Trigonometric Functions: Any Angle

Definition

To evaluate trigonometric functions for angles larger than $90^\circ$ or negative angles, we use a general coordinate approach. For an angle $\theta$ in standard position, let $(a, b)$ be any point on the terminal side and $r = \sqrt{a^2 + b^2}$. Then: $\sin \theta = \frac{b}{r}, \quad \cos \theta = \frac{a}{r}, \quad \tan \theta = \frac{b}{a} \quad (a \neq 0)$

• How to read: “Sine theta equals b over r; cosine theta equals a over r; tangent theta equals b over a.”
• Meaning: For any angle in standard position with terminal-side point $(a,b)$ at distance $r$ from the origin, trig values are coordinate ratios—extending beyond right triangles.

Why It Matters

Expanding trig to ‘any angle’ allows for the modeling of continuous rotation. This is the transition from ‘static triangles’ to ‘dynamic wheels,’ essential for anything involving motors, orbits, or alternating current.

Core Concepts

• Reference Angle ($\alpha$): The acute angle formed by the terminal side and the $x$-axis. The value of a trig function for $\theta$ is the same as for $\alpha$, except for the sign.
• Quadrant Signs:
  • QI: All positive.
  • QII: $\sin$ positive.
  • QIII: $\tan$ positive.
  • QIV: $\cos$ positive.
  • (A-S-T-C: “All Students Take calculus”).
• Coterminal Angles: Angles that share the same terminal side (e.g., $30^\circ$ and $390^\circ$). Their trig values are identical.
• Quadrantal Angles: Angles on the axes ($0, \pi/2, \pi, 3\pi/2$). Some functions (like $\tan \pi/2$) are undefined here because they would involve division by zero.

Connected Concepts

• Trigonometric Functions
• Trigonometric Functions: Right Triangle Approach
• Trigonometric Functions on the Unit Circle
• Solving Trigonometric Equations
• Trigonometric Identities
• Map-Territory Relation
• Symmetry