Final answer:
An angle θ is constructible if θ/2 is constructible based on the angle bisector theorem and trigonometric principles like sine and cosine functions.
Step-by-step explanation:
An angle θ is constructible if and only if the angle θ/2 is constructible. This statement is based on the angle bisector theorem, which states that an angle bisector in a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. By bisecting an angle, you can create two angles of θ/2, demonstrating the constructibility of the half angle if the original angle is constructible.
In terms of trigonometry, understanding the trigonometric functions, such as sine and cosine, is important when working with angle bisectors and constructible angles. For instance, if you can construct an angle θ, then you can find the sine and cosine values for θ. Given the sine and cosine values of θ, you can use trigonometric identities to find the sine and cosine values of θ/2, further implying the constructibility of θ/2.