Family Overview

In Plain Language

Topic 3.4 is about building the graphs of $y=\sin\theta$ and $y=\cos\theta$ from the unit circle.

• Sine graph: for each angle $\theta$, plot the $y$-coordinate (vertical displacement) of point $P$ on the unit circle. Domain: all real numbers (3.4.A.1).
• Sine behavior: as $\theta$ increases, outputs oscillate between $-1$ and $1$, tracking that vertical distance (3.4.A.2).
• Cosine graph: plot the $x$-coordinate (horizontal displacement) of $P$. Domain: all reals (3.4.A.3).
• Cosine behavior: outputs likewise oscillate between $-1$ and $1$, tracking horizontal distance from the $y$-axis (3.4.A.4).

Why It Matters

"Unwrapping" the unit circle into a wave is the bridge from 3.2–3.3 coordinates to later sinusoidal modeling (3.5–3.7). AP items ask students to connect a point on the circle to a point on the graph.

How Families Can Help

• Trace $P$ around the circle and plot $(\theta, y)$ for sine and $(\theta, x)$ for cosine.
• Mark key angles $0, \pi/2, \pi, 3\pi/2, 2\pi$ on both circle and graph.
• Emphasize: inputs can go past one revolution — the wave keeps repeating.

What Not to Do

• Do not swap which coordinate builds which graph.
• Do not restrict the domain to $[0,2\pi]$ as if that is the whole function.
• Do not claim outputs leave $[-1,1]$ for these parent functions.