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24 1. Trisecting Angles be replicated by an angle of the same size that has the x-axis as one of its sides. Figure 1.7.1 1.7.1. Proposition. If ∠PQR is constructible and has β radians, then there is a constructible line OB passing through the origin O such that if A is any point on the positive x-axis, then ∠AOB is a constructible angle with β radians. Proof. For convenience assume that Q, the vertex of the angle, lies in the first quadrant. Construct a line through O and parallel to QP let M be the constructible point at the intersection of this line and the circle centered at O with radius 1. Similarly construct a line through O and parallel to QR and let N be the point where this line meets the unit circle. See Figure 1.7.1. Clearly ∠MON has β radians. Now we need to “rotate” ∠MON so that the side OM sits on the x-axis. As in Figure 1.7.2, construct a line through N and perpendicular to OM let F be the constructible point where this line intersects OM. Note that |OF| = cos β. Construct the circle with center O and radius cos β let A be the constructible point where this circle meets the x-axis. Now construct the line perpendicular to the x-axis at the point A let B be the constructible point where this line meets the unit circle. Note that B = (cos β, sin β). Therefore the angle ∠AOB has β radians. 1.7.2. Lemma. A constructible angle having 3θ radians can be trisected if and only if cos θ is a constructible number. Proof. Suppose we are given an angle with 3θ radians and a = cos θ. As- sume that a is a constructible number with −1 a 1 (the case where a = ±1 being trivial). Let A = (a, 0). Draw the circle with center O and radius 1. Now draw the line perpendicular to the x-axis and passing through A. Let B be the point where this line intersects the circle thus