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Common Mistakes

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Frequent Errors

1. theta = arctan(y/x) without quadrant adjustment (3.13.A.3). 2. x = r sin theta instead of r cos theta (3.13.A.2). 3. r = x + y or r = |x| + |y| instead of sqrt(x^2+y^2). 4. Wrong sign on theta in Quadrant II/III/IV. 5. Complex polar written as r cos theta + r sin theta (missing i) (3.13.A.4). 6. Assuming only one (r, theta) for a point (3.13.A.1). 7. Mixing degrees and radians in same problem. 8. Plotting angle from wrong axis reference.

Why These Happen

Students memorize arctan without the full rect-to-polar workflow.

How to Fix Them

  • Always sketch the point and quadrant.
  • Verify with x = r cos theta, y = r sin theta.
  • For complex, write r(cos theta + i sin theta) explicitly.

Avoidance Scripts

  • "Quadrant before theta."
  • "Check with cos/sin back-substitution."
  • "r is distance: sqrt(x^2+y^2)."

Quick Self-Check

  • Does my theta place the point in the correct quadrant?
  • Do x and y recover after substitution?
  • Is complex polar form written with i correctly?

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