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Sine of the times

(unpublished)

High school trigonometry students who are learning to solve triangles for missing sides and missing angles, here is a problem I made up for you:

You and I and a famous adventurer with a hat and a bullwhip are in a bind. 20 yards straight west and behind him and two fellow explorers (me and you), across a gaping hole in the floor that has since opened up, a trap that he accidentally triggered after grabbing a priceless artifact is emanating a brilliant, 1 foot in diameter red laser beam in a direction 60 degrees east of north. The beam is shining on a crystal which is heating up and contains enough poison gas to flood the cavern we are currently in and all of the escape routes we can see several times over in only two or three seconds. -- we must break the beam entirely, even if only momentarily, to break the feedback loop and save us all. Because of the locations of the gaps in the floor all around us, we can't move closer or farther from the beam, and because of the way the bullwhip works, and the limited tools available, the best plan is to block the beam by attaching something to the end of the 40 foot bullwhip. You are currently the only adventurer undamaged enough to do the job, but don't know how to use the bullwhip well enough to control it at partial extension. In fact, the only way to block a 1 foot beam exactly 40 feet away from you, by swinging the bullwhip in such a way that it becomes completely outstretched by the time it travels through the same horizontal plane that you and the laser beam are in.

You look and see that unfortunately, swinging this apparatus in any direction between due east and due north is not possible because of falling debris.

Assume the center of the hat attaches to the end of the whip, the distance from the center of your arm's rotation to the nearest end of the bullwhip is 2 feet. Assume that the hat size is sufficient to block the beam if the center of the hat passes through the center of the beam, and that you can swing the whip such that it will be at its full length in the same plane as the beam and your path of travel.

Provide the compass heading of the direction in which you must swing the bullwhip to block the laser beam using its full length, accurate to within two decimal places.