Rossie is a simple robot in the plane, with Stat position at the origin O, facing the positive z-axis. An angle e is entered into Rossie's memory. He can take only two actions: S: Rossie steps one meter in the direction he is facing. R: Rossie stays in place and rotates counterelockwise through angle e. Notation: A string of symbols S and R (read from left to right) represents a sequence of Rossie's moves. For instance, SRRSS indicates that Rossie steps one meter along the r-axis, rotates through angle 20, and then steps two meters in that new direction. In the questions below, we consider only those sequences of actions that include at least one S. (a) For which e can a sequence of actions result in Rossie's return to Start? (Then by repeating that sequence of actions, Rossie will retrace the same path.) For example, with e= 2x/3 = 120, the actions SRSRSR cause Rossie to trace an equilateral triangle and return to Start. (b) Suppose e is the angle pictured below, with cos(0) = -1/3. Note that e is approximately 109.47. With this angle 0, explain why the actions SSSRSSRSSS cause Rossie to return to O. Those moves return Rossie to O but he is not at Start: He is not facing the positive z-axis. With that 6, is there some sequence of actions that returns Rossie to Start? Justify your answer. (c) Investigate the following question: Which angles allow Rossie return to O? (Nat uearily facing he positive asia) Provide more examples of such angles. Are there some angles e that alow Rosie to retum to 0, but only after tracing some path more complicated than a triangle or a regular polygon? (d) Are there some angles e for which Rossie can never return to O? Explain your reasoning.

Rossie is a simple robot in the plane, with Stat position at the origin O, facing the positive z-axis. An angle e is entered into Rossie's memory. He can take only two actions: S: Rossie steps one meter in the direction he is facing. R: Rossie stays in place and rotates counterelockwise through angle e. Notation: A string of symbols S and R (read from left to right) represents a sequence of Rossie's moves. For instance, SRRSS indicates that Rossie steps one meter along the r-axis, rotates through angle 20, and then steps two meters in that new direction. In the questions below, we consider only those sequences of actions that include at least one S. (a) For which e can a sequence of actions result in Rossie's return to Start? (Then by repeating that sequence of actions, Rossie will retrace the same path.) For example, with e= 2x/3 = 120, the actions SRSRSR cause Rossie to trace an equilateral triangle and return to Start. (b) Suppose e is the angle pictured below, with cos(0) = -1/3. Note that e is approximately 109.47. With this angle 0, explain why the actions SSSRSSRSSS cause Rossie to return to O. Those moves return Rossie to O but he is not at Start: He is not facing the positive z-axis. With that 6, is there some sequence of actions that returns Rossie to Start? Justify your answer. (c) Investigate the following question: Which angles allow Rossie return to O? (Nat uearily facing he positive asia) Provide more examples of such angles. Are there some angles e that alow Rosie to retum to 0, but only after tracing some path more complicated than a triangle or a regular polygon? (d) Are there some angles e for which Rossie can never return to O? Explain your reasoning.