The Unit Circle
The unit circle as the foundation of trigonometry — how it defines sin and cos for all angles, and the key values worth memorising.
What the Unit Circle Is
The unit circle is a circle of radius 1 centred at the origin in the coordinate plane.
Every point on it satisfies: x² + y² = 1
That’s just Pythagoras — for a right triangle with hypotenuse 1, the legs are x and y.
The unit circle is the canonical way to define sin and cos — not just for acute angles in triangles, but for any angle at all.
Defining Sin and Cos
Start at the point (1, 0) — the rightmost point of the circle. Rotate counterclockwise by angle θ. You land at a point (x, y).
That point is (cos θ, sin θ).
cos θ = x-coordinate
sin θ = y-coordinate
This works for any angle — positive (counterclockwise), negative (clockwise), greater than 360° (multiple rotations). The unit circle extends trig beyond triangles into a function defined for all real numbers.
The Four Quadrants
The circle is divided into four quadrants, and the signs of sin and cos change by quadrant:
| Quadrant | Angle range | cos (x) | sin (y) |
|---|---|---|---|
| I | 0° to 90° | + | + |
| II | 90° to 180° | − | + |
| III | 180° to 270° | − | − |
| IV | 270° to 360° | + | − |
Mnemonic — All Students Take Calculus (starting from Q1, going counterclockwise):
- All positive (Q1)
- Sin positive (Q2)
- Tan positive (Q3)
- Cos positive (Q4)
Key Angles and Values
The angles that matter — in degrees and radians, with exact sin and cos values:
| Degrees | Radians | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |
| 120° | 2π/3 | −1/2 | √3/2 | −√3 |
| 135° | 3π/4 | −√2/2 | √2/2 | −1 |
| 150° | 5π/6 | −√3/2 | 1/2 | −1/√3 |
| 180° | π | −1 | 0 | 0 |
| 270° | 3π/2 | 0 | −1 | undefined |
| 360° | 2π | 1 | 0 | 0 |
Memory pattern for sin at 0°, 30°, 45°, 60°, 90°:
sin: 0, 1/2, √2/2, √3/2, 1
↕
√0/2, √1/2, √2/2, √3/2, √4/2
It’s just √0, √1, √2, √3, √4 all divided by 2. cos is the same sequence reversed.
The Pythagorean Identity
Since every point on the unit circle satisfies x² + y² = 1, and x = cos θ, y = sin θ:
cos²θ + sin²θ = 1
This is the most fundamental trigonometric identity. Everything else derives from it.
Dividing through by cos²θ:
1 + tan²θ = sec²θ
Dividing by sin²θ:
cot²θ + 1 = csc²θ
Symmetry of the Circle
The circle has a lot of symmetry. Use it to find trig values for angles beyond the first quadrant.
Reference angle: the acute angle between the terminal side and the x-axis.
sin(150°) = sin(30°) = 1/2 (Q2: sin positive, ref angle 30°)
cos(150°) = −cos(30°) = −√3/2 (Q2: cos negative)
sin(210°) = −sin(30°) = −1/2 (Q3: sin negative, ref angle 30°)
cos(330°) = cos(30°) = √3/2 (Q4: cos positive, ref angle 30°)
Reflection identities
sin(−θ) = −sin(θ) (sin is odd)
cos(−θ) = cos(θ) (cos is even)
sin(π − θ) = sin(θ)
cos(π − θ) = −cos(θ)
sin(π + θ) = −sin(θ)
cos(π + θ) = −cos(θ)
Periodicity
Sin and cos repeat every 2π (360°):
sin(θ + 2π) = sin(θ)
cos(θ + 2π) = cos(θ)
Tan repeats every π (180°):
tan(θ + π) = tan(θ)
This periodicity is what makes trig functions the natural language for waves, oscillations, and anything that cycles.
Arc Length and Area, Revisited
With radians, arc length and sector area become clean:
Arc length = rθ
Sector area = ½r²θ
This is why radians are the natural unit for mathematics — the formulas have no conversion constants. Degrees are convenient for humans; radians are convenient for math.
The Circle as a Clock
One full revolution = 2π radians. The unit circle is literally a unit of angular measurement. Every periodic phenomenon — sound waves, light, AC current, planetary orbits — can be described as motion around the unit circle, with sin and cos giving the components.
The unit circle isn’t just a definition device. It’s why trig is the mathematics of anything that repeats.