Common Mistakes
Frequent Errors
1. Swapping sine (vertical) and cosine (horizontal). 2. Measuring angles from the $y$-axis instead of the positive $x$-axis. 3. Treating clockwise as positive. 4. Claiming $\tan(\pi/2)$ exists. 5. Confusing arc length on a non-unit circle with the radian measure without dividing by $r$. 6. Thinking coterminal angles have different sine/cosine. 7. Using triangle SOH-CAH-TOA in QI only and freezing in other quadrants. 8. Writing coordinates as $(\sin\theta,\cos\theta)$ on the unit circle.
Why These Happen
Students import right-triangle habits before locking the CED circle definitions tied to the terminal ray and point $P$.
How to Fix Them
- Always label $P=(x,y)$ and write $\cos\leftrightarrow x$, $\sin\leftrightarrow y$.
- Say aloud: "tan is slope of the terminal ray."
- For radians: "arc divided by radius; on unit circle, arc = angle."
Avoidance Scripts
- "Standard position starts on the positive $x$-axis."
- "Unit circle: $(\cos\theta,\sin\theta)$."
- "Tan undefined when the terminal ray is vertical."
Quick Self-Check
- Did I use the positive $x$-axis as the initial ray?
- Are my sin/cos matched to $y$/$x$ on the unit circle?
- Did I check whether tan is defined?