Question

Prove that an angle of t degrees is constructible if and only if is a constructible number.

Short answer

An angle of t degrees is constructible if and only if is a constructible number

Step-by-step solution

Properties of Constructible

1. If a,b,c,d are constructible numbers with $c\ne 0$ and d > 0then a+b,a-b,ab,a/cand $\sqrt{d}$are constructible.
2. The points (a,b) are constructible iff a and b are constructible.

Showing an angle of t cost degree is constructible. 

Let cos t is a constructible then $=\sqrt{1-{\cos }^{2}t}$is also constructible by the above result.

So , $\left(\cos t,\sin t\right)$is constructible. By joining the origin with this point we get an angle of t degrees is constructible.

Proving cost is a constructible number.

Let an angle of t degrees is constructible, then draw a line making the angle with the X-axis. Draw a circle of radius 1 that intersects the line in the point $\left(\cos t,\sin t\right)$ . So by the above result 2 , cos t is a constructible number.