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The maximum occurs at θ = atan(2). You can show this via
simple calculus. In the algebra-based introductory physics
course, you can find the maximum by plotting the function
sin(θ) + cos(θ)/2
and eyeballing the maximum, or using the spreadsheet
program's root-finding tool. I didn't evaluate this
arctangent in my head, but my computer tells me it is
63.43°, at which point the direct route wins by about
12 percent.