant back to acute triangles.
  • Next: waves, periodicity, and similarity scores.

Read more

Go deeper when curiosity hits - videos, essays, and primary trails.

Check your understanding

A short learning loop - try a few, learn from misses, mark complete when you have engaged. No timer, no scoreboard.

1.On the unit circle, the point at θ is…
2.cos²θ+sin²θ=1 is…
3.Why radius 1?
4.Sine Qua Non (Strogatz) shows…
5.Signs by quadrant…

Try at least 3 core prompts, or choose I'll return later.