It's possible to determine your orientation using an analog watch (i.e., one with hands) in place of a compass. The direction will be correct if the watch is set for true local time, without adjustments for daylight savings time. And the further you are from the equator, the more accurate this method will be.
Know the time, but don't have an analog watch? A simple drawing of a watch on a piece of paper or a clock face on the ground, as long as it shows the correct time (you only need worry about the hour hand), works just as well.
Can't see the sun clearly, or otherwise having difficulty determining its direction? As long as the sun is able to cast a shadow (which it can often do even through fog), you can determine its direction by holding a thin object (like a pencil or a stick) straight up and down; the resulting shadow will point away from the sun.
Wonder why you're bisecting angles? As the earth rotates once (each day), the hour hand goes around the clock face twice. So you need to divide the angle by two (i.e., in half).
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