Unit 8 · Core unit
A second way to locate a point, and the curves that are far easier to describe in it than in rectangular form.
What a strong answer looks like
A strong answer remembers that one point has infinitely many polar names, and converts using the standard relations rather than by inspection.
Know
A polar point gives a directed distance and an angle. A negative radius reflects the point through the pole, so many pairs name one point.
Apply
Add or subtract a half turn when changing the sign of the radius.
Watch out
Treating a negative radius as invalid rather than as a reflection.
Study move
Give three different polar names for one point.
Know
Rectangular coordinates come from the radius times cosine and sine; the radius squared is the sum of the squares of the coordinates.
Apply
Convert equations by multiplying through by the radius where it helps.
Watch out
Losing the quadrant when recovering an angle from a ratio.
Study move
Convert a polar equation into rectangular form and identify the curve.
Know
Circles, cardioids, limacons, and roses are identified from the constants and the multiplier of the angle. Rose petal count depends on the parity of that multiplier.
Apply
Test symmetry by substituting negative angle, half turn minus angle, or negative radius.
Watch out
Applying the even petal rule to an odd multiplier.
Study move
Predict the petal count for two rose equations and justify each.
Modeling drill
Convert a polar equation of the form a constant times cosine into rectangular form and describe the resulting circle.
