Common Mistakes
Frequent Errors
1. Using $x$-coordinates to graph sine (or $y$ for cosine). 2. Restricting domain to $[0,2\pi]$ only. 3. Starting cosine at the origin like sine. 4. Drawing waves taller than $1$ for $y=\sin\theta$ / $y=\cos\theta$. 5. Uneven spacing of $\pi/2$ marks on the horizontal axis. 6. Forgetting the functions take every value in $(-1,1)$ between extrema. 7. Confusing the input (angle) with the output (coordinate). 8. Treating one plotted cycle as a different function than the infinite continuation.
Why These Happen
Students plot from memory of a poster instead of constructing from the CED unit-circle displacements (3.4.A.1–A.4).
How to Fix Them
- Say: "sine tracks height of $P$; cosine tracks horizontal position of $P$."
- Build from a table before freehand drawing.
- Explicitly write Dom $= \mathbb{R}$ and range $[-1,1]$.
Avoidance Scripts
- "Sine = $y$ of $P$; cosine = $x$ of $P$."
- "Inputs keep going; the circle keeps spinning."
- "Parent sine and cosine stay between $-1$ and $1$."
Quick Self-Check
- Does my sine graph hit $1$ at $\pi/2$ and $0$ at $\pi$?
- Does my cosine graph hit $1$ at $0$ and $0$ at $\pi/2$?
- Did I state domain all reals?