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Topic 3.13 connects polar coordinates and rectangular coordinates—and complex numbers as points in the plane.

  • Polar: (r, theta) gives distance from origin and angle (3.13.A.1).
  • Convert: x = r cos theta, y = r sin theta; r = sqrt(x^2+y^2) with quadrant-aware theta (3.13.A.2/A.3).
  • Complex: a + bi and r(cos theta + i sin theta) (3.13.A.4).

Why It Matters

Polar thinking matches circular/spiral motion and sets up polar function graphs (3.14-3.15).

How Families Can Help

  • Sketch the point before and after converting.
  • Ask: "Which quadrant? What adjustment for theta?"
  • Relate r to distance from the origin.

What Not to Do

  • Do not use arctan(y/x) without quadrant check.
  • Do not swap cos/sin roles in x/y formulas.
  • Do not assume only one polar name for a point.

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