Trigonometry and Polar Coordinates
▶︎ Watch it animatedinteractive step-through · ~3 min · optional ⚙︎ Open the appletPolar Tracer · drag θ and compare signed r against distance |r|Polar coordinates name a point by distance and direction: walk r units along the ray at angle theta. Conversion is right-triangle work, x = r cos theta and y = r sin theta one way, r squared = x squared + y squared and a quadrant-checked arctangent the other way. Two conventions carry the topic: a negative r walks backward through the pole (a 180 degree about-face, not a reflection), and every point owns infinitely many polar names, so agreement is checked by converting, not by matching symbols.
The mistakes are convention mistakes: plotting a negative r on the forward ray, trusting a calculator arctangent without asking which quadrant the point is in, swapping the sine and cosine roles in conversion, and assuming polar names are unique. Each one is drilled in the lesson.
The work
3 ways in · any order
Lesson
Trigonometry and Polar Coordinates
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Distance and direction instead of over and up. The lesson converts both ways with exact values, teaches negative r as a walk backward through the pole, catalogs the many names of one point, and installs the quadrant check that corrects arctangent's restricted range. Ten scenarios close it out.
Diagnostic
10-item topic check
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Ten items spanning the Topic 3.13 misconception cluster: negative-r placement, non-unique names, and quadrant-blind conversions. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.