[1 Problem](https://artofproblemsolving.com#Problem)

[1.1 Solution 1](https://artofproblemsolving.com#Solution_1)
[1.2 Solution 2 (also simple)](https://artofproblemsolving.com#Solution_2_.28also_simple.29)
[1.3 Solution 3(Trig Bash)](https://artofproblemsolving.com#Solution_3.28Trig_Bash.29)
[1.4 Solution 4 (No Trig)](https://artofproblemsolving.com#Solution_4_.28No_Trig.29)
[1.5 Solution 5 (Intuitive and Quick)](https://artofproblemsolving.com#Solution_5_.28Intuitive_and_Quick.29)
[1.6 Solution 6](https://artofproblemsolving.com#Solution_6)
[1.7 Video Solution 1 by OmegaLearn](https://artofproblemsolving.com#Video_Solution_1_by_OmegaLearn)
[1.8 Video Solution (⚡Solved in 56 seconds⚡)](https://artofproblemsolving.com#Video_Solution_.28.E2.9A.A1Solved_in_56_seconds.E2.9A.A1.29)
[1.9 Video Solution](https://artofproblemsolving.com#Video_Solution)
[1.10 See also](https://artofproblemsolving.com#See_also)




Problem
Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta = \angle PAB=\angle QPC=\angle RQB=\cdots$ will produce in a length that is $120$ meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.)

$\textbf{(A)}~\arccos\frac{5}{6}\qquad\textbf{(B)}~\arccos\frac{4}{5}\qquad\textbf{(C)}~\arccos\frac{3}{10}\qquad\textbf{(D)}~\arcsin\frac{4}{5}\qquad\textbf{(E)}~\arcsin\frac{5}{6}$
By "unfolding" $APQRS$ into a straight line, we get a right angled triangle $ABS'$.
[asy] import olympiad; draw((-50,15)--(50,15)); draw((50,15)--(50,-15)); draw((50,-15)--(-50,-15)); draw((-50,-15)--(-50,15)); draw((-50,-15)--(-22.5,15)); draw((-22.5,15)--(5,-15)); draw((5,-15)--(32.5,15)); draw((32.5,15)--(50,-4.090909090909)); label("$\theta$", (-41.5,-10.5)); label("$\theta$", (-13,10.5)); label("$\theta$", (15.5,-10.5)); label("$\theta$", (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label("$D$",(-58,15)); label("$A$",(-58,-15)); label("$C$",(58,15)); label("$B$",(58,-15)); label("$S$",(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); dot((5,45)); dot((32.5,75)); dot((50,94.09090909090909)); draw((-22.5,15)--(50,94.09090909090909)); draw((50,-4.09090909090909)--(50,94.09090909090909)); label("$P$",(-22.5,23)); label("$Q$",(5,-23)); label("$R$",(32.5,23)); label("$Q'$",(5,35)); label("$R'$",(32.5,85)); label("$S'$",(58,94.09090909090909)); [/asy]
$cos(\theta)=\frac{100}{120}$
$\theta=\boxed{\textbf{(A) } \arccos\left(\frac{5}{6}\right)}$
~lptoggled