Sine Graph

Definition

The graph of $y = \sin x$ represents the oscillation of the $y$-coordinate of a point on the unit circle as it rotates. It is a periodic function, repeating its values every $2\pi$ radians.

Why It Matters

The sine graph is a fundamental mathematical model for periodic phenomena that begin at equilibrium. Whether tracking a pendulum’s swing or alternating current, it provides the signature of simple harmonic motion.

Core Concepts

• Standard Period: One full period from $0$ to $2\pi$.
  • How to read: “The value y is equal to the sine of x.”
  • Meaning: Starts at $(0,0)$, peaks at $(\pi/2, 1)$, crosses $(\pi, 0)$, bottoms at $(3\pi/2, -1)$, returns to $(2\pi, 0)$.
• Odd Symmetry: $\sin(-x) = -\sin(x)$
  • How to read: “The sine of negative x is equal to the negative sine of x.”
  • Meaning: The graph is symmetric under $180^\circ$ rotation about the origin.
• Five Key Points: Plotting the start, 1/4, 1/2, 3/4, and end of the period guarantees capturing max, min, and zeros.

Connected Concepts

• Cosine Graph
• Sinusoidal Graphs
• Periodic Functions
• Symmetry