Part 3/8:
What follows is a systematic comparison of the basic cardioid ( R = 1 + \sin \theta ) with shifted versions of this function, namely ( R = 1 + \sin (\theta - \frac{\pi}{6}) ) and ( R = 1 + \sin (\theta - \frac{\pi}{3}) ). Using a graphing calculator, the presenter visually demonstrates how shifting these polar equations alters their graphical representation:
When ( \theta = 0 ), the basic cardioid has coordinates ( (1, 0) ).
For the modified functions, substitutions show that at ( \theta = \frac{\pi}{6} ) and ( \frac{\pi}{3} ), the results remain equivalent, thus justifying that each transformation rotates the original cardioid by specified angles.