10 1. Trisecting Angles 4. Show that angles of 3π 4 radians and π radians can both be trisected. 5. Show that if angles of a radians and b radians can be trisected, so can the angle of a + b radians. 6. If angles of a radians and b radians can be trisected, can an angle of a − b radians be trisected? 1.4. The spiral of Archimedes As stipulated, we only allow the compass and straightedge as tools. If we allow other tools, the problem of trisecting an angle can be solved for all angles. Some additional physical tools have been invented through the years that permit trisections to be carried out. It is also possible to give some mathematical tools that will enable you to trisect an angle. One of these is the spiral of Archimedes, whose equation in polar coordinates is r = θ. See Figure 1.4.1. Figure 1.4.1 Wait! What is meant by the statement, “We are given the spiral r = θ?” This means that the spiral is drawn in the plane and it sits there. It is an object that appears just as the x and y axes appear. So if we draw a straight line, the points where this line meets the spiral are points we have constructed in the same sense that the points where a line meets a circle we have drawn are constructed. 1.4.1. Theorem. Given the spiral of Archimedes as well as a compass and straightedge, any angle can be trisected.

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