Trigonometric Functions on the Unit Circle

Definition

The Unit Circle is a circle with a radius of 1 centered at the origin $(0,0)$ in the Cartesian coordinate plane. Its equation is $x^2 + y^2 = 1$. In trigonometry, the unit circle is used to define trig functions for any angle $\theta$ by identifying the coordinates $(x, y)$ of the point where the terminal side of the angle intersects the circle.

• How to read: “x squared plus y squared equals one.”
• Meaning: Every point on the unit circle is at distance 1 from the origin—the foundation for defining trig on all angles.

$\cos \theta = x$ $\sin \theta = y$ $\tan \theta = \frac{y}{x} \quad (x \neq 0)$

• How to read: “Cosine theta equals x; sine theta equals y; tangent theta equals y over x.”
• Meaning: On the unit circle ($r = 1$), coordinates directly give cosine and sine; tangent is their ratio.

Why It Matters

The unit circle is the ‘Rosetta Stone’ of trigonometry. It links angles, coordinates, and ratios into a single, unified geometric object, providing the intuitive framework needed to understand how sine and cosine behave over all real numbers.

Core Concepts

• Radius and Coordinates: Because $r = 1$, the ratios $\frac{x}{r}$ and $\frac{y}{r}$ simplify to just $x$ and $y$.
  • How to read: “x over r; y over r.”
  • Meaning: On the unit circle ($r = 1$), the general coordinate ratios $x/r$ and $y/r$ collapse to $x$ and $y$ — coordinates read directly off the circle.
• Pythagorean Identity: The equation of the circle $x^2 + y^2 = 1$ directly yields the fundamental identity $\cos^2 \theta + \sin^2 \theta = 1$.
  • How to read: “Cosine squared theta plus sine squared theta equals one.”
  • Meaning: The circle equation in trig language—cosine and sine are always the legs of a unit right triangle.
• Quadrant Signs (ASTC):
  • Quadrant I: All functions are positive.
  • Quadrant II: Sine (and Cosecant) are positive.
  • Quadrant III: Tangent (and Cotangent) are positive.
  • Quadrant IV: Cosine (and Secant) are positive.
• Quadrantal Angles: Angles whose terminal sides lie on the axes ($0^\circ, 90^\circ, 180^\circ, 270^\circ$). Their coordinates are $(\pm 1, 0)$ or $(0, \pm 1)$.

Connected Concepts

• Trigonometric Functions: Right Triangle Approach
• Pythagorean Identities
• Reference Angles
• Quadrants
• Periodic Functions