Trigonometric Functions

Definition

Trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They describe periodic phenomena and are foundational to calculus, physics, and engineering.

Why It Matters

These functions are the ‘universal sensors’ for periodic phenomena. They provide the mathematical language to describe anything that repeats—from the swing of a pendulum to the vibration of an atom—forming the bedrock of modern physics.

Core Concepts

• Basic Ratios: Defined on a right triangle as Sine (opposite/hypotenuse), Cosine (adjacent/hypotenuse), and Tangent (opposite/adjacent).
• Unit Circle: The extension of these functions to all real numbers by mapping angles to coordinates $(x, y)$ on a circle of radius 1, where $x = \cos \theta$ and $y = \sin \theta$.
  • How to read: “x equals cosine theta; y equals sine theta.”
  • Meaning: On the unit circle, coordinates of the terminal-side point define cosine and sine for any real angle.
• Reciprocal Functions: Cosecant, Secant, and Cotangent, which are the reciprocals of sine, cosine, and tangent respectively.
• Periodicity: Functions like $\sin$ and $\cos$ repeat their values every $2\pi$ radians, modeling wave behavior.
  • How to read: “Period of sine and cosine equals two pi.”
  • Meaning: One full rotation around the unit circle returns to the same coordinates, producing periodic wave patterns.

Connected Concepts

• Radian Measure
• Unit Circle
• Periodic Functions
• Inverse Trigonometric Functions
• Trigonometric Identities