Polar Graph as Related to Transformations to Sine Waves
Relate transformations to sine waves in rectangular form to the polar equivalent. Slide the point on UB (red slider) to trace the polar graph starting from an angle of 0. There are two check boxes: The upper one color codes the polar graph by waves graphed. The lower one color codes by whether the radial coordinate is positive / negative. To "decongest" the graph, turn parts off (the wave s(t), the midline (line f), the crest line (line g), the trough line (line h)) To work with fractional frequencies, change the slider "freq" through the object properties menu (or command-E).
We often think of sine waves as starting at the "mid-line" (VT), ascending to the "crest" (VT + VS), passing back through the midline, descending to the "trough" (VT - VS) and, finally, ascending back to the midline to make one wave. Talk yourself through this same process for a polar graph. Note that x-intercepts on the wave correspond to places where the polar graph passes through or touches the pole. Note that the parts of the wave that are below the x-axis (y is negative), correspond to a negative radius on the polar graph. Consequently, theses points on the polar graph are a half-turn around from the angles being "plugged in" to the function at that time. For example, for s(t) = 3sin(2t), when the angle t is 3π/4 (135°), the radius is -3. This puts the placement of the point on the polar graph at the equivalent of a radius of +3 with an angle of 7π/4 (315°).