Cosine Graph

Definition

The graph of $y = \cos x$ represents the oscillation of the $x$-coordinate of a point on the unit circle as it rotates. It is a periodic function, essentially a sine wave shifted left by $\pi/2$.

Why It Matters

The cosine graph is the fundamental model for periodic phenomena that begin at a maximum or minimum value (like a compressed spring released from its peak).

Core Concepts

• Standard Period: One full period from $0$ to $2\pi$.
  • How to read: “The value y is equal to the cosine of x.”
  • Meaning: Starts at $(0,1)$, crosses $(\pi/2, 0)$, bottoms at $(\pi, -1)$, crosses $(3\pi/2, 0)$, returns to $(2\pi, 1)$.
• Even Symmetry: $\cos(-x) = \cos(x)$
  • How to read: “The cosine of negative x is equal to the cosine of x.”
  • Meaning: The graph is symmetric about the $y$-axis (mirror image left/right).
• Relationship to Sine: $\cos x = \sin(x + \pi/2)$. They are the same wave, just started at different points on the unit circle.

Connected Concepts

• Sine Graph
• Sinusoidal Graphs
• Periodic Functions
• Symmetry