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Calculus IParametric and polar

Polar coordinates, a new way to see curves.

Turn an angle, choose a distance, and watch a point trace a curve. Learn to convert coordinates and find tangents, then work through the class questions.

Points and coordinates · Step 1 of 7

Turn first, then move along the ray

Drag to move P; the sliders provide keyboard controls.xyO4−44−4P
Drag to move P; the sliders provide keyboard controls.
The pole is the origin O. The polar axis points to the right. In (r,θ){(r,\theta)}, the angle tells you which way to face and the signed distance tells you how far to move. Positive angles turn anticlockwise; negative angles turn clockwise. A negative distance sends you backwards.
Try it: Drag the point, or use the angle and distance sliders. Try a negative distance. The gold ray shows the angle; the blue segment shows where the point actually goes.
Angle θ (degrees)30°
Signed distance r2.00
(r,θ)=(2.00,30∘){\displaystyle (r,\theta)} = {\displaystyle (2.00,30^{\circ})}

Radians measure the same angle: 180∘=π{180^{\circ}} = {\pi} and 360∘=2π{360^{\circ}} = {2\pi}. Use radians for derivatives. At r=0{r} = {0}, every angle names the pole.

Check yourself
What does a negative r change?

Need a hand with polar curves or tangents?

Bring your practice questions. We work through them together until every type feels routine.

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  • In person, we meet at the SFU Burnaby campus or in a private meeting room near Brentwood in Burnaby. Online sessions run on Zoom. You choose when you book.